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NoncommutativeAlgebra and Geometry

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Edited by

Corrado De ConciniUniversity of RomeRome, Italy

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Nikolai VavilovSt. Petersburg State UniversitySt. Petersburg, Russia

Anatoly YakovlevSt. Petersburg State UniversitySt. Petersburg, Russia

NoncommutativeAlgebra and Geometry

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vii

Introduction

The international meeting at St. Petersburg was organized in honor of Prof. Dr. Z.Borevich, but there was no restriction on the topics of the lectures. A proceedings cov-ering all subjects of the meeting would therefore constitute a rather inhomogeneouscollection. The present volume, however, is mainly devoted to the contributions relat-ed to the ESF workshop organized in the framework of the scientific program“Noncommutative Geometry” of the European Science Foundation and integrated inthe Borevich meeting. The topics dealt with here may be classified as noncommuta-tive algebra.

The congenial atmosphere at the meeting combined with the city’s preparationsfor the anniversary festivities provided the perfect setting for a very fruitful meeting.Moreover, the combination of the ESF workshop and the Borevich meeting broughttogether many participants from East and West (now perhaps old-fashioned termi-nology) engaging in open discussions, hard work, and the occasional party. Most ofthis may be blamed on the local organizers, Vavilov and Yakovlev, whom we thankfor their great hospitality.

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ix

Contributors

Hans-Jochen BartelsUniversitat MannheimMannheim, Germany

Igor BurbanFachbereich MathematikKaiserslautern, Germany

Eloisa DetomiDipto di Matematica UniversitPadova, Italy

Yuriy DrozdKyiv Taras Shevchenko UniversityDepartment of Mechanics and Mathematics

Kyiv, Ukraine

G. Griffith ElderUniversity of Nebraska/OmahaDepartment of MathematicsOmaha, Nebraska

Eivind EriksenUniversity of WarwickInstitute of MathematicsCoventry, United Kingdom

Michiel HazewinkelCWIAmsterdam, The Netherlands

Lieven Le BruynUniversiteit AntwerpenDepartment of Wiskunde and Informatica

Antwerpen, Belgium

Lucchini, AndreaDipto di Matematica UniversityBrescia, Italy

Dmitry A. MalininBelarusian State Pedag. UniversityMinsk, Belarus

Janvière NdirahishaUniversity of Antwerp (UIA)Department of Math and ComputerScienceWilrijk, Belgium

Toukaiddine PetitUniversity of AntwerpDepartment of Math and Computer

ScienceAntwerp, Belgium

Tsetska G. RashkovaUniversity of RousseCenter of Applied Math andInformation

Rousse, Bulgaria

Wolfgang RumpUniversitat StuttgartInstitut f'ur Algebra und ZahStuttgart, Germany

Freddy Van OystaeyenUniversity of Antwerp/UIADepartment of MathematicsAntwerp/Wilrijk, Belgium

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xi

Table of Contents

Introduction .........................................................................................................................vii

Finite Galois Stable Subgroups of GLn ................................................................................1 HANS-JOCHEN BARTELS, DMITRY A. MALININ

Derived Categories for Nodal Rings and Projective Configurations ..............................23IGOR BURBAN, YURIY DROZD

Crowns in Profinite Groups and Applications ..................................................................47ELOISA DETOMI, ANDREA LUCCHINI

The Galois Structure of Ambiguous Ideals in Cyclic Extensions of Degree 8................63G. GRIFFITH ELDER

An Introduction to Noncommutative Deformations of Modules ....................................90EIVIND ERIKSEN

Symmetric Functions, Noncommutative Symmetric Functions and Quasisymmetric Functions II ...........................................................................................126 MICHIEL HAZEWINKEL

Quotient Grothendieck Representations .........................................................................147JANVIÈRE NDIRAHISHA, FREDDY VAN OYSTAEYEN

On the Strong Rigidity of Solvable Lie Algebras............................................................162TOUKAIDDINE PETIT

The Role of a Theorem of Bergman in Investigating Identities in Matrix Algebras with Symplectic Involution ...............................................................................................175TSETSKA G. RASHKOVA

The Triangular Structure of Ladder Functors ...............................................................184WOLFGANG RUMP

Non-commutative Algebraic Geometry and Commutative Desingularizations..........203LIEVEN LE BRUYN

Author Index ......................................................................................................................253

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FINITE GALOIS STABLE SUBGROUPS OF GLn

H. -J. BARTELS1 AND D. A. MALININ2

Abstract. Let K/Q be a finite Galois extension with maximal order OK and Galoisgroup Γ. We consider finite Γ-stable subgroups G ⊂ GLn(OK) and prove that theyare generated by matrices with coefficients in OKab , Kab the maximal abelian subex-tension of K over Q. This implies in particular a positive answer to a conjecture ofJ. Tate on the classification of p-divisible groups over Z and answers also a longstand-ing question of Y. Kitaoka on totally real scalar extensions of positive definite integralquadratic lattices.

Introduction

The starting point of our investigations was the following problem studied by Y. Kitaokaand the first named author around 1978 on the behaviour of the automorphism groups ofpositive definite quadratic Z-lattices under totally real scalar extensions. There was the

Question. If two positive definite quadratic Z-lattices become isomorphic over the ring OKof integers of a totally real field extension K of the rationals Q, are they already isomorphicover Z, the ring of rational integers?

Closely connected with this question was the following

Conjecture 1. Let K/Q be a finite totally real Galois extension and denote by OK thecorresponding ring of integers and let G ⊂ GLn(OK) be a finite subgroup stable underthe operation of the Galois group Γ = Gal(K/Q), then G ⊂ GLn(Z) holds, Z the ring ofrational integers.

There are several reformulations and generalizations of the above mentioned conjecture.One generalization is the following:

Consider an arbitrary not necessarily totally real finite Galois extensionK of the rationalsQ and a free Z-module M of rank n with basis m1, . . . ,mn. The group GLn(OK) acts ina natural way on OK ⊗M ∼= ⊕n

i=1OKmi. A finite group G ⊂ GLn(OK) is said to be ofA-type, if there exists a decomposition M =

⊕ki=1Mi such that for every g ∈ G there exists

a permutation Π(g) of 1, 2, . . . , k and roots of unity εi(g) such that εi(g)gMi = MΠ(g)i for1 ≤ i ≤ k. The following conjecture generalizes (and would imply) conjecture 1 and wouldalso give a positive answer to the above mentioned question:

Conjecture 2. Any finite subgroup of GLn(OK) stable under the Galois group Γ =Gal(K/Q) is of A-type.

For totally real fields K ± 1 are the only roots of 1 contained in K, and so conjecture 2reduces to conjecture 1.

Partial answers to these questions are given in [2], [3], [4], [8], [9], [10], [14], [16], [17], [19](compare also the references in mentioned articles).

1991 Mathematics Subject Classification. Primary 20C10, 11R33, 11S23, 11R29.

2 H. -J. BARTELS AND D. A. MALININ

In an earlier version of this paper (see [4]) it is shown that conjecture 2 is true in thecase of Galois field extension K/Q with odd discriminant. Also some partial answers aregiven in the case of field extensions K/Q which are un-ramified outside 2. The proof of themain part is essentially already contained in the article [17] of the second named author inslightly different formulation. While [17] focusses mainly on the proofs of conjecture 1 andcontains also some other related results, we observed that the proofs of conjecture 1 canimmediately be transfered in order to proof conjecture 2 in the mentioned cases. Using themethods of [2], [3] and discriminant estimations of A. Odlyzko [23] in order to exclude theexistence of certain Galois extensions having low ramification, the first named author provedin an unpublished note eighteen years ago, that conjecture 1 is true in the following cases:

i) Γ = Gal(K/Q) = PSL2(5) ∼= A5 the alternating group of order 60,ii) Γ = Gal(K/Q) = PSL2(7) the simple group of order 168,iii) K/Q is tamely ramified of degree ≤ 131iv) K/Q is tamely ramified of degree ≤ 233 assuming a generalized Riemann hypothesis

to be true.

The combination of this approach using discriminant estimations with the far reachingresults of [17] and [7] gave us the the following better results:

Conjecture 1 is true in the following cases:

i) [K : Q] ≤ 960 assuming the generalized Riemann hypothesis for the zeta function ofthe number field K, or if

ii) [K : Q] ≤ 480 unconditionally.

Conjecture 2 is true if [K : Q] < 288 unconditionally. See [4] for the details.After finishing the first version of our paper [4] we became aware of the recent work [20]

of M. Mazur on the same topic. It turned out that in a certain sense the partial results ofM. Mazur are complementary to our partial results. Using the the classification of finite flatgroup schemes over Z annihilated by a prime p for primes p ≤ 17 due to V. A. Abrashkin [1]and J.-M. Fontaine [6] the particular case of field extensions K/Q which are unramifiedoutside 2 follows in full generality from [20]. In this revised version of our paper we restricttherefore ourselves to the case of ramified primes p = 2. It should be noted that converselyour Main Theorem in combination with the work of M. Mazur has interesting consequencesfor the classification of finite flat commutative group schemes over Z annihilated by a primep: It answers a question of J. Tate [28] also for primes p ≥ 17 completing the partial resultsof Abrashkin [1] and Fontaine [6].

It is interesting to notice that the methods used in the proofs, namely the detailedstudy of the operation of the higher ramification groups of the Galois group on the givenGalois stable group G for the ramified primes in the field extension K over Q together withdiscriminant estimations, in order to eliminate ramification with large depth using trivialaction of higher ramification groups (compare [2] section 1), are similar to the methods usedby [1] and [6].

This paper is organized as follows: Section I contains the results and the propositions andlemmata used in the proofs. The proofs themselves are presented in Section II. As far as itis needed the necessary parts of the proofs from [17] are reproduced only slightly changedin this paper for the convenience of the reader.

Acknowledgement: The second author is grateful to DAAD for support. Helpfulcomments from an anonymous referee to an earlier version of this paper are also gratefullyacknowledged.

FINITE GALOIS STABLE SUBGROUPS OF GLn 3

Notation

Q,Qp,Z,Zp,OK denote the field of rationals and p-adic rationals, the ring of rational andp-adic rational integers respectively, and the ring of integers of an algebraic number field K.We consider O′K to be the intersection of valuation rings of all ramified prime ideals p ∈ OK(if K = Q). TrK/L denotes the trace map from K to L. GLn(R) denotes the general lineargroup over R. [E : F ] denotes the degree of the field extension E/F . Im denotes the unitm ×m-matrix, 0n,m and 0m are zero n ×m and m ×m-matrices, ei,j are square matriceshaving the only nonzero element 1 in the position (i, j), rankM and detM are rank anddeterminant of a matrix M . tM denotes a transposed matrix for M,diag(d1, d2, . . . , dm) isa block-diagonal matrix having diagonal components d1, d2, . . . , dn. We suppose that K is aGalois extension of the rationals Q. We denote by Γ the Galois group of a normal extensionK/F ; if needed we specify K/F as a subscript in ΓK/F . The symbols Γi(p) denote the i-thramification groups of the prime divisor p and Γ0(p) the inertia group in Γ, ei is the orderof Γi(p) for i ≥ 1, while e = e0 is the order of the inertia group. For Γ acting on G and anyσ ∈ Γ and g ∈ G we write gσ for the image of g under σ-action. If G is a finite linear group,F (G) denotes the field obtained by adjoining the matrix coefficients of all matrices g ∈ G.Throughout this paper ζm denotes a primitive m-th root of unity.

1. Statement of the main results

1.1. Let E/F be a normal extension of algebraic number fields, and let ΓE/F =Gal(E/F ) be its Galois group. We consider the problem of integral realizations of finitesubgroups G of the general linear group GLn(E) that are stable under the natural actionof ΓE/F on the matrices of the group G.

Let OF and OE denote the maximal orders of the number fields F and E respectively.Let us introduce the class C(F ) of fields normal over F that are obtained by adjoining toF all coefficients of matrices contained in some finite ΓE/F -stable group G ⊂ GLn(OE).

In [3] it is shown that if F = Q and the class C(Q) contains some field K = Q, thenC(Q) will also contain some field K1 = Q, K1 ⊂ K such that there exists only one primep ramified in K1. In this paper we use some properties of Galois groups for fields havingrestricted ramification. In general, the existence of global fields with a given Galois groupand prescribed local properties for ramification is a rather subtle question. L. Moret-Baillyproved the existence of extensions of number fields that have prescribed local structure oframification over a given set of prime divisors and unramified elsewhere for certain relativeextensions [22]. In our case we deal with absolute extensions of the rationals K/Q, andwe fix the only ramified prime p. Let Cp(Q) denote the class of fields in C(Q) with theunique ramified prime p. Nilpotent extensions of Q having this property were described byMarkshaitis in [18], but there are many examples of extensions in Cp(Q) that are not nilpo-tent, and also nonsolvable extensions unramified outside p; for this and also for non-existencetheorems compare [27], [7]. Both conjectures 1 and 2 are true for nilpotent extensions K/Q(see [3], [8]), and the proof of this fact uses the special structure of the Galois group ofnilpotent extensions unramified outside a prime p [18].

1.2. It is well known, that the problem of description of fields Q(G) can be reduced to thecase of commutative groups G of exponent p. Compare Proposition 1 in [17] and section 3of [19] and [20] chapter 4. The idea of this reduction appears already in [14], [15], [13] and [10]where it was used, in particular, to study conditions for coefficients of the representationsof nilpotent groups over integral rings providing their diagonalizability.

Hence, if there would be a counterexample to conjecture 1 or conjecture 2, there wouldexist also an elementary abelian p group G as a counterexample.


We use also reduction to the case of aGLn(Q)-irreducible groupG. Here a matrix groupGis reducible in GLn(R) or simply R-reducible (R a ring or a field) if there exist h ∈ GLn(R)such that

h−1Gh ⊂∣∣∣∣G1 ∗

0 G2,

∣∣∣∣ ,

and G is irreducible otherwise.We note that the reduction to the case of an irreducible group G can be done using the

following lemma:

Lemma 1.2.1. Let E/F be a normal extension of algebraic number fields with Galois groupΓE/F = Gal(E/F ) and let E1, F1 be rings with quotient fields E and F respectively. IfG ⊂ GLn(E1) is a finite ΓE/F -stable subgroup which has GLn(F1)-irreducible componentsG1, G2, . . . , Gr, then F (G) is the composite of the fields F (G1), F (G2), . . . , F (Gr).

The proof of this Lemma is given at the beginning of section II.1.3. The essential results of this note can be summarized as follows:

Main Theorem. Let K be a finite Galois extension of Q and G be a finite subgroup ofGLn(OK) that is stable under the natural action of the Galois group Γ of the field K. Then Gis of A-type and in particular G ⊂ GLn(OKab

) holds, Kab the maximal abelian subextensionof K over Q.

Let µp denote the multiplicative group scheme over Z of order p and αp the constantgroup scheme of order p (see [28] and [1]). Due to the results of [1] and [6] in conjunctionwith [20] one gets immediately the following

Corollary 1. If G is a finite flat commutative group scheme over Z annihilated by a primep, then it is a direct sum of copies of µp, αp and, if p = 2, the nontrivial element inExt(α2, µ2).

We can also express the result of the Main Theorem in the following form:

Corollary 2. A finite flat group scheme G over Z satisfies G(Q) = G(Qab),Q the algebraicclosure of Q and Qab the maximal abelian (over Q) subextension of Q.

For the proof of the Main Theorem we distinguish essentially two cases and for theirtreatment we need several results which are recorded in the subsequent sections 1.4 and 1.5.The first Proposition 1 gives a criterion for the existence of integral realizations of anabelian matrix group. It shows that the existence of G in question is possible only if certaindeterminants dk are divisible by the root of the discriminant D of a certain extension ofnumber fields (for the details see section 1.4 below). In the proof of the Main Theorem insection II we use this for a certain cyclic extension E/F which is tame with respect to afixed prime ideal (case I). Assume that E/Q is not abelian. Then we can make E/F to be aKummer extension via adjoining appropriate roots of 1. We use the explicit Kummer basisto find an index k for which

√D does not divide dk. The proof of the Main Theorem is

divided in to two parts depending on the ramification index e = e0 of Q(G). In the firstpart we use Proposition 1. In the second part we use lemma 1.5.2 and the Corollary 1.5.3of section 1.5.


We can sketch the scheme of the proof of the Main Theorem:

Let us outline the idea of the proof of the Main Theorem in more detail for the convenienceof the reader.

The outline of the proof of the Main Theorem.

In virtue of the argument of [3], lemmata 1 and 2 (compare also Theorem 2 in [19]),we can assume that K is unramified outside a prime p, so we can fix this prime. Since asalready remarked in the introduction the particular case of field extensions K/Q which areunramified outside 2 follows in full generality from [20], we can restrict ourself to the casep > 2. We can also assume that G is an abelian group of exponent p, and we can considerG to be irreducible under conjugation in GLn(Q) by Corollary 1.4.1. The proof of the MainTheorem consists of a reduction to special cases, and these special cases are treated withdifferent methods.

For number fields E,L be let O′E ,O′L denote the semilocal rings that are obtained byintersection of the valuation rings of all ramified prime ideals in the rings OE ,OL respec-tively. These semilocal rings are known to be principal ideal domains. Denote G0 = GΓ1(p)

the subgroup of elements in G that are fixed by the first ramification group Γ1(p) for someprime divisor p of p. Let e′0 be the ramification index of Q(G0) over Q with respect to p.Then e′0 e0/e1 (= the index of Γ1(p) in Γ0(p).)

Case I.Assume that e′0 does not divide p − 1. In this case we apply Proposition 1 to a certain

subgroup G0 ⊂ GΓ1(p) ⊂ GLn(O′E) for a certain cyclic Kummer extension E/F with aconvenient power basis πi, i = 0, . . . , t − 1 and with the explicit action of the generatingelement σ of order t of the Galois group on the uniformizing element π of O′E , namelyπσ = πζt, which is convenient for applying Proposition 1 explicitly. Here E and F are theramification field and the inertia field for some prime divisor p of p adjoined by a primitivet-root of 1, t = e′0.

Denote ΓE/F the Galois group of E/F . In case I we determine a ΓE/F -stable subgroupG0 ⊂ G0 which is generated by all conjugates hγ , γ ∈ ΓE/F of some element h ∈ G0. G0

can not be cyclic provided t = e′0 does not divide p − 1, and this is just the case where


the arguments in case II (see below) can not be applied. So we start the proof of the MainTheorem just from this most difficult case, and apply Proposition 1 to a subgroup G0 ⊂ G.We show that case I is impossible since the conditions of Proposition 1 never hold true forG0 and the extension E/F . In particular, if e′0 does not divide p−1 we have a contradictionwith the condition G ⊂ GLn(OE) which can not hold true since G0 ⊂ GLn(O′E).

Case II.Let us suppose that e′0 divides p− 1. In this case we can suppose without loss of gener-

ality, that K contains a p-th root of unity ζp (see Lemma 2.2.2 below). Using a localargument on the diagonalization of matrices which are congruent to In modulo the primeideal p (see Corollary 1.5.3 below) a certain subgroup G′1 in G is constructed such thatKΓ1(p)(G′1) is an extension of KΓ1(p) with ζp ∈ KΓ1(p)(G′1), tame ramification index p− 1and KΓ1(p)(G′1)/K

Γ1(p) is an elementary abelian Kummer extension. In a second step acareful study of the Galois-action of Γ0(p) on G′1 shows that the constructed group G′1 cannot exist. This gives then the desired contradiction.

1.4. In this section we formulate the mentioned criterion for the existence of an integralrealization of an abelian group G with the properties mentioned above.

Let E,L be finite Galois extensions of the number field F that are different from F withGalois groups ΓE/F and ΓL/F respectively. As above let O′E , O′L be the semilocal ringsthat are obtained by intersection of the valuation rings of all ramified prime ideals in therings OE , OL, and let O′F = F ∩ O′E . Let w1, w2, . . . , wt be a basis of O′E over O′F , andlet D be the discriminant of this basis. Suppose that some matrix g of prime order p hascoefficients in E and all ΓE/F -conjugates gγ , γ ∈ ΓE/F generate a finite abelian group G ofexponent p. Let σ1 = 1, σ2, . . . , σt denote all automorphisms of the Galois group ΓE/F ofthe field E over F .

Assume that L = E(ζ(1), ζ(2), . . . , ζ(n)) where ζ(1), ζ(2), . . . , ζ(n) are the eigenvalues ofthe matrix g, therefore L = E(ζp), ζp a primitive p-th root of unity. We will reserve thesame notations for some extensions of σi to L, and the automorphisms of L/F will bedenoted σ1, σ2, . . . , σr for some r t. Let E be a numberfield containing F (G) which isobtained by adjoining to F all coefficients of all g ∈ G. For a suitable choice of t elementsof ζ(1), ζ(2), . . . , ζ(n) say ζ(1), ζ(2), . . . , ζ(t) we can prove the following

Proposition 1. 1) Let G be generated by all gγ , γ ∈ ΓE/F and irreducible under GLn(F )-conjugation. Then G is conjugate in GLn(F ) to a subgroup of GLn(O′E) if and only ifall determinants

dk = det

∣∣∣∣∣∣∣∣∣

w1 . . . wk−1 ζ(1)wk+1 · · · wtwσ2

1 · · · wσ2k−1 ζ

σ2(2)w

σ2k+1 · · · wσ2

t

...wσt

1 · · · wσt

k−1 ζσt

(t) wσt

k+1 · · · wσtt

∣∣∣∣∣∣∣∣∣are divisible by

√D in the ring O′L.

2) If any of the three sets of conjugates gγ , γ ∈ ΓE/F , hγ , γ ∈ ΓE/F , (gh)γ , γ ∈ΓE/F generates G and the corresponding eigenvalues of g and h given in 1) areζg(1), ζ

g(2), . . . , ζ

g(t) and ζh(1), ζ

h(2), . . . , ζ

h(t) respectively, then the eigenvalues for the matrix gh

in 1) can be chosen as products ζ(1) = ζgh(1) = ζg(1)ζh(1), ζ(2) = ζgh(2) = ζg(2)ζ

h(2), . . . , ζ(t) = ζgh(t) =

ζg(t)ζh(t).

Note that the conditions of Proposition 1 are always true if E is unramified over F sinceDO′E = O′E in this case.


Corollary 1.4.1. If there is an abelian ΓE/F -stable subgroup G ⊂ GLn(O′E) of expo-nent p generated by gγ , γ ∈ ΓE/F such that E = F (G) = F , then the GLn(F )-irreduciblecomponents Gi ⊂ GLni

(E), i = 1, . . . , k of G are conjugate in GLni(F ) to subgroups

G′i ⊂ GLni(O′E) such that E = F (G1)F (G2) . . . F (Gk). In particular, F (Gi) = F for some

indices i.

The following corollary shows that the conditions of Proposition 1 hold true even if G isnot irreducible.

Corollary 1.4.2. Let E/F be a normal extension of number fields with Galois group ΓE/F .Let G ⊂ GLn(E) be an abelian ΓE/F -stable subgroup of exponent p generated by g andall matrices gγ , γ ∈ ΓE/F , and let E = F (G). Then G is conjugate in GLn(F ) to G′ ⊂GLn(O′E) if and only if all eigenvalues of matrices Bi, i = 1, . . . , t are contained in O′L,where L = E(ζp). The latter happens if and only if the criterion of Proposition 1, 1) holdstrue, i.e. all determinants

dk = det

∣∣∣∣∣∣∣∣∣

w1 . . . wk−1 ζ(1)wk+1 · · · wtwσ2

1 · · · wσ2k−1 ζ

σ2(2)w

σ2k+1 · · · wσ2

t

...wσt

1 · · · wσt

k−1 ζσt

(t) wσt

k+1 · · · wσtt

∣∣∣∣∣∣∣∣∣

are divisible by√D in the ring O′L.

Corollary 1.4.3. Let F = Q. If there is an abelian ΓE/Q-stable subgroup G ⊂ GLn(OE)of exponent p generated by gγ , γ ∈ ΓE/Q such that E = Q(G) = Q, then the GLn(Q)-irreducible components Gi ⊂ GLni

(E), i = 1, . . . , k of G are conjugate in GLni(Q) to

subgroups G′i ⊂ GLni(OE) such that E = Q(G1)Q(G2) . . .Q(Gk). In particular, Q(Gi) = Q

for some indices i.

1.5. For the proof of the Main Theorem (more precisely for the part of the proof dealingwith case II) we use a lemma which is a variation on a theme of Minkowski [21] and is –like in the earlier related work [2], [3] - the key ingredient in the proofs of Lemma 1.5.2 andthe Main Theorem. For the proof see [11]. Compare also [19], Proposition 1.

Lemma 1.5.1. Let J be an ideal in Dedekind ring S of characteristic χ, 0 = J = S, let gbe an n× n-matrix of finite order congruent to In(mod J).

(i) If χ = p > 0, then gpj

= In for some integer j. If χ = 0, then J contains a primenumber p and gp

j

= In, i ∈ Z. In particular, any finite group of matrices congruent toIn(mod J) is a p-group.

(ii) Let χ = 0, J = p be a prime ideal having the ramification index e with respect top, g ≡ In(mod pr) and mpi−1(p − 1) ≤ e/r < pi(p − 1), i ≥ 0,m = min1, i. Thengp

i

= In. In particular, any finite group of matrices congruent to In(mod pt) is trivialif e < t(p− 1).

Related to these properties is the following

Lemma 1.5.2. Let O be a Dedekind ring in an algebraic number field, and let ζp ∈ O. Letp = pe, e = p − 1. Let G be a finite subgroup of GLn(O) and g ≡ In(mod p) for all g ∈ G.Then G is conjugate in GLn(O) to an abelian group of diagonal matrices of exponent p.


Corollary 1.5.3. Let L be an extension of Q and p a prime ideal in the field L(ζp). Supposethat L is unramified at p and let Op denote the valuation ring of the ramified prime ideal pin L(ζp). Let Γ denote the Galois group of L(ζp) over L. If G is a finite Γ-stable subgroupof GLn(Op) consisting of matrices g, g ≡ In(mod p), then G is conjugate in GLn(L ∩ Op)to an abelian group of diagonal matrices of exponent p.

2. Proofs

2.1. Proof of Lemma 1.2.1. Let

h−1Gh ⊂

∣∣∣∣∣∣∣G1 ∗

. . .0 Gr

∣∣∣∣∣∣∣for h ∈ GLn(F1). If there exists g ∈ G such that gγ = g for some automorphism γ of F (G)over F (G1)F (G2) . . . F (Gr), then g′ = gγg−1 = In. The blocks Gi in h−1Gh are stableunder the action of γ, since h ∈ GLn(F1) and the elements of F (Gi) are fixed by γ. Because

h−1gh =

∣∣∣∣∣∣∣g1 ∗

. . .0 gr

∣∣∣∣∣∣∣and

(h−1gh)γ = h−1gγh =

∣∣∣∣∣∣∣g1 ∗′

. . .0 gr

∣∣∣∣∣∣∣are matrices having the same diagonal components, all eigenvalues of the matrix g′ = gγg−1

of finite order are 1 and hence g′ = In. This contradiction completes the proof ofLemma 1.2.1.

Proof of Proposition 1. One proof (namely of the first part) is given in the paper [17].The second part of proposition 1, which is important for the proof of the Main Theorem,follows from the construction given in [17]. But for convenience we give here a proof for theproposition, which is shorter than in [17].

Using the basis w1, . . . , wt of O′E over O′F we can write

gσj =t∑i=1

wiσjBi for j = 1, . . . , t

with semisimple matrices Bi ∈Mn(F ). Since the matrix W = [wσj

i ]j,i is nondegenerate, thematrices Bi can be expressed as a linear combination of gσj , i, j = 1, 2, . . . , t:

Bi =t∑

j=1

mijgσj ,


where [mij ] = W−1. Since by assumption the matrices gσj commute pairwise, all matrices Bialso commute with each other. The irreducibility of G implies that the minimal polynomialof Bi is irreducible over F for each i such that Bi is not zero (see [26], page 8, Corollary 3 forexample). So if one of the eigenvalues of Bi is in O′L then all of them are since they are Galoisconjugate. Using the dual basis w∗1 , . . . , w

∗t to w1, . . . , wt with respect to the traceform one

can see that the inverse matrix W−1 to W = [wσj

i ]j,i is of the form W−1 = [w∗σij ]j,i. In

order to prove the claim of the proposition, we need to determine whether or not matricesBi, i = 1, . . . , t are conjugate in GLn(F ) to matrices B′i ∈Mn(O′F ), since for the generatorg of G the equation

g = B1w1 +B2w2 + · · ·+Btwt,

holds with Bi ∈ Mn(F ) and w1, . . . , wt a basis of O′E over O′F . In fact each semisimplematrix Bi ∈Mn(F ) is conjugate in GLn(F ) to a matrix from Mn(O′F ) if and only if all itseigenvalues are contained in O′L (see Lemma 2.1.1 below).

Cramer’s rule now implies that w∗σj

i = (−1)i+jWi,jdet(W )−1, where Wi,j is the (i, j)-minor of W . Over the splitting field L there is a basis which consists of eigenvectors for G.Let u be one such common eigenvector with

gσiu = tiu.

Then ζ(i) := tσ−1

ii is an eigenvalue of g. It also follows, that u is an eigenvector for Bk

with eigenvalue

λk =t∑

j=1

mkjtj =t∑

j=1

(−1)j+kWj,kζσj

(j)det(W )−1.

The cofactor expansion for determinants implies λk = dk/detW and therefore the eigenval-ues of Bk are in O′L iff detW divides dk, which proves the criterion of Proposition 1 and - bydefinition of the eigenvalues ti - also the second statement modulo the proof of the following

Lemma 2.1.1. i) Let all eigenvalues λj , j = 1, 2, . . . , k of the semisimple matrices Bi ∈Mn(F ), i = 1 . . . , t be contained in the ring O′L for some field L ⊃ F . Then Bi are conjugatein GLn(F ) simultaneously to matrices that are contained in Mn(O′F ).

ii) Conversely, if the semisimple matrices Bi are contained in Mn(O′F ) and Bi are diag-onalizable over a field L ⊃ F , then their eigenvalues are contained in O′L.

Proof of Lemma 2.1.1. i) By the virtue of [26], chapter 1, sect. 1, corollary 2 we can considerA to be a field extending F . Let a1, a2, . . . , an be a basis of O′A over O′F . Then for any B ∈ Awe have B = b1a1 + · · · + bnan, and the elements bi ∈ F are contained in O′F iff B ∈ O′A.But all coefficients kij of the characteristic polynomials fi(x) = ki0 + ki1x + · · · + kinx

n

of the matrices Bi are contained in O′L, and kin = 1, so Bi ∈ A are integral over F . Itfollows that Bi = bi1a1 + · · · + binan, and bij ∈ O′F . If υ ∈ Fn is a non-zero vector in Fn,then a1υ, a2υ, . . . , anυ is a basis of Fn, and Biajυ = Σkcijkakυ, where cijk ∈ O′F . It followsthat for any i the matrix Ci = [cijk]k,j belongs to GLn(O′F ), and Ci is the matrix of theoperator Bi in the basis a1υ, a2υ, . . . , anυ of Fn. Therefore, Bi is conjugate in GLn(F ) toCi for any i = 1, . . . , t.

ii) Consider the characteristic polynomials fi(x) = ki0+ki1x+ · · ·+kinxn of the matricesBi. Since kin = 1 and all kij are in O′F all roots of f(x) are in O′L. This completes theproof of Lemma 2.1.1.


Remark. In the situation of Lemma 2.1.1, i) the F -algebra A = F [B1, . . . , Bt] is isomor-phic to the field L = F [λ1, . . . , λk] where λj , j = 1, 2, . . . , k are all eigenvalues of thematrices Bi, i = 1 . . . , t.

Proof of Corollary 1.4.1. If G ⊂ GLn(O′E) is a group of exponent p and g = B1w1 +B2w2 + · · ·+Btwt for a basis w1, . . . , wt of O′E over O′F , then Bi ∈Mn(O′F ), and it followsfrom Lemma 2.1.1 that the eigenvalues of Bj are contained in O′L. But eigenvalues arepreserved under conjugation, so the latter claim is also true for all components Gi. Wecan apply Proposition 1 to Gi, i = 1, . . . , k. It follows that Gi are conjugate to subgroupsG′i ⊂ GLni

(O′E). Now, Lemma 1.2.1 implies E = F (G1)F (G2) . . . F (Gk). This completesthe proof of Corollary 1.4.1.

Proof of Corollary 1.4.2. Let

C−1GC =

∣∣∣∣∣∣∣G1 ∗

. . .0 Gk

∣∣∣∣∣∣∣

for C ∈ GLn(F ) and irreducible components Gi ⊂ GLni(E), i = 1, . . . , k. Then for g =

B1w1 +B2w2 + · · ·+Btwt

C−1gC =

∣∣∣∣∣∣∣g1 ∗

. . .0 gk

∣∣∣∣∣∣∣= B′1w1 +B′2w2 + · · ·+B′twt

holds with B′i = C−1BiC. Let us consider the F -algebra A generated by all B′i, i = 1, . . . , tover F . Since A is semisimple, it is completely reducible. It follows that matrices B′i aresimultaneously conjugate in GLn(F ) to the block-diagonal form. Therefore, G is conjugatein GLn(F ) to a direct sum of its irreducible components Gi. Since E ⊂ F (Gi) for all i, andO′E contains all rings O′F (Gi)

, we can apply Proposition 1 to each of them. Proposition 1implies that each Gi is conjugate in GLni

(F ) to G′i ⊂ GLni(O′E) if and only if all eigenvalues

of matrices B′i, i = 1, . . . , t are contained in OLi′, where Li = F (Gi)(ζp) and this happens iff

dk = det

∣∣∣∣∣∣∣∣∣

w1 . . . wk−1 ζ(1)wk+1 · · · wtwσ2

1 · · · wσ2k−1 ζ

σ2(2)w

σ2k+1 · · · wσ2

t

...wσt

1 · · · wσt

k−1 ζσt

(t) wσt

k+1 · · · wσtt

∣∣∣∣∣∣∣∣∣

are divisible by√D in the ring O′L. But F (G) = F (G1)F (G2) . . . F (Gk) by the Lemma in

section 1.2, and so L = L1L2 . . . Lk. This completes the proof of Corollary 1.4.2.

Proof of Corollary 1.4.3. The argument of the proof of Corollary 1.4.1 remains true for therings of integers OE and Z in E and F = Q since Z is a principal ideal domain and OE hasa free basis over Z. Therefore, the rest of the proof of Corollary 1.4.3 reproduces the proofof Corollary 1.4.1 with OE and Z instead of O′E and O′F respectively.


2.2. Proof of the Main Theorem. Let us suppose that there exist a counterexample Gto the Main Theorem with corresponding Galois extension K/Q,K = Q(G) with Galoisgroup Γ := ΓK/Q. In virtue of Lemmas 1 and 2 in [3] or Theorem 2 in [19] we can assumethe field K to be unramified outside the fixed prime p. Since as already remarked abovethe particular case of field extensions K/Q which are unramified outside 2 follows in fullgenerality from [20], we can restrict our self to the case p > 2. Because of the Propositionin section 1.2 we can also suppose that G is an abelian group of exponent p and we canconsider G to be irreducible under conjugation in GLn(Q) by Corollary 1.4.3. Let usassume that G is a counterexample of minimal order of this kind. With the notation of thebeginning of this note let Γi(p) ⊂ Γ denote the i-th ramification groups of the prime divisorp for i ≥ 1 and Γ0(p) the inertia group in Γ. Let G0 = GΓ1(p) denote the subgroup ofelements in G that are fixed by the first ramification group Γ1(p) for some prime divisor pof p. Let e′0 be the ramification index of Q(G0) over Q with respect to p. Then e′0 e0/e1(=the index of Γ1(p) in Γ0(p).) We distinguish two cases: Case I : e′0 does not divide p − 1and Case II : e′0 is a divisor of p− 1.Case I. e′0 does not divide p− 1.

1) In this case, where e′0 does not divide p− 1, let us fix p and one of its ramified primedivisors say p. Let E1 and F1 denote the subfields of Γ1(p)-fixed elements and Γ0(p)-fixed elements of K respectively. We will prove that for p = 2 and a field K whichhas discriminant pj , j ∈ Z, all Γ0(p)/Γ1(p)-stable finite subgroups G of GLn(OE′

1)

are already in GLn(OF1) for E′1 = F1(GΓ1(p)) = F1(G0) ⊂ KΓ1(p) and F1 = KΓ0(p).We can extend the ground field F1 by adjoining ζt, t = e′0. Set E = E1(ζt) andF = F1(ζt). We obtain a cyclic extension E/F such that ζt ∈ F for t = e′0. SinceK is unramified outside p,Q(ζt) and K have intersection Q and therefore we canidentify the Galois group ΓE/F = Gal(E/F ) with the Galois group Gal(E1/F1). Withrespect to this extension of the corresponding Galois action to E/F we obtain a ΓE/F−

stable group G0 ⊂ GLn(OE). E/F is a tame extension with respect to p, t = e′0is its ramification index and p − 1 ≥ 2. We have the following conditions for localramification: p

e′0E = (p) = (ζp − 1)p−1 as ideals of the ring OEp(ζp), where pE is the

prime divisor of p in p-adic completion Ep of E. It is clear that([

e′02

]+ 1)

(p−1) > e′0.

Hence p[t/2]+1 does not divide (ζp − 1) as ideals of OE(ζp). We can also assume thatG is an abelian p-group of exponent p, and E = F because e′0 > 1 in the case I. Weuse the statement of Proposition 1 and its Corollary 1.4.2 for the rings O′E and O′Fand a basis 1, π, . . . , πt−1, such that πt ∈ F . If ΓE/F , the Galois group of E/F , isgenerated by an element σ of order t, we can consider the action of ΓE/F on the basis1, π, . . . , πt−1 in the following way: (πi)σ = πiζit . Then

det W = πt(t−1)/2∏

1i<jt(ζjt − ζit).

Let us consider the determinants of the matrices Wj that are obtained from W bychanging elements of j-th column of W = [(πi)σ

j

]i,j to appropriate p-roots ζ(1), ζ(2), . . . , ζ(t)of 1 that are the eigenvalues of the matrices gσ

i

, i = 1, 2, . . . , t for some g ∈ G, according toProposition 1. For simplicity let ζ = ζt, but reserve previous notation for ζp for the rest ofthis proof.

Recall, that G is supposed to be a minimal counterexample to the Main Theorem andthat K is unramified outside p. In the proof of the Case I we pick g ∈ G0 = GΓ1(p) and a


generator σ of the Galois group of E over F ; by our assumption, the order t of σ does notdivide p− 1. There is a matrix g ∈ G0 such that matrices gγ , γ ∈ Γ generate G. Indeed, ifmatrices gγ , γ ∈ Γ generated a proper subgroup G1 of G for any g ∈ G0, then G1 would bea group of A-type, since G is a minimal counterexample, and the order of e′0 would dividep− 1 (because Q(G1)/Q is unramified outside p and tamely ramified at p), contrary to theassumption of the Case I. Let us fix the above G and σ. We need the following auxiliarylemma which specifies the option of g for our proof of the case I:

Lemma 2.2.1. Let k be an integer such that 0 < k < p. There is a matrix g ∈ G0

such that matrices gγ , γ ∈ Γ generate G, and the group G is generated by all hγ , γ ∈ Γ,where h := gkgσ.

Proof of Lemma 2.2.1. Take a matrix g ∈ G0 such that matrices gγ , γ ∈ Γ generate G. Ifa group H generated by all hγ , γ ∈ Γ is a proper subgroup of G, it is a group of A-type,and it is fixed elementwise by the commutator subgroup Γ′ of Γ. Then gσ = g−kh = glh

for l ≡ −k(modp). We have gσ2

= gl2hlhσ, . . . , gσ

p−1= gl

p−1h0 = gh0 for some matrix

h0 having coefficients fixed by Γ′. Since h ∈ G0, G0 is fixed by Γ1(p) and K is unramifiedoutside p, we have h ∈ GLn(Q(ζp)). But ζσ

p−1

p = ζp, and we also have gσi(p−1)

= ghi0, sofor i = p we obtain gσ

p(p−1)= g. The same argument is true for elements g1, h1 such that

g1 = gτ ∈ G0(τ ∈ Γ) and h1 = gk1gσ1 taken instead of g, h. We have gσ

p(p−1)

1 = g1. But G0 iscovered by subgroups generated by all elements g1 = gτ since G is generated by elementsg1 = gγ , γ ∈ Γ. Therefore, σp(p−1) acts trivially on G0. But the order of σ is coprime to p.We conclude that the order of σ divides p − 1, which contradicts the assumption of theCase I. It follows that either the group H or the group H1 generated by all hγ1 , γ ∈ Γcoincides with G. In the latter case we can rename matrix g1 to g. This completes the proofof Lemma 2.2.1.

We distinguish the cases of odd and even t, the order of σ. If t is odd, we need a matrixg′ having at least one eigenvalue θi = ζ(i) = 1 (we use notations of Proposition 1) suchthat G is generated by all conjugates g′γ , γ ∈ Γ. For an even t we have to choose g′ =gkgσζsp . The choice of the eigenvalues ζ(i) (see Proposition 1) ensures that the product of thecorresponding eigenvalues are in accordance with the product of two matrices h1, h2 ∈ G(compare the proof of Proposition 1).

Now, we intend to replace G0 by a smaller subgroup G0 generated by a single elementof G0 which also satisfies the conditions of the Case I.G0 is covered by its ΓE/F -stable subgroups Gγ , where Gγ are generated by elements

(gγ)σi

, i = 1, 2, . . . , t for some γ ∈ Γ and any g such that gγ ∈ G0 and all gτ , τ ∈ Γ,generate G. By definition, Gγ is generated by the orbit of an element g having the aboveproperty. But if h satisfies the conditions of the above Lemma, the elements gτ , τ ∈ Γgenerate G for g = hγ

−1, so we can assume that Gγ is generated by elements hσ

i

, i = 1, . . . , tfor a given γ and some h ∈ G satisfying the conditions of the above Lemma. Since theramification index with respect to p of the composite of the fields F (Gγ), γ ∈ Γ, does notdivide p − 1, there is γ ∈ Γ such that the ramification index e(F (Gγ)/F ) of F (Gγ) doesnot divide p− 1. Let us briefly explain this claim. The field F (G0) is a composite of fieldsEi = F (Gγi

), and F (G0)/F is a cyclic totally ramified extension whose Galois group isgenerated by an element σ of order t equal to the ramification index of F (G0)/F in p.So Ei/F are also cyclic totally ramified extensions, and their Galois groups are generatedby elements σi of orders equal to the ramification indices ti of Ei/F . Therefore, if all tidivide p− 1, then the order of σ must also divide p− 1, because σ is a product of pairwisecommuting elements of orders ti. This completes the proof of our claim.


Let us fix γ and denote G0 = Gγ . The group G0 is not cyclic since the order of σ does notdivide p− 1 in the case I. Using Proposition 1 or, alternatively, Corollary 1.4.1 or Corollary1.4.2 of Proposition 1, we will prove that G0 ⊂ GLn(O′F ). Below we use ΓE/F -stability ofG0 in order to apply Proposition 1 to G0 ⊂ G0 generated by all (hγ)σ

i

, i = 1, 2, . . . , t for thefixed γ ∈ Γ. Since E/F is a cyclic Kummer extension, for E′ = F (G0) ⊂ E the extensionE′/F is also a cyclic Kummer extension, and there are an integer t dividing t, σ ∈ ΓE/Fand a basis 1, π, π2, . . . , πt−1 such that πt ∈ F, πσ = πζt and the Galois group ΓE′/F ofE′/F is generated by σ. Moreover, both extensions E/F and E′/F are totally ramified inp, and t is the ramification index of E′/F , so we have as earlier the following inequality:([

t2

]+ 1)

(p− 1) > t, and p[t/2]+1 does not divide (ζp − 1).

Since p is odd and t does not divide p − 1, we can assume that t > 2. We willconsider matrices

Mj

∣∣∣∣∣∣∣∣∣∣

1 π · · · πj−1 ζ(1) − 1 πj · · · πt−1

1 πζ · · · πj−2ζj−2 ζ(2) − 1 πjζj · · · πt−1ζt−1

...1πζt−1 · · · (πj−2)σ

t−1ζ(t) − 1 (πj)σ

t−1 · · · (πt−1)σt−1

∣∣∣∣∣∣∣∣∣∣,

j = 2, . . . , t that are obtained from Wj by subtracting first column of Wj from j-th columnof Wj . For even t we may suppose that only r n − 2 elements from ζ(1), ζ(2), . . . , ζ(t),the eigenvalues of h, are distinct from 1. Indeed, we can choose two elements g1 and g2 ofG0 generating a noncyclic subgroup of G0 in such a way that ζα1

p , ζα2p , . . . and ζβ1

p , ζβ2p , . . .

compose the full set of eigenvalues of g1 and g2 respectively and α1 = α2. Set

k =−(β1 − β2)α1 − α2

and h = ζsp · gk1g2 for s = −kα1 − β1,

since we are calculating αj , βj and k modulo p we can find an integer k with this properties.Then matrix h has two eigenvalues ζ(i) for different i, and the group generated by hγ , γ ∈

ΓE′(ζp)/F (ΓE′(ζp)/F denotes the Galois group of E′(ζp)/F ) is abelian of exponent p; we canstill apply the criterion of Proposition 1 to the group G0 generated by matrices hγ , γ ∈ΓE′/F . In other words, we can extend the group G0, if it is needed, by adjoining somescalar matrices and naturally extending Galois action to them, and this does not changeΓE/F -stability of G0. For convenience we still preserve our previous notation. We can applyour construction to the matrix h = ζsp · g0 for some g0 ∈ G0 and if we show that this matrixis not contained in GLn(O′E(ζp)), then g0 ∈ GLn(O′E), and this contradiction is exactly the

aim of our proof of the case 1). Denote Λ = [ζ(i−1)(j−1)]ti,j=1. Note that Λ is a symmetricmatrix. Let

det Wj = det Mj = θj1(ζ(1) − 1) + θj2(ζ(2) − 1) + · · ·+ θjt(ζ(t) − 1), where

θjk = (−1)j+kπt(t−1)/2−(j−1) · ζ−(j−1)(k−1)

t· c = πt(t−1)/2−(j−1) · λjk

t,

for

c = detΛ =∏

1i<jt(ζj − ζi).


and λjk = (−1)k+jζ−(j−1)(k−1) = λkj . Indeed, denote Λ−1 = [ ζ−(j−1)(i−1)

t]ti,j=1, and so

(ij)-th cofactor of Wj is (−1)j+i · ζ−(j−1)(i−1)

t· c. Let us consider the element δ from the

Galois group of Q(ζ)/Q such that δ : ζ → ζ−1, and so δ = 1, δ2 = 1. δ acts as a complexconjugation on t-th roots of 1. Note that for a t-root η of 1 ηδ = η iff η−1 = η or, equivalently,η = ±1. Let us determine some properties of the above elements λij under δ-action. Sincethe number of rows in Λ that are permuted under δ-action is equal to φ(t), the Eulerfunction, we have cδ = c if φ(t)/2 is even and cδ = −c if φ(t)/2 is odd. Furthermore,δ permutes i-th row and (t + 2 − i)-th row of the matrix Λ for 1 < i < 1 + t/2, and(−1)i+j = (−1)t−i+j = (−1)t(−1)i+j . Therefore, if both t and φ(t)/2 are even, or botht and φ(t)/2 are odd, then λδk,j = λk,t−j+2 = λt−k+2,j for 1 < j < 1 + t/2, otherwiseλδk,j = −λk,t−j+2 = −λt−k+2,j . In the general case we can claim that λδk,j = s · λk,t−j+2 =s · λt−k+2,j where s = s(t) = (−1)t+φ(t)/2 = ±1 depends only on t.

Let t be even, and let Λ1 = [λij ]i,j = [(−1)i+jζ−(i−1)(j−1)]i,j . Then Λ−11 = [λi,j ]−1

i,j =

[(−1)i+j · ζ(i−1)(j−1)

t]i,j , and it follows that cofactors of λij are equal to aij = ζ(i−1)(j−1)

t, and

so all aij ≡ 0(modq), in particular, a1j = t−1. Let C = [cij ] be a (t − 1) × (t − 1)- matrix

obtained via eliminating the first row and the first column of Λ. Taking an expansion of a1i

by t2 -th row of C we obtain: t−1 = ci1Ai1+ci2Ai2+· · ·+ci,t−1Ai,t−1 where Aiu are cofactors

of the elements ciu in the i-th row of C. It follows that for some m Aim ≡ 0(modq). Now itis possible to fix integers j = 1 and m. We can use matrices g1 = g and g2 = gσ for gettinga matrix g′ whose eigenvalues associated with j-th and m-th blocks are ζ(j) = ζ(m) = 1(see Proposition 1, 2)) and the above Lemma. For this purpose take the eigenvalues ζα1

p

and ζα2p of g1 and the eigenvalues ‘ζβ1

p and ζβ2p of g2 associated with j-th and m-th blocks

respectively. If ζα1p = ζα2

p , set g′ = ζα1p g, otherwise set g′ = ζspg

k1g2 for s = −kα1 − β1

and k = −(β1−β2)α1−α2

. Now we can apply Proposition 1 to the group G0 generated by all

hσi

, i = 1, . . . , t for h = g′.Let us consider a prime ideal q in the ring of integers O of the field Qp(ζp, ζ) such that

q divides p. Let us suppose that ζ(l) = 1 and the elements

(ζ(1) − 1)λi1ζ(l) − 1

+(ζ(2) − 1)λi2ζ(l) − 1

+ · · ·+ (ζ(t) − 1)λitζ(l) − 1

, i = 1, 2, . . . , t

are divisible by (ζ(l) − 1) in the ring O, then the system of congruences

x1λ11 + x2λ12 + · · ·+ xtλ1t ≡ 0(mod q)x1λ21 + x2λ22 + · · ·+ xtλ2t ≡ 0(mod q)...x1λt1 + x2λt2 + · · ·+ xtλtt ≡ 0(mod q)

(S )

has a nontrivial solution

x1 = 1, x2 =ζ(2) − 1ζ(l) − 1

, x3 =ζ(3) − 1ζ(l) − 1

, · · · , xt =ζ(t) − 1ζ(l) − 1

.

Let us eliminate the first and the (t/2 + 1)-th congruences from system (S), coefficientsof which are equal to (λi1, λi2, . . . , λit) = (1, 1, . . . , 1) for i = 1 and (1,−1, 1,−1, . . . , 1,−1),


for i = t/2 + 1. We obtained a system containing r = t − 2 congruences in r = t − 2variables, since two variables xj , xm that correspond to ζ(j) = 1, ζ(k) = 1 do not appear inthe system (S). The determinant of the matrix of this system is a r × r-minor N of thematrix [ζ−(i−1)(j−1)]i,j , and the above choice of j = 1,m (such that ζ(j) = 1, ζ(m) = 1)allows us to assume that det N = 0, since det N = (−1)i+mAim ≡ 0(modq) as it was provedabove. But in this case the system has the unique solution (0, . . . , 0). This contradicts thefact that all xi in question which are different from 0 are invertible elements of the ring ofintegers O of the field Qp(ζp, ζ). Therefore, we can claim that

sj =t∑

k=1

(ζ(k) − 1)λjk ≡ 0(mod(ζ(l) − 1)2)

for some j, where summands (ζ(1)− 1)λj1 and (ζ(m)− 1)λjm are equal to 0. Since r = t− 2and in virtue of the mentioned equality λδk,j = s · λk,t−j+2 = s · λt−k+2,j , where δ2 = 1, wecan consider some j that satisfies inequalities 2 + t/2 j t. Let us calculate det Mj :

dj = det Mj = π1+2+···+(t−1)−(j−1)(t∑i=1

(ζ(i) − 1)λji) = π(t(t− 1)/2− (j − 1)) · sj .

We can calculate the determinant detW with respect to the basis 1, π, . . . , πt−1:

det W = πt(t−1)/2∏

1i<jt(ζjt − ζit).

Taking into account that πj−1 does not divide (ζp − 1) for j ≥ 2 + t2 and comparing

determinants√D = detW and dj , we obtain that dj · (

√D)−1 can not be contained in

O′L, L = E(ζp). By Proposition 1 and its Corollary 1.4.2 this implies that the above matrixg′ ∈ GLn(O′L) and so G0 ⊂ GLn(O′E). This is a contradiction.

If t is odd, the same argument is valid, and we can find an index j such that (t+ 3)/2 ≤j ≤ n and detWj/detW ∈ O′l. Hence the previous proof remains unchanged if we eliminatethe first and the (t + 1)/2-th congruences of the above system (S). However, for odd t itwould be enough to eliminate only the first equation of the system (S).

Case II. e′0 divides p− 1.Now we can consider the case II. We recall the notation from the beginning of the proof

of the Main Theorem. So K = Q(G) is Galois over Q, unramified outside the prime p, p > 2and G0 = GΓ1(p) is the subgroup of elements in G that are fixed by the first ramificationgroup Γ1(p) for some prime divisor p of p, and e′0 denotes the ramification index of Q(G0)over Q with respect to p. For case II we suppose, that e′0 is a divisor of p − 1. Firstly weneed the following

Lemma 2.2.2. The only ramified prime in the extension Q(G0)(ζp)/Q is p, the ramificationindex e(Q(G0)(ζp)/Q) of a ramified prime ideal in Q(G0)(ζp) lying over p∩OQ(G0) is p−1.

Proof of lemma 2.2.2. For the calculation of the ramification index we consider the corre-sponding local situation. Therefore, let Qp denote the p-adic numbers and Q(G0)υ thecompletion of Q(G0) with respect to the valuation υ defined by the prime ideal p∩OQ(G0).


According to the assumptions in case II the ramification index e′0 of Q(G0)υ/Qp divides p−1,while the ramification of Qp(ζp)/Qp is p−1. The compositum Qp(ζp)·Q(G0)υ = Q(G0)υ(ζp)of these two extensions is tamely ramified over Qp with a ramification index t(p− 1), wherethe natural number t divides the ramification index e′0 and therefore divides also p − 1.We claim, that t = 1. For this purpose let Lυ denote the maximal over Qp unramifiedextension in Q(G0)υ(ζp). Then Q(G0)υ(ζp)/Lυ is a totally ramified cyclic Galois extension.Therefore, there is only one subgroup of index t in the Galois group of this cyclic extension.Galois theory give us a uniquely determined subfield of Q(G0)υ(ζp) over Lυ with ramifi-cation index t. But in case t > 1 we would have two such extensions: one is a subfield ofQp(ζp) ·Lυ. This contradiction shows that the ramification index of the composite field cannot exceed p− 1.

According to this Lemma 2.2.2 we see that adjoining a p-th root of unity ζp to Kand extending the Galois operation to this larger field does not influence the validityof condition II, e′0 is still a divisor of p − 1. So we can and do assume ζp ∈ K withoutloss of generality. As it was already mentioned in the beginning of the proof of the MainTheorem we can assume that G is GLn(Q)-irreducible (using corollary 1.4.3) and that Gis a counterexample to the Main Theorem with minimal order. Therefore, also in case IIlet G ⊂ GLn(OK) be a group of the minimal order such that the extension Q(G)/Q is notabelian. For the treatment of case II we distinguish two subcases:

case II a): Γ1(p) is trivial, i.e. K is tamely ramified over Q.and

case II b): Γ1(p) is not trivial, i.e. K is wildly ramified over Q.We start with case II a), since we can use an argument of the proof of case I. There

we have seen: if the group generated by all gγ , γ ∈ Γ for a g ∈ G is not cyclic, then someelement h = ζspg

k1g2 has an eigenvalue 1 (for the notation of g1, g2 see above the proof of

case I). We have the following conditions:

([e′02

]+ 1)

(p− 1) > e′0,

and: p[t/2]+1 does not divide (ζp − 1) for t = e′0 = p− 1.The argument of the proof of Case I implies that the conditions of Proposition 1 are not

satisfied for the group generated by all hγ , γ ∈ Γ. Therefore, gγ = ga for all g ∈ G and anyγ ∈ Γ0(p). Moreover, a is the same for all g. Indeed, if gγ = ga and gγ1 = gb1, with a = b,then the elements (gg1)γ , γ ∈ Γ would generate a noncyclic group. So we have gγσ = gσγ

for any γ ∈ Γ0(p), σ ∈ Γ. This implies gγ = gσγσ−1

. If G is generated by all gγ , γ ∈ Γ, thisimplies the coincidence of all inertia groups Γ0(p). Since Γ0(p) is cyclic, it follows that Gmust be of A-type.Now we consider case II b), where K is wildly ramified.

We assumed ζp ∈ K. Since Q(ζp) is a tame extension of Q, Γ1(p) operates trivially on thep-th roots of unity ζp, hence KΓ1(p) contains also ζp. Take now in Corollary 1.5.3L = KΓ0(p),then this field is unramified over Q for the prime divisor p of p. Corollary 1.5.3 shows: upto conjugation in GLn(Op ∩KΓ0(p)), where Op is the valuation ring of of KΓ0(p)(ζp) at p,the group

G0(p) = g ∈ G0, g ≡ In(modp)


consists of diagonal matrices. The group G(p) := g ∈ G, g ≡ In(mod p) is a nontrivialp-group and therefore G0(p) = In is not trivial as the subgroup of Γ1(p)-fixed elements ofa nontrivial p-group. G is abelian and therefore in the centralizer of every matrix h ∈ G0(p).If in particular h = diag(l1In1 , . . . , lkInk

), then g = diag(g1, . . . , gk), gi ∈ GLni(O′K) holds

for every g ∈ G and therefore we can split G into GLn(Op∩KΓ0(p))-irreducible components.In this decomposition we choose an irreducible component G′ ⊂ GLm(O′K) of G with asuitable natural number m such that G′ has nontrivial Γ1(p)-action. Moreover it is worthmentioning, that the described decomposition is stable under the operation of Γ0(p) (seeCorollary 1.5.3), in particular Γ0(p) operates on the group G′.

If G′0 denotes the subgroup of Γ1(p)-fixed elements of G′, then the group

G′0(p) := g ∈ G′0, g ≡ Im(modp)

consists of scalar matrices. The conditions on the ramification of case II are also satisfiedfor G′ and G′0 instead of G and G0. But now the group G′0(p) is equal to the group

µ := ζIm, ζp = 1.

Let us now consider the Galois-equivariant homomorphism

ψ = ψm : G′ → GLmp(K)

given by ψ(g) = g⊗p

. The kernel of ψ is the set of all scalar matrices contained in G′. Thiskernel is not trivial, since G′0(p) Kerψ. Hence we have:

There is an exact sequence

1 −→ µ −→ G′ −→ ψ(G′) −→ 1

of Γ0(p)-groups.The aim of our proof is the construction of a certain group G′1 ⊂ G′ ⊂ GLm(K) such

that: KΓ1(p)(G′1) is an extension of KΓ1(p) with ζp ∈ KΓ1(p)(G′1), tame ramification indexe′0 = p−1 and KΓ1(p)(G′1)/K

Γ1(p) is an elementary abelian Kummer extension. In a secondstep a careful study of the Galois-action of Γ0(p) on G′1 will then show that the constructedgroup G′1 can not exist. This gives then the desired contradiction.

First step: Construction of G′1.We have H := ψ(G′)Γ1(p) = Im since both ψ(G′) and Γ1(p) are p-groups. For later use

we notice, that

(i) H is Γ0(p)- stable, since Γ1(p) is a normal subgroup of Γ0(p), and(ii) the action of Γ0(p) on H is given by the cyclotomic character.

More precisely, we have for h ∈ H and δ ∈ Γ0(p)hδ = hχ(δ)

. Here χ(δ) denotes the uniqueinteger modulo p such that ζδ = ζχ

(δ)holds for all p-th root of unity ζ and δ ∈ Γ0(p). This

is an immediate consequence of Corollary 1.5.3.Now, if there exist a g ∈ ψ−1(H) having nontrivial Γ1(p)-action, then define G′1 as

the subgroup of ψ−1(H) generated by all gδ, δ ∈ Γ0(p). If such an element g does notexist in ψ−1(H), we can suppose, that ψ(G′) has nontrivial Γ1(p)-action (since otherwise


g with the needed property would exist). Now consider a suitable irreducible componentG′′ of ψ(G′) having non-trivial Γ1(p)-action and apply the corresponding map ψ′ to G′′.For simplicity we call this map ψ′ also simply ψ. If ψ(G′′) is fixed elementwise by Γ1(p),again we have the needed element g ∈ G′′ with non-trivial Γ1(p)-action, and we can defineG′′1 in G′′ correspondingly. Otherwise, we take an irreducible component G′′′ ψ(G′′)having non-trivial Γ1(p)-action etc.. Since the order of the groups G′, G′′, G′′′, . . . isbecoming smaller and smaller (the kernel of the different maps ψ is not trivial), we willhave at last G(i) to be fixed by Γ1(p) with the least possible i, so we have the neededelement g ∈ G(i−1) with non-trivial Γ1(p)-action. Instead of G′1 we consider then thesubgroup of ψ−1(ψ(G(i−1))Γ1(p)) generated by all gδ, δ ∈ Γ0(p). For simplicity let us callthese groups again G′1, G

′ and call also the degree of the corresponding linear groupagain m.

step 2: study of the Galois-action of Γ0(p) on G′1 and on KΓ0(p)(G′1).For g ∈ G′1 and for γ ∈ Γ1(p) we have ψ(gγ)ψ(g)−1 = ψ(g)γψ(g−1) = ψ(g)ψ(g)−1 = Im.

This implies gγ = gζ for any γ ∈ Γ1(p) with a suitable p-th root of unity ζ = ζγ .Let σ be an element of Γ0(p), whose image in Γ0(p)/Γ1(p) is a generator of Γ0(p)/Γ1(p)

and take g ∈ G′1.There are two possibilities: g−1gσ ∈ GLm(KΓ1(p)) or g−1gσ is not fixed by the ramifica-

tion group Γ1(p).In the first of these two cases we claim that gσ = gζσ for a suitable p-th root of unity ζσ.

Let us prove this and show how to get the desired contradiction in that case. For this purposenotice that d := g−1gσ ≡ Im(mod p) and therefore using Corollary 1.5.3 we can diagonalizethis matrix d over GLm(Op ∩KΓ0(p)). But since G′ is irreducibel over GLm(Op ∩KΓ0(p))it follows, that d = ζσIm, for a suitable root of unity ζσ.

Now we have gσ = gζσ and at the same time gγ = gζγ for any γ ∈ Γ1(p). Since Γ1(p)operates trivially on the p-th roots of unity ζ we obtain: gσ = gγ

k

, for some integer k andtherefore the two Galois automorphisms σ and γk coincides on KΓ0(p)(G′1) since g is anygenerator of G′1. This gives the contradiction in the case, where g−1gσ ∈ GLm(KΓ1(p)).

In the alternative case g0 := g−1gσ is not fixed by the ramification group Γ1(p). Nowconsider the group G ⊆ G′1 generated by all elements gδ0, δ ∈ Γ0(p). Since for any δ ∈ Γ0(p)we have

ψ(g0δ) = ψ(g0)δ = ψ(g0)χ(δ) = ψ(gχ(δ)0 ),

it follows that gδ0 = gχ(δ)

0 ζδ with suitable p-th roots of unity ζδ depending on the Galoisautomorphism δ. Therefore the group G is generated by g0 and ζpIm and the order of G is p2.

Define K := KΓ0(p)(G), which is Galois over KΓ0(p) by definition of G. We study theGalois-action on K (like on KΓ0(p)(G′1) in the first case). For this purpose we denote by

Γ0(p) and Γ1(p) the corresponding inertia respectively ramification groups of the exten-

sion K/KΓ0 (p). We have Γ1(p) = 1 since the Γ1(p)-action on G is not trivial. We then

claim firstly, that p is the highest p-power dividing the order of Γ0(p). The Galois group

Γ0(p) of K/KΓ0(p) is contained in the group of linear automorphism of G (considered as a2-dimensional vector space over the field Fp of p elements), so its order divides the orderof GL2(Fp), which equals to (p2 − 1)(p2 − p). This implies that p2 does not divide the

order of Γ0(p), so the Galois group of K/KΓ1(p) is cyclic of order p, as claimed above.


Hence K = KΓ1(p)( p√u) with u ∈ KΓ1(p). Now σ(KΓ1(p)) = KΓ1(p) since Γ1(p) is a normal

subgroup of Γ0(p). Therefore KΓ1(p)( p√u) = KΓ1(p)( p

√uσ), and one concludes:

p√uσ( p√u)−1 ∈ KΓ1(p) KΓ1(p).

Since g−10 gγ0 = ζγIm for all γ ∈ Γ1(p) we have g0 = p

√ug1 with g1 ∈ KΓ1(p). It follows that

g−10 gσ0 ∈ GLm(KΓ1(p))

and we can apply Corollary 1.5.3 to this element. Like in the first of the considered twocases with g0 instead of g we can conclude that gσ0 = g0ζσ for a suitable p-th root of unityζσ. The contradiction follows then analogously to the first case (see above).

2.3. Proof of Lemma 1.5.2. It is a generalization of the well known argument proposedby Minkowski [21]. The outline of our proof is given in [13]. It is easy to prove that Gis abelian of exponent p. Let Op be the valuation ring of p and π a prime element. Letg1 = In + πB1, g2 = In + πB2 for some g1, g2 ∈ G. Then g−1

i ≡ In − πBi(mod π2), i = 1, 2and h = g1g2g

−11 g−1

2 ≡ In(mod π2). It follows from Lemma 1.5.1, (ii) that h = In, andthe same Lemma 1.5.1, (ii) shows that gp = In for any g ∈ G. First of all, G is conjugateover Op to a group of triangular matrices, since G is abelian and Op is a local ring, see [5]Theorem (73.9) and the remarks in [5] on page 493. On the other hand, we can describeexplicitely the matrix M such that

M−1gM = diag(λ1, λ2, . . . , λn)

is a diagonal matrix for a triangular matrix g of order p which is congruent to In(mod p).Indeed, let g ∈ G and

g =

∣∣∣∣∣∣∣∣∣

ζ(1)It1 P 12 . . . P 1

k

0 ζ(2)It2 . . . P 2k

.... . .

...0 · · · ζ(k)Itk

∣∣∣∣∣∣∣∣∣,

and let

S =

∣∣∣∣∣∣∣∣∣

It1 0 . . . A1

0 It2 · · · A2

.... . .

...0 . . . Itk

∣∣∣∣∣∣∣∣∣for t1 + t2 + · · ·+ tk = n and t1 ≤ t2 ≤ · · · ≤ tk, ζ(i), i = 1, 2, . . . , k are appropriate p-rootsof 1. We consider

S−1gS =

∣∣∣∣∣∣∣∣∣

ζ(1)It1 ∗ . . . M1k

0 ζ(2)It2 . . . M2k

.... . .

...0 · · · ζ(k)Itk

∣∣∣∣∣∣∣∣∣,


and we find the system of conditions for providing M ik = 0ti,tk , the zero ti × tk-matrix. We

have the following system of conditions:

ζ(1)(1− ζ(k)ζ−1(1) )A1 + P 1

2A2 + · · ·+ P 1k−1Ak−1 + P 1

k = 0t1,tk. . .

ζ(k−2)Ak−2(1− ζ(k)ζ−1(k−2)) + P k−2

k−1Ak−1 + P k−2k = 0tk−2,tk

ζ(k−1)Ak−1(1− ζ(k)ζ−1(k−1)) + P k−1

k = 0tk−1,tk .

The condition g ≡ In(mod p) implies P ji ≡ 0tjti(mod p), and we can find Ai, 1 ≤ i ≤ k−1sequentially using the results of previous steps:

Ak−1 = − P k−1k

ζ(k−1)(1− ζ(k)ζ−1(k−1))

,

Ak−2 = − (P k−2k + P k−2

k−1Ak−1)

ζ(k−2)(1− ζ(k)ζ−2(k−2))

,

Ak−3 = − (P k−3k + P k−3

k−1Ak−1 + P k−3k−2Ak−2)

ζ(k−3)(1− ζ(k)ζ−1(k−3))

,

and so on. Now, using induction on the degree n we can find a matrix M that transformsg to a diagonal form as required.

Since G is an abelian group of exponent p this allows to prove our claim locally overthe ring Op. We use statement (81.20) in [5] for proving our result globally for the givenDedekind ring (compare for this also the proof of (81.20) and (75.27) in [5]).

Remark. Another proof of the fact that G is elementary abelian can be found in [29], sect.4 and [30], p. 187.

Proof of Corollary 1.5.3. We can assume that for some matrix g ∈ G and a generator σ of Γthe condition gσ = gα, 1 < α < p, is fulfilled. Indeed, by Lemma 1.5.2 G is an abelian groupof exponent p, so it can be considered as an FpΓ - module over the field Fp of p elements.Since Γ is a cyclic group of order p− 1 generated by an element σ this element determinesan automorphism of G and all its eigenvalues are contained in Fp. In fact, its matrix isdiagonalizable over Fp because the order of σ is prime to p. Hence we can take g ∈ G to be aneigenvector of this automorphism and so gσ = gα, 1 < α < p since not all eigenvalues are 1.Now Lemma 1.5.2 provides the existence of a matrix M ∈ GLn(Op) such that M−1GM is agroup of diagonal matrices. We shall show that α coincides with the integer β, ζσp = ζβp , 1 <β < p. Let us suppose that M−1gM = h = diag(λ1In1 , λ2In2 , . . . , λmInm

), λj ∈ L(ζp), thenhσ = hβ and (Mσ)−1gσMσ = hβ . Since M−1gαM = hα and gσ = gα, it is obvious that

(Mσ)−1MhαM−1Mσ = hβ .

As Γ coincides with the inertia group of the ideal p and M ∈ GLn(Op), it follows that Mσ ≡M(mod p). Therefore, the congruence M−1Mσ ≡ In(mod p) is valid and conjugation by


matrix M−1Mσ maps diagonal elements of hα to diagonal elements of hβ . But if α = β, thenthe matrix M−1Mσ must have at least one diagonal element dii = 0, which is impossible.We proved our claim, and α = β. We obtained also that M−1Mσ = λ = diag(d1, d2, . . . , dm)for some nj × nj-matrices dj . Let us introduce the following matrix:

M1 =1

p− 1(Mσ1 +Mσ2 + · · ·+Mσp−1), M1 = [mij ], mij ∈ Op,

σ1, σ2, . . . , σp−1 are all elements of Γ. It is clear, that M1 ≡ M(mod p) and det M1 ≡detM(mod p). It follows that M1 ∈ GLn(Op). Furthermore, M1 is stable under elementwiseΓ-action, so all mij are Γ-stable and mij ∈ L. Hence M1 ∈ GLn(L). Since Mσ = Mλ, itfollows that M−1

1 GM1 is contained in the group of diagonal matrices, as it was claimed.

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[21] H. Minkowski, Uber den arithmetischen Begriff der Aquivalenz und uber die endlichen Gruppenlinearer ganzzahliger Substitutionen, J. reine angew. Math. 100 (1887), 449–458.

[22] L. Moret-Bailly, Extensions de corps globaux a ramification et groupe de Galois donnes, C.R.Acad. Sci. Paris, Serie 1 311 (1990), 273–276.

[23] A. Odlyzko, Discriminant bounds, unpublished Tables from November 29 (1976), seehttp://www.research.att.com/∼amo/unpublished/discr.bound.table.

[24] I. Schur, Elementarer Beweis eines Satzes von L. Stickelberger, Math. Z. 29 (1929), 464–465.[25] J.-P. Serre, Corps locaux, Hermann, Paris, 1962.[26] D.A. Suprunenko, R.I. Tyshkevich, Commutative Matrices, Academic Press, New York and

London, 1968.[27] J. Tate, The Non-Existence of Certain Galois Extensions of Q Unramified Outside 2, Contem-

porary Mathematics 174 (1994), 153–156.[28] J. Tate, p-Divisible Groups (1967), in: Conf. Local Fields (Dreibergen), Springer Verlag, Berlin

and New York, 158–183.[29] A. Weiss, Rigidity of p-adic p-torsion, Annals of Math. 127 (1988), 317–322.[30] A. Weiss, Torsion in integral group rings, J. fur die Reine und angew. Math. 145

(1991), 175–187.

1Fakultat fur Mathematik und Informatik, Universitat Mannheim, Seminar-gebaude A5,

D-68131 Mannheim, Germany

E-mail address: [email protected] State Pedag. University, Sovetskaya str. 18, 220050 Minsk, Belarus

E-mail address: [email protected]

DERIVED CATEGORIES FOR NODAL RINGS AND PROJECTIVECONFIGURATIONS

IGOR BURBAN AND YURIY DROZD

Contents

Introduction 231. Backstrom rings 242. Nodal rings 253. Examples 293.1. Simple node 293.2. Dihedral algebra 323.3. Gelfand problem 334. Projective configurations 365. Configurations of type A and A 376. Application: Cohen–Macaulay modules over surface singularities 43References 45

Introduction

This paper is devoted to recent results on explicit calculations in derived categories ofmodules and coherent sheaves. The idea of this approach is actually not new and waseffectively used in several questions of module theory (cf. e.g. [10, 12, 13, 7]). Neverthelessit was somewhat unexpected and successful that the same technique could be applied toderived categories, at least in the case of rings and curves with “simple singularities.” Wepresent here two cases: nodal rings and configurations of projective lines of types A andA, when these calculations can be carried out up to a result, which can be presented inmore or less distinct form, though it involves rather intricate combinatorics of a specialsort of matrix problems, namely “bunches of semi-chains” [4] (or, equivalently, “clans”[8]). In Sections 1 and 4 we give a general construction of “categories of triples,” whichare a connecting link between derived categories and matrix problems, while in Sections 2and 5 this construction is applied to nodal rings and configurations of types A. Section 3contains examples of calculations for concrete rings and Section 5 also presents those fornodal cubic. We tried to choose typical examples, which allow to better understand thegeneral procedure of passing from combinatorial data to complexes. Section 6 contains anapplication to Cohen–Macaulay modules over surface singularities, which was in fact theorigin of investigations of vector bundles over projective curves in [13].

More detailed exposition of these results can be found in [5, 6, 14].

2000 Mathematics Subject Classification. 16E05, 16D90.It is a survey of a research supported by the CRDF Award UM 2-2094 and by the DFG Schwerpunkt

“Globale Methoden in der komplexen Geometrie”.

24 IGOR BURBAN AND YURIY DROZD

1. Backstrom rings

We consider a class of rings, which generalizes in a certain way local rings of ordinary multi-ple points of algebraic curves. Following the terminology used in the representations theoryof orders, we call them Backstrom rings. Since in the first three sections we are investigat-ing a local situation, all rings there are supposed to be semi-perfect [3] and noetherian. Wedenote by A-mod the category of finitely generated A-modules and byD(A) the derived cate-gory D−(A-mod) of right bounded complexes over A-mod. As usually, it can be identifiedwith the homotopy categoryK−(A-pro) of (right bounded) complexes of (finitely generated)projective A-modules. Moreover, since A is semi-perfect, each complex from K−(A-pro) ishomotopic to a minimal one, i.e. to such a complex C• = (Cn, dn) that Im dn ⊆ radCn−1

for all n. If C• and C ′• are two minimal complexes, they are isomorphic in D(A) if and onlyif they are isomorphic as complexes; moreover, any morphism C• → C ′• in D(A) can bepresented by a morphism of complexes, and f is an isomorphism if and only if the latter one is.

Definition 1.1. A ring A is called a Backstrom ring if there is a hereditary ring H ⊇ A(also semi-perfect and noetherian) and a (two-sided) H-ideal I ⊂ A such that both R = H/Iand S = A/I are semi-simple.

For Backstrom rings there is a convenient approach to the study of derived categories.Recall that for a hereditary ring H every object C• from D(H) is isomorphic to the directsum of its homologies. Especially, any indecomposable object from D(H) is isomorphic toa shift N [n] for some H-module N , or, the same, to a “short” complex 0→ P ′ α−→ P → 0,where P and P ′ are projective modules and α is a monomorphism with Imα ⊆ radP(maybe P ′ = 0). Thus it is natural to study the category D(A) using this information aboutD(H) and the functor T : D(A)→ D(H) mapping C• to H⊗A C•.1

Consider a new category T = T (A) (the category of triples) defined as follows:

• Objects of T are triples (A•, B•, ι), where– A• ∈ D(H);– B• ∈ D(S);– ι is a morphism B• → R ⊗H A• from D(S) such that the induced morphismιR : R⊗S B• → R⊗H A• is an isomorphism in D(R).

• A morphism from a triple (A•, B•, ι) to a triple (A′•, B′•, ι′) is a pair (Φ, φ), where

– Φ : A• → A′• is a morphism from D(H);– φ : B• → B′• is a morphism from D(S);– the diagram

B•ι−−−−→ R⊗H A•

φ

1⊗Φ

B′•ι′−−−−→ R⊗H A′•

(1.1)

commutes in D(S).

One can define a functor F : D(A) → T (A) setting F(C•) = (H ⊗A C•,S ⊗A C•, ι),where ι : S ⊗A C• → R ⊗H (H ⊗A C•) R ⊗A C• is induced by the embedding S → R.The values of F on morphisms are defined in an obvious way.

1Of course, we mean here the left derived functor of ⊗, but when we consider complexes of projective

modules, it restricts indeed to the usual tensor product.

DERIVED CATEGORIES FOR NODAL RINGS 25

Theorem 1.2. The functor F is a full representation equivalence, i.e. it is

• dense, i.e. every object from T is isomorphic to an object of the form F(C•);• full, i.e. each morphism F(C•)→ F(C ′•) is of the form F(γ) for some γ : C• → C ′•;• conservative, i.e. F(γ) is an isomorphism if and only if so is γ;

As a consequence, F maps non-isomorphic objects to non-isomorphic and indecomposableto indecomposable.

Note that in general F is not faithful : it is possible that F(γ) = 0 though γ = 0 (cf.Example 3.1.3 below).

Sketch of the proof. Consider any triple T = (A•, B•, ι). We may suppose that A• is aminimal complex from K−(A-pro), while B• is a complex with zero differential (since S issemi-simple) and the morphism ι is a usual morphism of complexes. Note that R⊗H A• isalso a complex with zero differential. We have an exact sequence of complexes

0 −→ IA• −→ A• −→ R⊗H A• −→ 0.

Together with the morphism ι : B• → R⊗H A• it gives rise to a commutative diagram inthe category of complexes Com−(A-mod)

0 −−−−→ IA• −−−−→ A• −−−−→ R⊗H A• −−−−→ 0∥∥∥ α

ι

0 −−−−→ IA• −−−−→ C• −−−−→ B• −−−−→ 0,

where C• is the preimage in A• of Im ι. The lower row is also an exact sequence of complexesand α is an embedding. Moreover, since ιR is an isomorphism, IA• = IC•. It implies thatC• consists of projective A-modules and H⊗A C• A•, wherefrom T FC•.

Let now (Φ, φ) : FC• → FC ′•. We suppose again that both C• and C ′• are minimal, whileΦ : H⊗AC• → H⊗AC

′• and φ : S⊗AC• → S⊗AC

′• are morphisms of complexes. Then the

diagram (1.1) is commutative in the category of complexes, so Φ(C•) ⊆ C ′• and Φ inducesa morphism γ : C• → C ′•. It is evident from the construction that F(γ) = (Φ, φ). Moreover,if (Φ, φ) is an isomorphism, so are Φ and φ (since our complexes are minimal). ThereforeΦ(C•) = C ′•, i.e. Im γ = C ′•. But ker γ = ker Φ∩C• = 0, thus γ is an isomorphism too.

2. Nodal rings

We apply these considerations to the class of rings first considered in [10], where thesecond author has shown that they are unique pure noetherian rings such that the classifi-cation of their modules of finite length is tame (all others being wild).

Definition 2.1. A ring A (semi-perfect and noetherian) is called a nodal ring if it is purenoetherian, i.e. has no minimal ideals, and there is a hereditary ring H ⊇ A, which issemi-perfect and pure notherian such that

1) radA = radH; we denote this common radical by R.2) lengthA(H⊗A U) ≤ 2 for every simple left A-module U and

lengthA(V ⊗A H) ≤ 2 for every simple right A-module V .

Note that condition 2 must be imposed both on left and on right modules.


It is known that such a hereditary ring H is Morita equivalent to a direct product ofrings H(D, n), where D is a discrete valuation ring (maybe non-commutative) and H(D, n)is the subring of Mat(n,D) consisting of all matrices (aij) with non-invertible entries aijfor i < j. Especially, H and A are semi-prime (i.e. without nilpotent ideals)

Example 2.2. 1. The first example of a nodal ring is the completion of the local ringof a simple node (or a simple double point) of an algebraic curve over a field k. It isisomorphic to A = k[[x, y]]/(xy) and can be embedded into H = k[[x1]] × k[[x2]] asthe subring of pairs (f, g) such that f(0) = g(0): x maps to (x1, 0) and y to (0, x2).Evidently this embedding satisfies conditions of Definition 2.1.

2. The dihedral algebra A = k〈〈x, y 〉〉/(x2, y2) is another example of a nodal ring. Inthis case H = H(k[[t]], 2) and the embedding A→ H is given by the rule

x →(

0 t0 0

), y →

(0 01 0

).

3. The “Gelfand problem” is that of classification of diagrams with relations

2x+

1x− y−

3y+

x+x− = y+y−.

If we consider the case when x+x− is nilpotent (the main part of the problem), suchdiagrams are just modules over the ring A, which is the subring of Mat(3,k[[t]])consisting of all matrices (aij) with a12(0) = a13(0) = a23(0) = a32(0) = 0. The arrowsof the diagram correspond to the following matrices:

x+ → te12, x− → e21, y+ → te13, y− → e31,

where eij are matrix units. It is also a nodal ring with H being the subring ofMat(3,k[[t]]) consisting of all matrices (aij) with a12(0) = a13(0) = 0 (it is Moritaequivalent to H(k[[t]], 2)).

4. The classification of quadratic functors, which play an important role in algebraictopology, reduces to the study of modules over the ring A, which is the subring ofZ2

2 ×Mat(2,Z2) consisting of all triples

(a, b,

(c1 2c2c3 c4

))with a ≡ c1(mod 2) and b ≡ c4(mod 2),

where Z2 is the ring of p-adic integers [11]. It is again a nodal ring: one can takefor H the ring of all triples as above, but without congruence conditions; then H =Z2

2 ×H(Z2, 2).

Certainly, a nodal ring is always Backstrom, so Theorem 1.2 can be applied. More-over, in nodal case the resulting problem belongs to a well-known type. For the sake ofsimplicity, we consider now the situation, when A is a D-algebra finitely generated asD-module, where D is a discrete valuation ring with algebraically closed residue field k. Wedenote by U1, U2, . . . , Us indecomposable non-isomorphic projective (left) modules over A


and by V1, V2, . . . , Vr those over H. Condition 2 from Definition 2.1 implies that there arethree possibilities:

1) H⊗A Ui Vj for some j and Vj does not occur as a direct summand in H⊗A Uk fork = i;

2) H⊗A Ui Vj ⊕ Vj′ (j = j′) and neither Vj nor Vj′ occur in H⊗A Uk for k = i;3) there are exactly two indices i = i′ such that H⊗A Ui H⊗A Ui′ Vj and Vj does

not occur in H⊗A Uk for k /∈ i, i′ .

We denote by Hj the indecomposable projective H-module such that Hj/RHj Vj .Since H is a semi-perfect hereditary order, any indecomposable complex from D(H) is

isomorphic either to 0 → Hkφ−→ Hj → 0 or to 0 → Hj → 0 (it follows, for instance,

from [9]). Moreover, the former complex is completely defined by either j or k and thelength l = lengthH Cokerφ. We shall denote it both by C(j,−l, n) and by C(k, l, n + 1),while the latter complex will be denoted by C(j,∞, n), where n denotes the place of Hj

(so the place of Hk is n + 1). We denote by Z the set (Z \ 0 ) ∪ ∞ and consider theordering ≤ on Z, which coincides with the usual ordering separately on positive integersand on negative integers, but l <∞ < −l for any l ∈ N. Note that for each j the submod-ules of Hj form a chain with respect to inclusion. It immediately implies the followingresult.

Lemma 2.3. There is a homomorphism C(j, l, n) → C(j, l′, n), which is an isomorphismon the n-th components, if and only if l ≤ l′ in Z. Otherwise the n-th component of anyhomomorphism C(j, l, n)→ C(j, l′, n) is zero modulo R.

We transfer the ordering from Z to the set Ej,n =C(j, l, n) | l ∈ Z

, so the

latter becomes a chain with respect to this ordering. We also denote by Fj,n the set (i, j, n) |Vj is a direct summand of H⊗A Ui . It has at most two elements. We alwaysconsider Fj,n with trivial ordering. Then a triple (A•, B•, ι) from the category T (A) isgiven by homomorphisms φijnjln : di,j,nUi → rj,l,nVj , where (i, j, n) ∈ Fjn, the left Ui comesfrom Bn and the right Vj comes from direct summands rj,l,nC(j, l, n) of A•. Note thatif both C(j,−l, n) and C(k, l, n + 1) correspond to the same complex (then we writeC(j,−l, n) ∼ C(k, l, n+1)), we have rj,−l,n = rk,l,n+1. We present φijnjln by its matrix M ijn

jln .Then Lemma 2.3 implies the following

Proposition 2.4. Two sets of matricesM ijnjln

and

N ijnjln

describe isomorphic triples

if and only if one of them can be transformed to the other by a sequence of the following“elementary transformations”:

1) For any given values of i, n, simultaneously M ijnjln → M ijn

jln S for all j, l such that(ijn) ∈ Fj,n, where S is an invertible matrix of appropriate size.

2) For any given values of j, l, n, simultaneously M ijnjln → S′M ijn

jln for all (i, j, n) ∈ Fjn

and M i,k,n−sgn lk,−l,n−sgn l → S′M i,k,n−sgn l

k,−l,n−sgn l for all (i, k, n− sgn l) ∈ Fk,n−sgn l, where S′ is aninvertible matrix of appropriate size and C(j, l, n) ∼ C(k,−l, n − sgn l). If l = ∞, itjust means M ijn

j∞n → SM ijnj∞n.

3) For any given values of j, l′ < l, n, simultaneously M ijnjln → M ijn

jln + RM ijnjl′n for all

(i, j, n) ∈ Fj,n, where R is an arbitrary matrix of appropriate size. Note that, unlike


the preceding transformation, this one does not touch the matrices M i,k,n−sgn lk,−l,n−sgn l such

that C(j, l, n) ∼ C(k,−l, n− sgn l).

This sequence must contain finitely many transformations for every fixed values of j and n.

Therefore we obtain representations of the bunch of semi-chains Ejn,Fjn in the sense of[4], so we can deduce from this paper a description of indecomposables in D(A). We arrangeit in terms of strings and bands, often used in representation theory.

Definition 2.5. 1. We define the alphabet X as the set⋃j,n(Ej,n ∪ (j, n) ). We define

symmetric relations ∼ and − on X by the following exhaustive rules:(a) C(j, l, n)− (j, n) for all l ∈ Z;(b) C(j,−l, n) ∼ C(k, l, n+ 1) defined as above;(c) (j, n) ∼ (k, n) (k = j) if Vj ⊕ Vk H⊗A Ui for some i;(d) (j, n) ∼ (j, n) if Vj H⊗A Ui H⊗A Ui′ for some i′ = i.

2. We define an X-word as a sequence w = x1r1x2r2x3 . . . rm−1xm, where xk ∈ X, rk ∈−,∼ such that(a) xkrkxk+1 in X for 1 ≤ k < m;(b) rk = rk+1 for 1 ≤ k < m− 1.We call x1 and xm the ends of the word w.

3. We call an X-word w full if(a) r1 = rm−1 = −(b) x1 ∼ y for each y = x1;(c) xm ∼ z for each z = xm.Condition (a) reflects the fact that ιR must be an isomorphism, while conditions (b,c)come from generalities on bunches of semi-chains [4].

4. A word w is called symmetric, if w = w∗, where w∗ = xmrm−1xm−1 . . . r1x1 (theinverse word), and quasisymmetric, if there is a shorter word v such that w = v ∼v∗ ∼ · · · ∼ v∗ ∼ v.

5. We call the end x1 (xm) of a word w special if x1 ∼ x1 and r1 = − (respectively,xm ∼ xm and rm−1 = −). We call a word w(a) usual if it has no special ends;(b) special if it has exactly one special end;(c) bispecial if it has two special ends.Note that a special word is never symmetric, a quasisymmetric word is always bispe-cial, and a bispecial word is always full.

6. We define a cycle as a word w such that r1 = rm−1 =∼ and xm − x1. Such a cycle iscalled non-periodic if it cannot be presented in the form v − v − · · · − v for a shortercycle v. For a cycle w we set rm = −, xqm+k = xk and rqm+k = rk for any q, k ∈ Z.

7. A (k-th) shift of a cycle w, where k is an even integer, is the cycle w[k] =xk+1rk+1xk+2 . . . rk−1xk. A cycle w is called symmetric if w[k] = w∗ for some k.

8. We also consider infinite words of the sorts w = x1r1x2r2 . . . (with one end) andw = . . . x0r0x1r1x2r2 . . . (with no ends) with restrictions(a) every pair (j, n) occurs in this sequence only finitely many times;(b) there is an n0 such that no pair (j, n) with n < n0 occurs.We extend to such infinite words all above notions in the obvious manner.

Definition 2.6 (String and band data). 1. String data are defined as follows:(a) a usual string datum is a full usual non-symmetric X-word w;(b) a special string datum is a pair (w, δ), where w is a full special word and δ ∈ 0, 1 ;


(c) a bispecial string datum is a quadruple (w,m, δ1, δ2), where w is a bispecial wordthat is neither symmetric nor quasisymmetric, m ∈ N and δ1, δ2 ∈ 0, 1 .

2. A band datum is a triple (w,m, λ), where w is a non-periodic cycle, m ∈ N and λ ∈ k∗;if w is symmetric, we also suppose that λ = 1.

The results of [4, 8] imply

Theorem 2.7. Every string or band datum d defines an indecomposable object C•(d) fromD(A), so that

1) Every indecomposable object from D(A) is isomorphic to C•(d) for some d.2) The only isomorphisms between these complexes are the following:

(a) C(w) C(w∗);(b) C(w,m, δ1, δ2) C(w∗,m, δ2, δ1);(c) C(w,m, λ) C(w[k],m, λ) C(w∗[k],m, 1/λ) if k ≡ 0 (mod 4);(d) C(w∗,m, λ) C(w[k],m, 1/λ) C(w∗[k],m, λ) if k ≡ 2 (mod 4).

3) Every object from D(A) uniquely decomposes into a direct sum of indecompos-able objects.

The construction of complexes C•(d) is rather complicated, especially in the case, whenthere are pairs (j, n) with (j, n) ∼ (j, n) (e.g. special ends are involved). So we only showseveral examples arising from simple node, dihedral algebra and Gelfand problem.

3. Examples

3.1. Simple node. In this case there is only one indecomposable projective A-module(A itself) and two indecomposable projective H-modules H1,H2 corresponding to the firstand the second direct factors of the ring H. We have H ⊗A A H H1 ⊕ H2. So the∼-relation is given by:

1) (1, n) ∼ (2, n);2) C(j, l, n) ∼ C(j,−l, n− sgn l) for any l ∈ Z \ 0 .

Therefore there are no special ends at all. Moreover, any end of a full string must be of theform C(j,∞, n). Note that the homomorphism in the complex corresponding to C(j,−l, n)and C(j, l, n + 1) (l ∈ N) is just multiplication by xlj . Consider several examples of stringsand bands.

Example 3.1. 1. Let w be the cycle

C(2, 1, 1) ∼ C(2,−1, 0)− (2, 0) ∼ (1, 0)− C(1,−2, 0) ∼ C(1, 2, 1)−− (1, 1) ∼ (2, 1)− C(2, 4, 1) ∼ C(2,−4, 0)− (2, 0) ∼ (1, 0)−− C(1,−1, 0) ∼ C(1, 1, 1)− (1, 1) ∼ (2, 1)− C(2,−3, 1) ∼ C(2, 3, 2)−− (2, 2) ∼ (1, 2)− C(1, 2, 2) ∼ C(1,−2, 1)− (1, 1) ∼ (2, 1)


Then the band complex C•(w, 1, λ) is obtained from the complex of H-modules

H2x2

H2

H1

x21

H1

H2

x42

H2

H1x1

H1

H2

x32

H2

H1

x21

H1

λ

by gluing along the dashed lines (they present the ∼ relations (1, n) ∼ (2, n)). Allglueings are trivial, except the last one marked with ‘λ’; the latter must be twistedby λ. It gives the A-complex

Ay

A

A

λx2

y3

A

x2

y4

A

A

x

(3.1)

Here each column presents direct summands of a non-zero component Cn (in our casen = 2, 1, 0) and the arrows show the non-zero components of the differential. Accordingto the embedding A → H, we have to replace x1 by x and x2 by y. Gathering alldata, we can rewrite this complex as

A

λx2

0y3

−−−−−→ A⊕A⊕A

y 0x2 y4

0 x

−−−−−−→ A⊕A ,

though the form (3.1) seems more expressive, so we use it further. If m > 1, one onlyhas to replace A by mA, each element a ∈ A by aE, where E is the identity matrix,


and λa by aJm(λ), where Jm(λ) is the Jordan m ×m cell with eigenvalue λ. So weobtain the complex

mA

x2Jm(λ)

0y3E

−−−−−−−−−→ mA⊕mA⊕mA

yE 0x2E y4E0 xE

−−−−−−−−−→ mA⊕mA .

2. Let w be the word

C(1,∞, 1)− (1, 1) ∼ (2, 1)− C(2, 2, 1) ∼ C(2,−2, 0)− (2, 0) ∼∼ (1, 0)− C(1,−3, 0) ∼ C(1, 3, 1)− (1, 1) ∼ (2, 1)− C(2,−1, 1) ∼∼ C(2, 1, 2)− (2, 2) ∼ (1, 2)− C(1, 1, 2) ∼ C(1,−1, 1)− (1, 1) ∼∼ (2, 1)− C(2, 2, 1) ∼ C(2,−2, 0)− (2, 0) ∼ (1, 0)− C(1,∞, 0)

Then the string complex C•(w) is

Ay2

A

Ay

x

Ax3

Ay2

A

Note that for string complexes (which are always usual in this case) there are nomultiplicities m and all glueings are trivial.

3. Set a = x+ y. Then the factor A/aA is represented by the complex A a−→ A, whichis the band complex C•(w, 1, 1), where

w = C(1, 1, 1) ∼ C(1,−1, 0)− (1, 0) ∼ (2, 0)−− C(2,−1, 0) ∼ C(2, 1, 1)− (2, 1) ∼ (1, 1).

Consider the morphism of this complex to A[1] given on the 1-component by multi-plication A x−→ A. It is non-zero in D(A), but the corresponding morphism of triplesis (Φ, 0), where Φ arises from the morphism of the complex H a−→ H to H[1] givenby multiplication with x1. But Φ is homotopic to 0: x1 = e1a, where e1 = (1, 0) ∈ H,thus (Φ, 0) = 0 in the category of triples.


4. The string complex C•(l, 0), where w is the word

C(1,∞, 0)− (1, 0) ∼ (2, 0)− C(2,−1, 0) ∼ C(2, 1, 1)− (2, 1) ∼∼ (1, 1)− C(1,−2, 1) ∼ C(1, 1, 2)− (1, 2) ∼ (2, 2)− C(2,−1, 2) ∼∼ C(2, 1, 3)− (2, 3) ∼ (1, 3)− C(1,−2, 3) ∼ C(1, 2, 4)− · · · ,

is

. . . A x2

−→ Ay−→ A x2

−→ Ay−→ A −→ 0.

Its homologies are not left bounded, so it does not belong to Db(A-mod).

3.2. Dihedral algebra. This case is very similar to the preceding one. Again there is onlyone indecomposable projective A-module (A itself) and two indecomposable projective H-modules H1,H2 corresponding to the first and the second columns of matrices from thering H, and we have H⊗A A H H1 ⊕H2. The main difference is that now the uniquemaximal submodule of Hj is isomorphic to Hk, where k = j. So the ∼-relation is givenby:

1) (1, n) ∼ (2, n);2) C(j, l, n) ∼ C(j,−l, n−sgn l) if l ∈ Z\ 0 is even, and C(j, l, n) ∼ C(j′,−l, n−sgn l),

where j′ = j, if l ∈ Z \ 0 is odd.

Again there are no special ends. The embeddingsHk → Hj are given by right multiplicationswith the following elements from H:

H1 → H1 − by tre11 (colength 2r),

H1 → H2 − by tre12 (colength 2r − 1),

H2 → H1 − by tre21 (colength 2r + 1),

H2 → H2 − by tre22 (colength 2r).

When gluing H-complexes into A-complexes we have to replace them respectively

tre11 − by (xy)r,

tre22 − by (yx)r,

tre12 − by (xy)r−1x,

tre21 − by (yx)ry.

The glueings are quite analogous to those for simple node, so we only present the results,without further comments.

Example 3.2. 1. Consider the band datum (w, 1, λ), where

w = C(1,−2, 0) ∼ C(1, 2, 1)− (1, 1) ∼ (2, 1)− C(2,−5, 1) ∼∼ C(1, 5, 2)− (1, 2) ∼ (2, 2)− C(2, 4, 2) ∼ C(2,−4, 1)− (2, 1) ∼∼ (1, 1)− C(1, 3, 1) ∼ C(2,−3, 0)− (2, 0) ∼ (1, 0).


The corresponding complex C•(w,m, λ) is

mA xyE mA

mA

(xy)2xE

(yx)2E mA

xyxJm(λ)

2. Let w be the word

C(2,∞, 0)− (2, 0) ∼ (1, 0)− C(1,−1, 0) ∼ C(2, 1, 1)− (2, 1) ∼ (1, 1)− C(1, 3, 1) ∼∼ C(2,−3, 0)− (2, 0) ∼ (1, 0)− C(1,−3, 0) ∼ C(2, 3, 1)− (2, 1) ∼ (1, 1)− C(1,∞, 1).

Then the string complex C•(w) is

A e21

t2e12

A

A te21 A

3. The factor A/R is described by the infinite string complex C•(w)

. . . e21 A te12 A e21 A.

. . . te12 A e21 A

te12

The corresponding word w is

· · · − C(2, 1, 2) ∼ C(1,−1, 1)− (1, 1) ∼ (2, 1)−− C(2, 1, 1) ∼ C(1,−1, 0)− (1, 0) ∼ (2, 0)− C(2,−1, 0) ∼∼ C(1, 1, 1)− (1, 1) ∼ (2, 1)− C(2,−1, 1) ∼ C(1, 1, 2)− · · ·

3.3. Gelfand problem. In this case there are 2 indecomposable projective H-modules H1

(the first column) and H2 (both the second and the third columns). There are 3 indecom-posable A-projectives Ai (i = 1, 2, 3); Ai correspond to the i-th column of A. We haveH⊗A A1 H1 and H⊗A A2 H⊗A A3 H2. So the relation ∼ is given by:

1) (2, n) ∼ (2, n);2) C(j, l, n) ∼ C(j,−l, n− sgn l) if l is even;3) C(j, l, n) ∼ C(j′,−l, n− sgn l) (j′ = j) if l is odd.

So a special end is always (2, n).


Example 3.3. 1. Consider the special word w:

(2, 0)− C(2,−2, 0) ∼ C(2, 2, 1)− (2, 1) ∼ (2, 1)− C(2,−4, 1) ∼∼ C(2, 4, 2)− (2, 2) ∼ (2, 2)− C(2, 2, 2) ∼ C(2,−2, 1)−− (2, 1) ∼ (2, 1)− C(2,−1, 1) ∼ C(1, 1, 2)− (1, 2)

The complex C•(w, 0) is obtained by gluing from the complex of H-modules

H2 2 H2

H2 4 H2

H2 2 H2

H1 1 H2

Here the numbers inside arrows show the colengths of the corresponding images. Wemark dashed lines defining glueings with arrows going from the bigger complex (withrespect to the ordering in Ej,n) to the smaller one. When we construct the corre-sponding complex of A-modules, we replace each H2 by A2 and A3 starting with A2

(since δ = 0; if δ = 1 we start from A3). Each next choice is arbitrary with the onlyrequirement that every dashed line must touch both A2 and A3. (Different choiceslead to isomorphic complexes: one can see it from the pictures below.) All horizontalmappings must be duplicated by slanting ones, carried along the dashed arrow fromthe starting point or opposite the dashed arrow with the opposite sign from the endingpoint (the latter procedure will be marked by ‘−’ near the duplicated arrow). So weget the A-complex

A2 2 A2

A3

−

4

4

2

2

A3

2

A2 2

2

A2

A1

−1

1 A3

All mappings are uniquely defined by the colengths in the H-complex, so we just markthem with ‘l.’


2. Let w be the bispecial word

(2, 2)− C(2, 2, 2) ∼ C(2,−2, 1)− (2, 1) ∼ (2, 1)− C(2, 2, 1) ∼∼ C(2,−2, 0)− (2, 0) ∼ (2, 0)− C(2,−4, 0) ∼ C(2, 4, 1)−− (2, 1) ∼ (2, 1)− C(2, 6, 1) ∼ C(2,−6, 0)− (2, 0)

The complex C•(w,m, 1, 0) is the following one:

aA3 ⊕ bA2 M1

−−M1

mA3

2

−

2mA2

2 −

2 mA3

mA3 4 mA2

mA2

4

M2 aA2 ⊕ bA3

where a = [(m + 1)/2], b = [m/2], so a + b = m. (The change of δ1, δ2 transpose A2

and A3 at the ends.) All arrows are just αlE, where αl is defined by the colength l,except of the “end” matrices Mi. To calculate the latter, write αlE for one of them(say, M1) and αlJ for anothher one (say, M2), where J is the Jordan m×m cell witheigenvalue 1, then put the odd rows or columns into the first part of Mi and the evenones to its second part. In our example we get

M1 = α2

1 0 0 0 00 0 0 1 00 1 0 0 00 0 0 0 10 0 1 0 0

, M2 = α6

1 1 0 0 00 0 1 1 00 0 0 0 10 1 1 0 00 0 0 1 1

.

(We use columns for M1 and rows for M2 since the left end is the source and the rightend is the sink of the corresponding mapping.)

3. The band complex C•(w, 1, λ), where w is the cycle

(2, 1) ∼ (2, 1)− C(2,−2, 1) ∼ C(2, 2, 2)− (2, 2) ∼ (2, 2)−− C(2, 4, 2) ∼ C(2,−4, 1)− (2, 1) ∼ (2, 1)− C(2, 6, 1) ∼∼ C(2,−6, 0)− (2, 0) ∼ (2, 0)− C(2,−4, 0) ∼ C(2, 4, 1)

is


mA2 2

−

2

mA2

4λ

−4λ

mA3

2

4

−

4

−

2

mA2

6

mA3 6 mA2

mA3

4λ −

4λ mA3

Superscript ‘λ’ denotes that the corresponding mapping must be twisted by Jm(λ).

4. The projective resolution of the simple A-module U1 is

A2 1 A1

A1

−

1

1 A3

1

It coincides with the usual string complex C•(w), where w is

(1, 0)− C(1,−1, 0) ∼ C(2, 1, 1)− (2, 1) ∼ (2, 1)− C(2,−1, 1) ∼ C(1, 1, 2)− (1, 2).

The projective resolution of U2 (U3) is A1 → A2 (respectively A1 → A3), which is thespecial string complex C•(w, 0) (respectively C•(w, 1)), where

w = (2, 0)− C(2,−1, 0) ∼ C(1, 1, 1)− (1, 1).

Note that gl.dimA = 2.

4. Projective configurations

We can “globalize” the results of the preceding sections. The simplest way is toconsider the so called projective configurations, which are a sort of global analogues ofBackstrom rings.

Definition 4.1. LetX be a projective curve over k, which we suppose reduced, but possiblyreducible. We denote by π : X → X its normalization; then X is a disjoint union ofsmooth curves. We call X a projective configuration if all components of X are rationalcurves (i.e. of genus 0) and all singular points p of X are ordinary. The latter means thatif π−1(p) = y1, y2, . . . , ym , the image of OX,p in

∏mi=1OX,yi

contains∏mi=1 mi, where mi

is the maximal ideal of OX,yi.


We denote by S = p1, p2, . . . , ps the set of singular points of X and by S = y1, y2, . . . , yr its preimage in X. We also put O = OX , O = OX and denote by J theconductor of O in O, i.e. the maximal sheaf of π∗O-ideals contained in O. Set S = O/Jand R = π∗O/J O/π−1J . Both these sheaves have 0-dimensional support S, so wemay (and shall) identify them with the algebras of their global sections. In the case ofprojective configurations both these algebras are semi-simple, namely S =

∏si=1 k(pi) and

R =∏ri=1 k(yi).

Let D(X) = D−(CohX) be the right bounded derived category of coherent sheavesover X. As X is a projective variety, it can be identified with the category of fractionsK−(VBX)[Q−1], where K−(VBX) is the category of right bounded complexes of vectorbundles (or, the same, locally free coherent sheaves) over X modulo homotopy and Q isthe set of quasi-isomorphisms in K−(VBX). So we always present objects from D(X)and from D(X) as complexes of vector bundles. We denote by T : D(X) → D(X) theleft derived functor Lπ∗. Again if C• is a complex of vector bundles, TC• coincides withπ∗C•.

Just as in Section 1, we define the category of triples T = T (X) as follows:

• Objects of T are triples (A•, B•, ι), where– A• ∈ D(X);– B• ∈ D(S);– ι is a morphism B• → R ⊗O A• from D(S) such that the induced morphismιR : R⊗S B• → R⊗O A• is an isomorphism in D(R).

• A morphism from a triple (A•, B•, ι) to a triple (A′•, B′•, ι′) is a pair (Φ, φ), where– Φ : A• → A′• is a morphism from D(X);– φ : B• → B′• is a morphism from D(S);– the diagram

B•ι−−−−→ R⊗O A•

φ

1⊗Φ

B′•ι′−−−−→ R⊗O A′•

(4.1)

commutes in D(S).

We define a functor F : D(X) → T (X) setting F(C•) = (π∗C•,S ⊗O C•, ι), where ι :S ⊗O C• → R ⊗O (π∗C•) R ⊗O C• is induced by the embedding S → R. Just as inSection 1 the following theorem holds (with almost the same proof, see [6]).

Theorem 4.2. The functor F is a representation equivalence, i.e. it is dense and conser-vative.

Remark. We do not now whether it is full, though it seems to be true.

5. Configurations of type A and A

As it was shown in [13], even classification of vector bundles is wild for almost all projec-tive curves. Among singular curves the only exceptions are projective configurations oftype A and A. These curves only have ordinary double points (so no three components


have a common point). Moreover, in A case irreducible components X1,X2, . . . , Xs andsingular points p1, p2, . . . , ps−1 can be so arranged that pi ∈ Xi ∩ Xi+1, while in A casethe components X1,X2, . . . , Xs and the singular points p1, p2, . . . , ps can be so arrangedthat pi ∈ Xi ∩ Xi+1 for i < s and ps ∈ Xs ∩ X1. Note that in A case s > 1, while inA case s = 1 is possible: then there is one component with one ordinary double point(a nodal plane cubic). These projective configurations are global analogues of nodal rings,and the calculations according Theorem 4.2 are quite similar to those of Section 2. Wepresent here the A case and add remarks explaining which changes should be done forA case.

If s > 1, the normalization of X is just a disjoint union⊔si=1Xi; for uniformity, we

write X1 = X if s = 1. We also denote Xqs+i = Xi. Note that Xi P1 for all i.Every singular point pi has two preimages p′i, p

′′i in X; we suppose that p′i ∈ Xi corre-

sponds to the point ∞ ∈ P1 and p′′i ∈ Xi+1 corresponds to the point 0 ∈ P1. Recallthat any indecomposable vector bundle over P1 is isomorphic to OP1(d) for some d ∈ Z.So every indecomposable complex from D(X) is isomorphic either to 0 → Oi(d) → 0or to 0 → Oi(−lx) → Oi → 0, where Oi = OXi

, d ∈ Z, l ∈ N and x ∈ Xi. Thelatter complex corresponds to the indecomposable sky-scraper sheaf of length l andsupport x . We denote this complex by C(x,−l, n) and by C(x, l, n + 1). The complex0 → Oi(d) → is denoted by C(p′i, dω, n) and by C(p′′i−1, dω, n). As before, n is theunique place, where the complex has non-zero homologies. We define the symmetricrelation ∼ for these symbols setting C(x,−l, n) ∼ C(x, l, n + 1) and C(p′i, dω, n) ∼C(p′′i−1, dω, n).

Let Zω = (Z ⊕ 0 ) ∪ Zω, where Zω = dω | d ∈ Z . We introduce an ordering on Zω,which is natural on N, on −N and on Zω, but l < dω < −l for each l ∈ N, d ∈ Z. Then ananalogue of Lemma 2.3 can be easily verified.

Lemma 5.1. There is a morphism of complexes C(x, z, n) → C(x, z′, n) such that its nthcomponent induces a non-zero mapping on Cn(x) if and only if z ≤ z′ in Zω.

We introduce the ordered sets Ex,n = C(x, z, n) | z ∈ Zω with the orderinginherited from Zω, We also put Fx,n = (x, n) and (p′i, n) ∼ (p′′i−1, n) for all i, n.Lemma 5.1 shows that the category of triples T (X) can be again described in termsof the bunch of chains Ex,n, Fx,n . Thus we can describe indecomposable objects interms of strings and bands just as for nodal rings. We leave the corresponding defini-tions to the reader; they are quite analogous to those from Section 2. If we considera configuration of type A, we have to exclude the points p′s, p

′′s and the correspond-

ing symbols C(p′s, z, n), C(p′′s , z, n), (p′s, n), (p′′s , n). Thus in this case C(p′′s−1, dω, n) andC(p′1, dω, n) are not in ∼ relation with any symbol. It makes possible finite or one-side infinite full strings, while in A case only two-side infinite strings are full. Notethat an infinite word must contain a finite set of symbols (x, n) with any fixed n;moreover there must be n0 such that n ≥ n0 for all entries (x, n) that occur in thisword.

If x /∈ S and z /∈ Zω, the complex C(x, z, n) vanishes after tensoring by R, so gives noessential input into the category of triples. It gives rise to the n-th shift of a sky-scrapersheaf with support at the regular point x. Therefore in the following examples we onlyconsider complexes C(x, z, n) with x ∈ S. Moreover, we confine most examples to the cases = 1 (so X is a nodal cubic). If s > 1, one must distribute vector bundles in the picturesbelow among the components of X.


Example 5.2. 1. First of all, even a classification of vector bundles is non-trivial in Acase. They correspond to bands concentrated at 0 place, i.e. such that the underlyingcycle w is of the form

(p′s, 0) ∼ (p′′s , 0)− C(p′′s , d1ω, 0) ∼ C(p′1, d1ω, 0)−− (p′1, 0) ∼ (p′′1 , 0)− C(p′′1 , d2ω, 0) ∼ C(p′2, d2ω, 0)−− (p′2, 0) ∼ (p′′2 , 0)− C(p′′2 , d3ω, 0) ∼ · · · ∼ C(p′s, drsω, 0)

(obviously, its length must be a multiple of s, and we can start from any place p′k, p′′k).

Then C•(w,m, λ) is actually a vector bundle, which can be schematically described asthe following gluing of vector bundles over X.

• d1 •λ

• d2 •

• d3 •

...

• drs •

Here horizontal lines symbolize line bundles over Xi of the superscripted degrees,their left (right) ends are basic elements of these bundles at the point ∞ (respec-tively 0), and the dashed lines show which of them must be glued. One musttake m copies of each vector bundle from this picture and make all glueings triv-ial, except one going from the uppermost right point to the lowermost left one(marked by ‘λ’), where the gluing must be performed using the Jordan m × mcell with eigenvalue λ. In other words, if e1, e2, . . . , em and f1, f2, . . . , fm arebases of the corresponding spaces, one has to identify f1 with λe1 and fk withλek + ek−1 if k > 1. We denote this vector bundle over X by V(d,m, λ), whered = (d1, d2, . . . , drs); it is of rankmr and of degreem

∑ri=1 di. If r = s = 1, this picture

becomes

• dλ

•

If r = m = 1, we obtain all line bundles: they are V((d1, d2, . . . , ds), 1, λ) (of degree∑si=1 di). Thus the Picard group is Zs × k∗.


In A case there are no bands concentrated at 0 place, but there are finite strings ofthis sort:

C(p′′1 , d1ω, 0)− (p′1, 0) ∼ (p′′1 , 0)− C(p′′1 , d2ω, 0) ∼∼ C(p′2, d2, 0)− (p′2, 0) ∼ (p′′2 , 0)− C(p′′2 , d3, 0) ∼· · · ∼ C(p′s−1, ds−1ω, 0)− (p′s−1, 0) ∼ (p′′s−1, 0)− C(p′′s−1, dsω, 0)

So vector bundles over such configurations are in one-to-one correspondence withintegral vectors (d1, d2, . . . , ds); in particular, all of them are line bundles and thePicard group is Zs. In the picture above one has to set r = 1 and to omit the lastgluing (marked with ‘λ’).

2. From now on s = 1, so we write p instead of p1. Let w be the cycle

(p′′, 1) ∼ (p′, 1)− C(p′,−2, 1) ∼ C(p′, 2, 2)− (p′, 2) ∼ (p′′, 2)−− C(p′′, 3ω, 2) ∼ C(p′, 3ω, 2)− (p′, 2) ∼ (p′′, 2)− C(p′′, 3, 2) ∼∼ C(p′′,−3, 1)− (p′′, 1) ∼ (p′, 1)− C(p′, 1, 1) ∼ C(p′,−1, 0)−− (p′, 0) ∼ (p′′, 0)− C(p′′,−2, 0) ∼ C(p′′, 2, 1).

Then the band complex C•(w,m, λ) can be pictured as follows:

• 2 •λ

• 3 •

• 3 •

• 1 •

• 2 •

Again horizontal lines describe vector bundles over X. Bullets and circles correspond tothe points∞ and 0; circles show those points, where the corresponding complex givesno input into R⊗OA•. Horizontal arrows show morphisms in A•; the numbers l insidegive the lengths of factors. Dashed and dotted lines describe glueings. Dashed lines(between bullets) correspond to mandatory glueings arising from relations (p′, n) ∼


(p′′, n) in the word w, while dotted lines (between circles) can be drawn arbitrarily;the only conditions are that each circle must be an end of a dotted line and the dottedlines between circles sitting at the same level must be parallel (in our picture theyare between the 1st and 3rd levels and between the 4th and 5th levels). The degreesof line bundles in complexes C(x, z, n) with z ∈ N ∪ (−N) (they are described by thelevels containing 2 lines) can be chosen as d− l and d with arbitrary d (we set d = 0),otherwise (in the second row) they are superscripted over the line. Thus the resultingcomplex is

V((−2, 3,−3),m, 1) −→ V((0, 0,−1,−2),m, λ) −→ V((0, 0),m, 1)

(we do not precise mappings, but they can be easily restored).

3. If s = 1, the sky-scraper sheaf k(p) is described by the complex

· · · • • 1 •

· · · • 1 • • 1 •

· · · • 1 • • 1 •

· · · • • 1 •

which is the string complex corresponding to the word

. . . C(p′,−1, 2)− (p′, 2) ∼ (p′′, 2)− C(p′′, 1, 2) ∼ C(p′′,−1, 1)−− (p′′, 1) ∼ (p′, 1)− C(p′, 1, 1) ∼ C(p′,−1, 0)− (p′, 0) ∼∼ (p′′, 0)− C(p′′,−1, 0) ∼ C(p′′, 1, 1)− (p′′, 1) ∼ (p′, 1)−− C(p′,−1, 1) ∼ C(p′, 1, 2)− (p′, 2) ∼ (p′′, 2)− C(p′′,−1, 2) . . .

4. The band complex C(w,m, λ) , where w is the cycle

(p′, 0) ∼ (p′′, 0)− C(p′′,−3ω, 0) ∼ C(p′,−3ω, 0)−− (p′, 0) ∼ (p′′, 0)− C(p′′, 0ω, 0) ∼ C(p′, 0ω, 0)− (p′, 0) ∼∼ (p′′, 0)− C(p′′,−1, 0) ∼ C(p′′, 1, 1)− (p′′, 1) ∼ (p′, 1)−− C(p′′, 2, 1) ∼ C(p′,−2, 0)− (p′, 0) ∼ (p′′, 0)− C(p′′,−4, 0) ∼∼ C(p′′, 4, 1)− (p′′, 1) ∼ (p′, 1)− C(p′, 5, 1) ∼ C(p′,−5, 0)−− (p′, 0) ∼ (p′′, 0)− C(p′′, 0ω, 0) ∼ C(p′, 0ω, 0)

describes the complex


• -3 •λ

• 0 •

• 1 •

• 2 •

• 4 •

• 5 •

• 0 •or

V((0, 0),m, 1)⊕ V((0, 0),m, 1) −→ V((−3, 0, 1, 2, 4, 5, 0),m, λ).

Its homologies are zero except the place 0, so it correspond to a coherent sheaf. Onecan see that this sheaf is a “mixed” one (neither torsion free nor sky-scraper).

Note that this time we could trace dotted lines another way, joining the first freeend with the last one and the second with the third.

• -3 •λ

• 0 •

• 1 •

• 2 •

• 4 •

• 5 •

• 0 •

It gives an isomorphic object in D(X)

V((0, 0, 0, 0),m, 1) −→ V((−3, 0, 1, 5, 0),m, λ)⊕ V((2, 4),m, 1).


Remark. In [6] we used another encoding of strings and bands for projective configurations,which is equivalent, but uses more specifics of the situation. In this paper we prefer to use auniform encoding, which is the same both for nodal rings and for projective configurations.

6. Application: Cohen–Macaulay modules over surface singularities

The results on vector bundles over projective configurations can be applied to studyCohen–Macaulay modules over normal surface singularities. Recall some related notions.Let A be a noetherian local complete domain of Krull dimension 2, which is normal (i.e.integrally closed in its field of fractions), X = SpecA and o be the unique closed points of X(corresponding to the maximal ideal m of A). We call A or X a normal surface singularity.A resolution of this singularity is a morphism of schemes π : Y → X such that

• Y is smooth;• π is projective and birational;• the restriction of π onto Y = Y \ π−1(o) is an isomorphism Y → X = X \ o .

We denote by E = π−1(o)red and call it the exceptional curve of the resolution. It is indeeda projective curve. Let E1, E2, . . . , Es be its irreducible components. We call effective cyclesnon-zero divisors on Y of the form Z =

∑si=1 kiEi with ki ≥ 0 and consider such a cycle

as a projective curve (non-reduced if some ki > 1), namely the subscheme of Y definedby the sheaf of ideals OY (−Z). Obviously Zred =

⋃ki>0Ei. In [17] C. Kahn established

a one-to-one correspondence between Cohen–Macaulay modules over A and some vectorbundles over a special effective cycle Z, called a reduction cycle. We shall not present herehis result in full generality, but only in the case, when the singularity is minimally elliptic,which means, by definition, that A is Gorenstein and dimk H1(Y,OY ) = 1 [19]. We alsosuppose that the resolution π : Y → X is minimal, i.e. cannot be factored through anyother non-isomorphic resolution. Then Kahn’s result can be stated as follows

Theorem 6.1 ([17]). Let A be a minimally elliptic surface singularity and Z be the funda-mental cycle of its minimal resolution, i.e. the smallest effective cycle such that (Z.Ei) ≤ 0for all i. There is one-to-one correspondence between Cohen–Macaulay modules over A andvector bundles F over Z such that F G ⊕ nOZ , where

1) G is generically spanned, i.e. global sections from Γ(E,G) generate G everywhere,except maybe finitely many closed points;

2) H1(E,G) = 0;3) n ≥ dimk H0(E,G(Z)).

Especially, indecomposable Cohen–Macaulay A-modules correspond to vector bundles F G ⊕ nOZ , where either G = 0, n = 1 or G is indecomposable, satisfies the above condi-tions (a,b) and n = dimk H0(E,G(Z)). (The vector bundle OZ corresponds to the regularA-module, i.e. A itself.)

Kahn himself deduced from this theorem and the results of Atiyah [1] a description ofCohen–Macaulay modules over simple elliptic singularities, i.e. such that E is an ellipticcurve (smooth curve of genus 1). Using the results of Section 5, one can obtain an analogousdescription for cusp singularities, i.e. such that E is a projective configuration of type A.Briefly, one gets the following theorem (for more details see [14]).


Theorem 6.2. There is a one-to-one correspondence between indecomposable Cohen–Macaulay modules over a cusp singularity A, except the regular module A, and vectorbundles V(d,m, λ), where d = (d1, d2, . . . , drs) satisfies the following conditions2:

• d > 0, i.e. di ≥ 0 for all i and d = (0, 0, . . . , 0);• no shift of d, i.e. a sequence (dk+1, . . . , drs, d1, . . . , dk), contains a subsequence

(0, 1, 1, . . . , 1, 0), in particular (0, 0);• no shift of d is of the form (0, 1, 1, . . . , 1).

Moreover, from Theorem 6.1 and the results of [13] one gets the following

Theorem 6.3 ([14]). If a minimally elliptic singularity A is neither simple elliptic nor cusp,it is Cohen–Macaulay wild, i.e. the classification of Cohen–Macaulay A-modules includesthe classification of representations of all finitely generated k-algebras.

As a consequence of Theorem 6.2 and the Knorrer periodicity theorem [18, 20], one alsoobtains a description of Cohen–Macaulay modules over hypersurface singularities of typeTpqr, i.e. factor-rings

k[[x1, x2, . . . , xn]]/(xp1 + xq2 + xr3 + λx1x2x3 +Q) (n ≥ 3, 1/p+ 1/q + 1/r ≤ 1),

where Q is a non-degenerate quadratic form of x4, . . . , xn, and over curve singularities oftype Tpq, i.e. factor-rings

k[[x, y]]/(xp + yq + λx2y2) (1/p+ 1/q ≤ 1/2).

The latter fills up a flaw in the result of [12], where one has only proved that the curvesingularities of type Tpq are Cohen–Macaulay tame, but got no explicit description ofmodules.

Recall that a normal surface singularity A is Cohen–Macaulay finite, i.e. has only a finitenumber of non-isomorphic indecomposable Cohen–Macaulay modules, if and only if it is aquotient singularity, i.e. A k[[x, y]]G, where G is a finite group of automorphisms [2, 15].Just in the same way one can show that all singularities of the form A = BG, where Bis either simple elliptic or cusp, are Cohen–Macaulay tame, and obtain a description ofCohen–Macaulay modules in this case. We call such singularities elliptic-quotient. There isan evidence that all other singularities are Cohen–Macaulay wild, so Table 1 completelydescribes Cohen–Macaulay types of isolated singularities (we mark by ‘?’ the places, wherethe result is still a conjecture).

2There was a mistake in the preprint [14], where we claimed that d > 0 is enough for V(d, m, λ) to

satisfy Kahn’s conditions. It has been improved in the final version. We are thankful to Igor Burban who

has noticed this mistake.


Table 1.

Cohen–Macaulay types of singularities

CM type curves surfaces hypersurfaces

finite dominate quotient simpleA-D-E (A-D-E)

tame dominate elliptic-quotient TpqrTpq (only ?) (only ?)

wild all other all other ? all other ?

References

[1] M. Atiyah. Vector bundles over an elliptic curve. Proc. London Math. Soc. 7 (1957), 414–452.[2] M. Auslander. Rational singularities and almost split sequences. Trans. Amer. Math. Soc. 293

(1986), 511–531.[3] H. Bass. Finitistic dimension and a homological generalization of semi-primary rings. Trans.

Amer. Math. Soc. 95 (1960), 466–488.[4] V. M. Bondarenko. Representations of bundles of semi-chained sets and their applications.

Algebra i Analiz 3, No. 5 (1991), 38–61 (English translation: St. Petersburg Math. J. 3(1992), 973–996).

[5] I. I. Burban and Y. A. Drozd. Derived categories of nodal rings. J. Algebra 272 (2004), 46–94.[6] I. I. Burban and Y. A. Drozd. Coherent sheaves on rational curves with simple double points

and transversal intersections. Duke Math. J. 121 (2004), 189–229.[7] I. I. Burban, Y. A. Drozd and G.-M. Greuel. Vector bundles on singular projective curves.

Applications of Algebraic Geometry to Coding Theory, Physics and Computation. KluwerAcademic Publishers, 2001, 1–15.

[8] W. Crawley-Boevey. Functorial filtrations, II. Clans and the Gelfand problem. J. London Math.Soc. 1 (1989), 9–30.

[9] Y. A. Drozd. Modules over hereditary orders. Mat. Zametki 29 (1981), 813–816.[10] Y. A. Drozd. Finite modules over pure Noetherian algebras. Trudy Mat. Inst. Steklov Acad.

Nauk USSR 183 (1990), 56–68. (English translation: Proc. Steklov Inst. of Math. 183(1991), 97–108.)

[11] Y. A. Drozd. Finitely generated quadratic modules. Manuscripta matem. 104 (2001), 239–256.[12] Y. A. Drozd and G.-M. Greuel. Cohen–Macaulay module type. Compositio Math. 89

(1993), 315–338.[13] Y. A. Drozd and G.-M. Greuel. Tame and wild projective curves and classification of vector

bundles. J. Algebra 246 (2001), 1–54.[14] Y. A. Drozd, G.-M. Greuel and I. V. Kashuba. On Cohen–Macaulay modules on surface

singularities. Preprint MPI 00–76. Max–Plank–Institut fur Mathematik, Bonn, 2000 (to appearin Moscow Math. J.).

[15] H. Esnault. Reflexive modules on quotient surface singularities. J. Reine Angew. Math. 362(1985), 63–71.

[16] R. Hartshorn. Algebraic Geometry. Springer–Verlag, New York, 1977.


[17] C. Kahn. Reflexive modules on minimally elliptic singularities. Math. Ann. 285 (1989),141–160.

[18] H. Knorrer. Cohen–Macaulay modules on hypersurface singularities. I. Invent. Math. 88(1987), 153–164.

[19] H. Laufer. On minimally elliptic singularities. Am. J. Math. 99 (1975), 1257–1295.[20] Y. Yoshino. Cohen–Macaulay Modules over Cohen–Macaulay Rings. Cambridge University

Press, 1990.

Kyiv Taras Shevchenko University, University of Kaiserslautern and Institute of Mathemat-

ics of the National Academy of Sciences of Ukraine



CROWNS IN PROFINITE GROUPS AND APPLICATIONS

ELOISA DETOMI AND ANDREA LUCCHINI

In [6] Gaschutz introduced the notion of crown associated with a complemented chieffactor H/K of a finite soluble group G; the crown is a certain normal factor of G, whichcollects all complemented chief factors of G which are G-isomorphic to H/K. He employedthis notion in the construction of a characteristic conjugacy class of subgroups, the prefrat-tini subgroups. Later this notion has been generalized to all finite groups (see for example[10] and [8]): it has been defined the crown associated with a non-Frattini chief factor of anarbitrary finite group.

In [4] the notion of crown have been applied to study some properties of the probabilisticzeta function of a finite group. Let we recall how this function is defined. For a finite groupG and a non-negative integer t let ProbG(t) be the probability that t random elementsgenerate G. In [7] Hall proved that

ProbG(t) =∑H≤G

µ(H)|G : H|t

where µ is the Mobius function of the subgroup lattice ofG. Hence ProbG(t) can be exhibitedas a finite Dirichlet series

∑n∈N ann

−t with an ∈ Z and an = 0 unless n divides |G|. So,in view of Hall’s formula, we can speak of ProbG(s) for an arbitrary complex number s.The function ProbG(s) is the multiplicative inverse of a zeta function for G, as describedby Mann [11] and Boston [1]. What is shown in [4] is that the properties of the crowns ofa finite group G can be used to study the factors of ProbG(s) in the ring of finite Dirichletseries with integer coefficients.

In the present paper we revise the notion of crown in the contest of profinite groups. Weprove that it is possible to extend the definitions and the results known in the finite case,to arbitrary profinite groups. Moreover, when G is a finitely generated profinite group, itis possible to associate to G an infinite formal Dirichlet series, generalizing the definitiongiven in the finite case: we apply the crowns to study some properties of this series.

1. G-equivalence and crowns

Recall that a profinite group is a compact Hausdorff topological group whose opensubgroups form a base for the neighborhoods of the identity; these groups are exactly thoseobtained as inverse limits of finite groups. In this paper we are mainly interested in profinitegroups, so, unless stated otherwise, “groups” means profinite groups, “subgroups” meansclosed subgroups and the homomorphisms are assumed to be continuous. Recall that a(closed) subgroup is open if and only if it has finite index in G.

In [8] an equivalence relation among irreducible G-groups is described in the particularcase when G is a finite group; we generalize this notion to the case when G is a profi-nite group.

48 ELOISA DETOMI AND ANDREA LUCCHINI

Definition 1. Let G be a profinite group and let A and B be two finite irreducible G-groups.We say that they are G-equivalent and put A ∼G B, if there are two continuous isomor-phisms φ : A→ B and Φ : AG→ BG such that the following diagram commutes:

1 −−−−→ A −−−−→ AG −−−−→ G −−−−→ 1φΦ

∥∥∥1 −−−−→ B −−−−→ BG −−−−→ G −−−−→ 1

It is immediate that this is an equivalence relation. Notice that if φ : A → B is aG-isomorphism then (ag)Φ = aφg, a ∈ A, g ∈ G, defines an isomorphism Φ : AG → BGwhich makes the above diagram commutative. That is, two G-isomorphic G-groups areG-equivalent. Conversely, if A and B are abelian and G-equivalent then A and B are alsoG-isomorphic. Indeed for any g ∈ G there exists bg ∈ B with gΦ = bgg, so for any a ∈ Awe have (ag)φ = (ag)Φ = (aΦ)g

Φ= (aφ)bgg = (aφ)g. But for nonabelian G-groups the

G-equivalence is strictly weaker than G-isomorphism; for example the two minimal normalsubgroups of G = Alt(5)2 are G-equivalent without being G-isomorphic.

Now assume that A and B are finite irreducible G-groups and consider C = CG(A) ∩CG(B). Since the actions of G on A and B are assumed to be continuous, CG(A) and CG(B)are open subgroups of G, so in particular C is an open normal subgroup of G and G/C is afinite group. The following lemma reduces the study of our equivalence relation to the casewhen G is a finite group.

Lemma 2. A and B are G-equivalent if and only if they are G/C-equivalent.

Proof. The statement is obvious when A and B are abelian, since in that case G-equivalentis the same as G-isomorphic. So we may assume that A and B are nonabelian.

First assume A ∼G B. For any c ∈ C there exists bc ∈ B with cΦ = bcc. If a ∈ A, wehave aφ = (ac)φ = (ac)Φ = (aΦ)c

Φ= (aφ)bcc = (aφ)bc , hence bc ∈ Z(Aφ) = Z(B) = 1. This

proves that cφ = c for any c ∈ C. But then it is well defined an isomorphism Ψ : AG/C →BG/C by the position (agC)Ψ = (ag)ΦC which makes commutative the following diagram:

1 −−−−→ A −−−−→ AG/C −−−−→ G/C −−−−→ 1φΨ

∥∥∥1 −−−−→ B −−−−→ BG/C −−−−→ G/C −−−−→ 1.

Hence A ∼G/C B.Now assume that A ∼G/C B and let Ψ : AG/C → BG/C be an isomorphism which

makes commutative the diagram:

1 −−−−→ A −−−−→ AG/C −−−−→ G/C −−−−→ 1φΨ

∥∥∥1 −−−−→ B −−−−→ BG/C −−−−→ G/C −−−−→ 1.

CROWNS IN PROFINITE GROUPS AND APPLICATIONS 49

For any g ∈ G, there exists bg ∈ B such that (gC)Ψ = bggC. Define Φ : AG → BG bysetting aΦ = aφ if a ∈ A, gΦ = bgg if g ∈ G; it is easy to check that Φ is well defined andthe following diagram is commutative:

1 −−−−→ A −−−−→ AG −−−−→ G −−−−→ 1φΦ

∥∥∥1 −−−−→ B −−−−→ BG −−−−→ G −−−−→ 1.

Hence A ∼G B.

We will say that a section H/K is a chief factor of G if H and K are closed normalsubgroups of G with K < H and for any closed normal subgroup X of G with K ≤ X ≤ Heither X = K or X = H. Notice that if H/K is a chief factor of G, then there exists anopen normal subgroup of N of G with H/K ∼=G HN/KN ; indeed H (as well as K) is theintersection of all the open normal subgroups that contain it and so, as H = K, we getHN = KN for at least one open normal subgroup N of G. This implies that a chief factorH/K is finite and that the action of G on H/K is continuous and irreducible.

Our first aim is to study the G-equivalence relation between the chief factors of G. Recallthat a finite group L is said to be primitive if it has a maximal subgroup with trivialnormal core. The socle soc(L) of a primitive group L can be either an abelian minimalnormal subgroup (I), or a nonabelian minimal normal subgroup (II), or the product of twononabelian minimal normal subgroups (III); we say respectively that G is primitive of typeI, II, III and in the first two cases we say that L is monolithic.

As in the case of finite groups (see [5], [8]) the G-equivalence relation on chief factors ofG is strictly related to the primitive epimorphic images of G. We have:

Lemma 3. Let G be a profinite group. Two chief factors are G-equivalent as G-groupsif and only if they are G-isomorphic either between them or to the two minimal normalsubgroups of a finite primitive epimorphic image of type III of G.

Proof. This is true if G is finite (see [8]) and Lemma 2 allows us to reduce the proof to thefinite case.

A chief factor H/K is called Frattini factor if H/K ≤ Frat(G/K). Notice that if H/Kis a Frattini factor, then HN/KN is Frattini for every normal closed subgroup N of G. Inparticular, by considering a finite image of G, we get that a Frattini chief factor is abelian.Now we are ready to give two crucial definitions.

Let A be a finite irreducible G-group. We set

IG(A) = g ∈ G | g induces an inner automorphism in A.

Notice that IG(A) contains CG(A), so it is an open normal subgroup of G.Next let XG(A) be the set of open normal subgroups N of G with the properties that

N ≤ IG(A), IG(A)/N ∼G A and IG(A)/N is non-Frattini. We define

RG(A) =⋂

N∈XG(A)

N


if the set XG(A) is nonempty, otherwise we set RG(A) = IG(A). The quotient groupIG(A)/RG(A) is called the A-crown of G or the crown of G associated with A.

Note that two G-equivalent G-groups, A and B, define the same crown; indeed IG(A) =IG(B) and so RG(A) = RG(B). Moreover, since RG(A) and IG(A) are closed normalsubgroups of G, the quotient groups G/RG(A) and IG(A)/RG(A) are profinite groups; ifXG(A) = ∅, then the family of subgroups N/RG(A) where N is an intersection of finitelymany subgroups in XG(A) is a fundamental system of open neighborhoods of the identityin both G/RG(A) and IG(A)/RG(A).

We want to study the structure of G/RG(A). First note that RG(A) = IG(A) if and onlyif A is equivalent to a non-Frattini chief factor of G; so we restrict our attention to this case.

Let ρ : G → Aut(A) be defined by g → gρ, where gρ : a → ag for all a ∈ A. Themonolithic primitive group associated with A is defined as

LG(A) =

GρA ∼= (G/CG(A))A if A is abelian,Gρ ∼= G/CG(A) otherwise.

Observe that LG(A) is a finite primitive group of type I or II, and soc(LG(A)) ∼= A. Notethat two G-equivalent G-groups may have different centralizers in G, but their associatedmonolithic primitive groups are isomorphic.

To simplify our notation we identify A with soc(LG(A)) and we set I = IG(A), R =RG(A), L = LG(A), and X = XG(A). Moreover let Y be the set of normal subgroups of Gobtained as intersection of finitely many subgroups in X ; we remark that, if X = ∅, thenG/R is the inverse limit of the family of finite groups G/N for N ∈ Y (as well as I/R isthe inverse limit of the family I/N for N ∈ Y). We want to describe the structure of G/Nwhen N ∈ Y. To do that we recall a definition:

Definition 4. (see [3]) Let now L be a monolithic primitive group and let A be its uniqueminimal normal subgroup. For each positive integer k, let Lk be the k-fold product of L.The crown-based power of L of size k is the subgroup Lk of Lk defined by

Lk = (l1, . . . , lk) ∈ Lk | l1 ≡ · · · ≡ lk mod A.

Clearly, soc(Lk) = Ak, Lk/ soc(Lk) ∼= L/A and the quotient group of Lk over any minimalnormal subgroup is isomorphic to Lk−1, for k > 1. Moreover any normal subgroup of Lkeither contains or is contained in soc(Lk).

The utility of the previous definition in our study of the group G/R is explained by thenext lemma (see Proposition 9 in [4]):

Lemma 5. If Y ∈ Y then G/Y ∼= Lk where k is the smallest cardinality of a subsetN1, . . . , Nk of X with Y = N1 ∩ · · · ∩ Nk. Moreover I/Y = soc(G/Y ) and any chieffactor H/K of G with Y ≤ K < H ≤ I is non-Frattini and G-equivalent to A.

Corollary 6. If N is a closed normal subgroup of G and R ≤ N then either I ≤ N orN ≤ I. Moreover if N is open and R ≤ N < I then N ∈ Y.Proof. As N is closed and Y/RY ∈Y is a fundamental system of open neighborhoods of theidentity in G/R, we have N =

⋂Y ∈Y NY . Now NY/Y is a normal subgroup of the finite

group G/Y which is isomorphic to Lk for an integer k by the previous lemma. It followsthat I/Y = soc(G/Y ) and also either NY ≤ I or NY > I. In the first case we concludeN ≤ I. Otherwise, NY > I for every Y ∈ Y and thus N =

⋂Y ∈Y NY ≥ I.


For any set Ω, the cartesian product LΩ, endowed with the product topology, is a profinitegroup. Let now consider the subgroup

LΩ = (lω)ω∈Ω ∈ LΩ | lω1 ≡ lω2 mod A for any ω1, ω2 ∈ Ω.

LΩ is a closed subgroup of LΩ, so it can be viewed as a profinite group; indeed, LΩ is theinverse limit of the family of finite groups LI , where I is a finite subset of Ω.

Let now D be the set of subsets ∆ of Hom(G,L) satisfying:

(1) for any φ ∈ ∆, kerφ ∈ X ;(2) for any finite subset I = φ1, . . . , φk of ∆, the position gφI = (gφ1 , . . . , gφk), defines

an homomorphism φI : G → Lk; in particular gφ1 ≡ gφ2 mod A for any g ∈ G andany φ1, φ2 ∈ ∆;

(3) for any finite subset I of ∆, the homomorphism φI is surjective.

This definition implies that if ∆ ∈ D, then the functions φI , where I is a finite subset of∆, are compatible surjections from G to the inverse system LI; thus the correspondinginduced mapping of profinite groups Φ : G → L∆ is onto. Moreover, kerφI ∈ Y and soker Φ =

⋂φ∈∆ kerφ is an intersection of elements of X .

We may order the elements of D by inclusion. By Zorn’s lemma, D has at least onemaximal element.

Lemma 7. If ∆ is a maximal element of D then⋂φ∈∆ kerφ = R.

Proof. For any φ ∈ ∆ let Nφ = kerφ ∈ X . Suppose that S =⋂φ∈∆Nφ = R. Then

there exists N ∈ X with S ≤ N. Moreover there is an epimorphism α : G → L withkerα = N. Fix φ ∈ ∆; the map G/(Nφ ∩ N) → L2 defined by g(Nφ ∩ N) → (gφ, gα)is injective; by Lemma 5, G/(Nφ ∩ N) ∼= L2, hence there exists β ∈ Aut(L) such thatβ−1gαβ ≡ gφ mod soc(L) for any g ∈ G. Let γ : G → L be defined by gγ = β−1gαβ.Now let ∆ = ∆ ∪ γ. We claim that ∆ ∈ D. The only thing that remains to prove is thatfor any finite subset I = φ1, . . . , φk of ∆, the homomorphism φI : G → Lk+1 defined byg → (gφ1 , . . . , gφk , gγ) is surjective. By Lemma 5 and the fact that φI is surjective, eitherφI is surjective or G/(Nφ1 ∩ · · · ∩Nφk

) ∼= G/(Nφ1 ∩ · · · ∩Nφk∩N) ∼= Lk. But in the latter

case S ≤ Nφ1 ∩ · · · ∩Nφk≤ N, a contradiction.

Let w0(G) denote the local weight of the profinite group G, i.e. the smallest cardinalityof a fundamental system of open neighborhoods of 1 in G.

Theorem 8. G/R is homeomorphic to LΩ, for a suitable choice of the set Ω. If X isinfinite, then |Ω| = |X |.Proof. By Lemma 7, G/R is homeomorphic to LΩ, where Ω is a maximal element of D.Since a base of neighborhoods of 1 in G/R is given by the subgroups N/R, for N ∈ Y,if X is infinite, then |X | = |Y| = w0(G/R). On the other hand, w0(G/R) = w0(LΩ) isthe cardinality of the set of the finite subsets of Ω, which is precisely the cardinality of Ωwhenever Ω is infinite.

In [4] it is proved that ifG is a finite group, then the cardinality of the set Ω which appearsin the previous theorem coincides with the number of non-Frattini factors G-equivalentto A in any chief series of G. We want to prove that a similar result holds for arbitraryprofinite groups.


First we recall that any profinite group G has a chain of closed normal subgroups

Gµ = 1 ≤ · · · ≤ Gλ ≤ · · · ≤ G0 = G

indexed by the ordinals λ ≤ µ such that

• Gλ/Gλ+1 is a chief factor of G, for each λ < µ;• if λ is a limit ordinal then Gλ =

⋂ν<λGν .

Such a chain will be called chief series of G. Note that |µ| is an invariant of G, since, ifG is infinite, then |µ| = w0(G).

Lemma 9. Let H/K be a chief factor of G. If R ≤ K < H ≤ I then H/K is non-Frattiniand G-equivalent to A.

Proof. Since Y induces a fundamental system of open neighborhoods of the identity inI/R, we get that K =

⋂N∈Y KN and H =

⋂N∈Y HN . Thus there exists N ∈ Y such

that KN = HN and so H/K ∼=G HN/KN . By Lemma 5 HN/KN is non-Frattini andG-equivalent to A, and thus the same holds for H/K.

Lemma 10. Let H/K be a chief factor of G. Then H/K is non-Frattini and G-equivalentto A if and only if RH/RK = 1 and RH ≤ I.

Proof. If RK = RH ≤ I, then, by Lemma 9, RH/RK is non-Frattini and G-equivalent toA; hence also H/K satisfies these properties.

Now assume that H/K is a non-Frattini chief factor G-equivalent to A. As H/K ≤Frat(G/K), there exists a closed maximal subgroup, say M, which contains K but not H.Let N be the normal core of M in G; since H and K are normal in G, from K ≤ M andH ≤M we deduce K ≤ N and H ≤ N. In particular HN/N is a minimal normal subgroupof the primitive group G/N and it is G-isomorphic to H/K, hence G-equivalent to A. Notethat, by the definition of I = IG(A), I/N is the socle of the primitive group G/N . Theneither I/N = HN/N and N ∈ X or G/N is primitive of type III and, by Lemma 3, N ∈ Y.In particular R ≤ N = NK < NH ≤ I and so RK = RH ≤ I.

Theorem 11. Let Gλλ≤µ be a chief series of G and let Θ be the set of factors Gλ/Gλ+1

which are non-Frattini and G-equivalent to A. The cardinality δG(A) of Θ does not dependon the choice of the chief series. Moreover if G/R ∼= LΩ then |Ω| = δG(A).

Proof. We obtain a chain of closed normal subgroups Hλλ≤µ with H0 = G and Hµ = Rby defining Hλ = RGλ. For any λ ≤ µ, either Hλ = Hλ+1 or Hλ/Hλ+1 is a chief factorof G/R. Moreover the set of non trivial factors Hλ/Hλ+1 coincides with the set of factorsof a chief series of G/R. By Corollary 6 either Hλ ≤ I or Hλ ≥ I. Let ν be the smallestordinal with Hν ≤ I. Now Lemma 10 implies that if λ < ν, Gλ/Gλ+1 cannot be non-Frattiniand G-equivalent to A; moreover, if λ ≥ ν, then Hλ/Hλ+1 = 1 if and only if Gλ/Gλ+1 isnon-Frattini and G-equivalent to A.

So, Hλ/Hλ+1 | Hλ/Hλ+1 = 1 is a chief series of G/R which passes through I/R andhas the property that the elements of Θ are in bijective correspondence with the non trivialfactors Hλ/Hλ+1 contained in I/R; in particular |Θ| does not depend on the choice ofthe series.

Finally, since G/R ∼= LΩ implies I/R ∼= AΩ, we conclude that |Θ| = |Ω|.


Theorem 12. If G is finitely generated then δG(A) is finite for every finite irreducibleG-group A.

Proof. If X ∈ XG(A), then by definition I/X ∼= A and so |G : X| = |G : I| · |A| = n foran integer n. As G is finitely generated, the number of subgroups of index n is finite; thus|XG(A)| is finite and consequently R has finite index in G. Therefore G/R ∼= LΩ for a finiteset Ω and the result follows from the previous theorem.

2. Factors of the Dirichlet series PG(s)

Let G be a finitely generated profinite group, that is a profinite group topologicallygenerated by a finite number of elements. Recall that for every integer n, the number ofopen subgroups of index n in G is finite. So we can define a formal Dirichlet series PG(s)as follows:

PG(s) :=∑n∈N

αnns

with αn :=∑

|G:H|=nµ(H)

where µ(H) denotes the Mobius function of the lattice of open subgroups of G, defined by∑H≤K µG(K) = 0 unless H = G, in which case the sum is 1.When G is finite, PG(s) is actually the series ProbG(s) defined in the introduction; in

particular, when t is an integer, PG(t) is the probability that t random elements generate G.Given a normal subgroup N of G we define a formal Dirichlet series PG,N (s) as follows:

PG,N (s) :=∑n∈N

bnns

with bn :=∑

|G:H|=nHN=G

µ(H).

Notice that PG(s) = PG,G(s). Moreover, N admits a proper supplement in G if and onlyif it is not contained in the Frattini subgroup of G; it follows easily that PG,N (s) = 1 if andonly if N ≤ Frat(G).

Let A(s) =∑n an/n

s and B(s) =∑n bn/n

s be two formal Dirichlet series. We denoteby A(s) ∗B(s) the convolution product of A(s) and B(s), i.e. the Dirichlet series

∑n cn/n

s

with cn =∑d|n adbn/d.

Theorem 13. If G is a finitely generated profinite group and N is a closed normal subgroupof G then PG(s) = PG/N (s) ∗ PG,N (s).

Proof. The result is already known when G is a finite group (see for example [2] section2.2), so we need an argument to reduce to the finite case. Let n ∈ N; we have to prove thatthe coefficients of 1/ns in PG(s) and PG/N (s) ∗ PG,N (s) are equal, that is:

∑|G:H|=n

µ(H) =∑d|n

∑N≤H1≤G|G:H1|=d

µ(H1)

∑H2N=G|G:H2|=n/d

µ(H2)

(2.1)

Let Xn be the intersection of the open subgroups of G with index at most n; as G is finitelygenerated, Xn has finite index in G. Thus

PG/Xn(s) = PG/NXn

(s) ∗ PG/Xn,NXn/Xn(s). (2.2)


Now (2.1) follows from (2.2) since the terms in (2.1) are equal to the coefficients of 1/ns inthe two series in (2.2); indeed if |G : H| ≤ n then Xn ≤ H and µ(H) = µ(H/Xn).

If G is a finite group, by taking a chief series

Σ : 1 = Nl < . . .N1 < N0 = G

and iterating the above formula, we obtain an expression of PG(s) as a product indexed bythe non-Frattini chief factors in the series:

PG(s) =∏

Ni/Ni+1 ≤Frat(G/Ni+1)

PG/Ni+1,Ni/Ni+1(s). (2.3)

In [4] it is proved that if G is a finite group, the factors in (2.3) are independent of thechoice of the series Σ. Moreover it is also described how the factors in (2.3) look like. Let Lbe a finite monolithic primitive group, and let A be its socle. We define

PL,1(s)=PL,A(s),

PL,i(s)=PL,A(s)− (1 + q + · · ·+ qi−2)γ|A|s , (2.4)

where γ = |CAutA(L/A)| and q = |EndLA| if A is abelian, q = 1 otherwise. In [4] it isproved that if G is a finite group, then the factors of PG(s) corresponding to the non-Frattini factors in a chief series are all of the kind PL,i(s), for suitable choices of L and i.In particular:

Theorem 14. ([4], Theorem 18) Let G be a finite group. Then

PG(s) =∏A

∏

1≤i≤δG(A)

PLA,i(s)

(2.5)

where A runs over the set of irreducible G-groups G-equivalent to a non-Frattini chief factorof G, and LA is the monolithic primitive group associated with A. Moreover, the factoriza-tion of PG(s) corresponding to the non-Frattini factors in a chief series Σ of G is precisely(2.5), independently of the choice of Σ.

We want to generalize this result to the case when G is a finitely generated profinitegroup. First of all, we get the following result:

Theorem 15. Let G be a finitely generated profinite group and let A = H/K be a non-Frattini chief factor of G. If R = RG(A), we have:

PG/K,H/K(s)=PG/RK,RH/RK(s)

= PLA,k(s)

where LA is the monolithic primitive group associated with A and k = δG/RK(A).


Proof. Let Sn = X ≤ G | K ≤ X ≤ G, XH = G, |G : X| = n and µ(X) = 0; noticethat PG/K,H/K(s) =

∑n∈N αn/n

s with αn =∑X∈Sn

µ(X). But µ(X) = 0 only if X isan intersection of closed maximal subgroups of G. Moreover in the proof of Lemma 10 wehave seen that if M is a closed maximal subgroup of G with K ≤ M and H ≤ M thenR ≤ M ; hence RK ≤ X for any X ∈ Sn. This implies immediately that PG/K,H/K(s) =PG/RK,RH/RK(s).

Now, by Theorems 12 and 11, we get that k = δG/RK(A) is finite and G/RK ∼= Lk. More-over RH/RK is a minimal normal subgroup of G/RK and is equivalent to A. Therefore,by [4] Theorem 17, we conclude that PG/RK,RH/RK(s) = PL,k(s).

Let now G be an infinite finitely generated profinite group. In that case ω0(G) = ℵ0 andG has a chief series of length ℵ0

Σ : G = G0 ≥ · · · ≥ Gi ≥ · · · ≥ Gℵ0 = 1; (2.6)

to each chief factor Gi/Gi+1 it is associated the finite Dirichlet series

Pi(s) = PG/Gi+1,Gi/Gi+1(s). (2.7)

Now, by Theorem 13, for any i ∈ N we get

PG/Gi+1(s) = P0(s) ∗ P1(s) ∗ · · · ∗ Pi(s).

So, one is tempted to say that PG(s) is the product of the infinite factors Pi(s)i∈N.Unfortunately, since the formal series PG(s) is not necessarily convergent, PG(s) is not afunction and a sentence like “PG(s) = limi→∞ P0(s) . . . Pi(s)” has no meaning if we think toa convergence of complex functions. However we can prove that the formal Dirichlet seriesPG(s) is uniquely determined as an “infinite convolution” of the factors Pi(s)i∈N and thatthe set of these factors is independent of the choice of the chief series Σ. To do that we givesome definitions.

Let P = Pω(s)ω∈Ω be a family of finite Dirichlet series, let say Pω(s) =∑αωn/n

s, withthe property that αω1 = 1, for every ω. We say that the family P is suitable for convolutionif for every m > 1 the set

Ωm = ω ∈ Ω | αωn = 0 for some 1 < n ≤ m

is finite. If P has this properties, then, for any m > 1,

P∗m(s) =∏ω∈Ωm

Pω(s)

is a well-defined and finite Dirichlet series, say P∗m(s) =∑n cn,m/n

s. Then we define the(infinite) convolution product of P to be

P∗(s) =∑n∈N

γn/ns where γ1 = 1 and γn = cn,n if n > 1.


Note that if P is suitable for convolution and ∆ ⊆ Ω, then the family Q = Pω(s)ω∈∆

is again suitable for convolution and the following holds:

Lemma 16. Let Q∗(s) =∑n δn/n

s and let m > 1. If Ωm ⊆ ∆, then γn = δn for anyn ≤ m.

For example, whenever n ≤ m, the first n terms of P∗n(s) and P∗m(s) are equal.Now let ΩG be the set of pairs (A, i) where A runs over a set of representatives for the

equivalence classes of finite irreducible G-groups and 1 ≤ i ≤ δG(A). If ω = (A, i) ∈ ΩGdefine Pω(s) = PLA,i(s) as in (2.4); finally let

PG = Pω(s)ω∈ΩG. (2.8)

Given a chief series Σ of G, let ΩΣ be the set of non-Frattini chief factors in this seriesand let

PΣ = PG/K,H/K(s)H/K∈ΩΣ.

Theorem 17. Let G be a finitely generated profinite group and let Σ be a chief series of G.There is a bijection φ : ΩΣ → ΩG such that if H/K ∈ ΩΣ then PG/K,H/K(s) = Pφ(H/K)(s).The two families PΣ and PG are suitable for convolution and PG(s) = P∗Σ(s) = P∗G(s).

Proof. Let A be a finite irreducible G-group with δG(A) = 0. By Theorem 11 there areexactly δ = δG(A) non-Frattini factors H1,A/K1,A, . . . , Hδ,A/Kδ,A in the chief series Σ withHi,A/Ki,A ∼G A, for 1 ≤ i ≤ δ. We may assume

Kδ,A < Hδ,A < · · · < Ki,A < Hi,A < · · · < K1,A < H1,A.

The map Hi,A/Ki,A → (A, i) induces a bijection φ : ΩΣ → ΩG. Moreover, by Theorem 15,PHi,A/Ki,A

(s) = PLA,i(s) = Pφ(Hi,A/Ki,A)(s). This proves the first part of the statement andthat P∗Σ(s) = P∗G(s) for every chief series Σ.

To complete the proof it now suffices to prove that there exists a chief series Σ such thatPΣ is suitable for convolution and that PG(s) = P ∗

Σ(s). For any integer n define Xn to be the

intersection of the open subgroups H of G with |G : H| ≤ n. Since G is finitely generated,Xn is an open normal subgroup of G. Moreover

⋂nXn = 1, hence we may produce a chief

series Σ by refining the series Xnn∈N.Now fix an integer m. Let H/K ∈ ΩΣ and PG/K,H/K(s) =

∑n βn/n

s. If βn = 0 forsome 1 = n ≤ m, then there exists an open subgroup Y/K of G/K with G = HY and|G : Y | = n; this implies Xn ≤ K, otherwise H ≤ Xn and, as Xn ≤ Y , we get G =HY = Y, a contradiction. Thus Xm ≤ Xn ≤ K. As G/Xm is finite, there are only finitelymany factors H/K ∈ ΩΣ with Xm ≤ K. This proves that the family PΣ is suitable forconvolution. Moreover, if Qm is the subfamily of PΣ indexed by the factors H/K ∈ ΩΣ

satisfying Xm ≤ K, then Lemma 16 gives that the coefficients bm and cm in the two series

Q∗m(s) =∑n

bnns, P ∗Σ(s) =

∑n

cnns

are equal.


On the other hand Q∗m(s) = PG/Xm(s); thus, by definition of Xm, the coefficient am of

PG(s) =∑m am/m

s is

am =∑

|G:X|=mµ(X) =

∑|G/Xm:X/Xm|=m

µ(X/Xm) = bm

and we conclude that am = cm. Since this holds for every integer m, the theorem isproved.

3. The probabilistic zeta function

Since a profinite group G has a natural compact topology, it has also a Haar measure,which is determined uniquely by the algebraic structure of G. We normalize this measureso that G has measure 1, and is thus a probability space. This allows us to define, forany positive integer t, ProbG(t) as the measure of the subset (g1, . . . , gt) ∈ Gt | g1, . . . , gttopologically generate G.

If G is finite, the function φG(t) = Prob(G)|G|t is the Eulerian function of G, which givesthe number of ordered t-uples (g1, . . . , gt) generating G. This function was introduced andstudied by P. Hall [7], who proved in particular

φG(t) =∑H≤G

µ(H)|H|t. (3.1)

This implies that if G is finite, then the Dirichlet series PG(s) defined in the previoussection is a complex function with the property that, for any integer t, PG(t) = ProbG(t) =φG(t)/|G|t; the complex function ζG(s) = 1/PG(s) is called the probabilistic zeta functionassociated to the finite group G.

In [11] Mann proposed the problem of looking for a complex function interpolating thevalues ProbG(t) | t ∈ N. Of course in order to discuss this question one must focushis attention on finitely generated profinite groups with the property that ProbG(t) > 0for some t ∈ N (otherwise the interpolating function we are looking for is just the zerofunction); the groups with this property are called positively finitely generated (PFG). Itis worth mentioning that a finitely generated profinite group is not necessarily PFG. Forexample Kantor and Lubotzky [9] proved that the free profinite group of rank d is not PFGif d ≥ 2. The conjecture proposed by Mann in [11] is the following: to each PFG group Gthere corresponds naturally a “zeta function” ζG(s) which is an analytic function definedin some right half plane of the complex numbers, such that ζG(t) = ProbG(t)−1, for all

sufficiently large integers t. For example, if Z denotes the profinite completion of a cyclicinfinite group, then

ProbZ(t) =∑n

µ(n)nt

=(∑

n

1nt

)−1

= ζ(t)−1

where ζ is the Riemann zeta function. Hence in this case ζ(s) is the function we are look-ing for.


Before discussing Mann’s conjecture, we want to prove that it is possible to decidewhether a finitely generated profinite group G is PFG from the knowledge of the family PGof finite Dirichlet series defined in (2.8). First recall:

Proposition 18 (Mann [11] Theorem 1). If G is a finitely generated profinite group andt is an integer, then ProbG(t) = infN ProbG/N (t), where N varies over all open normalsubgroups of G.

In particular, if Σ : G = G0 < · · · < Gℵ0 = 1 is a chief series of G then

ProbG(t) = infn∈N

ProbG/Gn(t) = lim

n→∞PG/Gn(t), (3.2)

since Gnn∈N is a base of neighborhoods of 1 in G, ProbG/Gn(t) ≥ ProbG/Gn+1(t) and

PG/Gn(t) = ProbG/Gn

(t). This suggests that given an integer t there may be a relationbetween ProbG(t) and the infinite product of the numbers Pω(t), where Pω(s) ∈ PG asdefined in (2.8).

Theorem 19. A finitely generated profinite group G is PFG if and only if the infinite prod-uct

∏ω∈ΩG

Pω(t) is absolutely convergent for some positive integer t; in that case ProbG(t) =∏ω∈ΩG

Pω(t) for sufficiently large positive integers.

Proof. If G is finite, the result follows from Theorem 14. So, let G be infinite and letΣ : G = G0 > . . . > Gℵ0 = 1 be a chief series of G. Let us denote the Dirichlet seriesPG/Gi+1,Gi/Gi+1(s) by Pi(s); notice that Pi(s) = 1 whenever Gi/Gi+1 is a Frattini factor.Now G is PFG if and only if for some integer t we have

0 = ProbG(t) = limn→∞PG/Gn

(t) = limn→∞P0(t) . . . Pn−1(t). (3.3)

Since 0 < Pi(t) < 1 for any i ∈ N, the condition limn→∞ P0(t) . . . Pn−1(t) = 0 is equivalentto the absolute convergence of the infinite product

∏n∈N Pn(t).

As the value of an absolutely convergent product does not change if the factors arereordered, and the Frattini factors do not influence the product, from Theorem 17 we deducethat the infinite product

∏n∈N Pn(t) is absolutely convergent if and only if

∏ω∈ΩG

Pω(t) isabsolutely convergent.

The previous theorem says that if G is PFG then the infinite product∏ω∈ΩG

Pω(t) isabsolutely convergent for any sufficiently large integer t. Unfortunately from this result noinformation can be obtained about the behaviour of the product

∏ω∈ΩG

Pω(s) when s is acomplex number. Mann [11] proved that if G is prosoluble then

∏ω∈ΩG

Pω(s) is absolutelyconvergent in some right half plane of the complex plane. We conjecture that this holds foran arbitrary PFG group G. This would give the possibility of defining the probabilistic zetafunction of G as the multiplicative inverse of the infinite product

∏ω∈ΩG

Pω(s).

4. Recognizing PFG groups

Mann and Shalev proved that PFG groups can be characterized by the behaviour of thefunction mn(G) which is defined as the number of closed maximal subgroups of G withindex n.


Theorem 20 (Mann and Shalev [12] Theorem 4). A finitely generated profinite group G isPFG if and only if G has polynomial maximal subgroup growth, i.e. there exists a constantc such that for all n, the number mn(G) is at most nc.

This criterion can be translated in term of “multiplicity” of the chief factors ofG.We needfirst some definition. For a finite primitive group L let λ(L) be the minimum of the index|L : X| where X runs over the set of core-free maximal subgroups of L. Let M be the setof closed maximal subgroups of G; as M ∈M is open, its normal core MG has finite indexin G and G/MG is a primitive group. We define

K= N G | N = MG for some M ∈M,κn(G)= |N ∈ K | λ(G/N) = n|.

It was announced by Pyber that, using the classification of the finite simple groups, thefollowing result can be proved:

Theorem 21 (Pyber). There exists a constant b such that for every finite group G andevery n ≥ 2, G has at most nb core-free maximal subgroups of index n. In fact, b = 2 will do.

Using this result we can deduce easily:

Lemma 22. κn(G) ≤ mn(G) ≤ n2∑m≤n κm(G).

Proof. The first inequality is trivial. We prove the second one. Let M ∈ Mn; sinceλ(G/MG) ≤ n, the normal subgroup MG can be chosen in at most

∑m≤n κm(G) different

ways. Given N = MG, by Theorem 21, there are at most n2 core-free maximal subgroupsof index n containing N.

Combining Theorem 20 and Lemma 22 we obtain:

Corollary 23. Let G be a finitely generated profinite group. The following are equivalent:

(1) G is PFG;(2) there exists a constant c1 such that mn(G) ≤ nc1 for all n ∈ N;(3) there exists a constant c2 such that κn(G) ≤ nc2 for all n ∈ N.

Now we study κn(G) using the G-equivalent relation among finite irreducible G-groupsdescribed in the first section. In the following G will be a finitely generated profinite group.Let N be an element of K; the quotient group G/N is a finite primitive group and itsminimal normal subgroups are all equivalent to the same irreducible G-group, say A; indeedeither G/N is monolithic or G/N is primitive of type III and, by Lemma 3, its two minimalnormal subgroups are G-equivalent. By definition IG(A)/N is the socle of G/N and so eitherN ∈ XG(A) or N is the intersection of two different elements of XG(A); anyway, RG(A) ≤ Nand G/N ∼= Li where L = LG(A) is the monolithic primitive group associated with A andi = 1, 2. When G/N ∼= L2, A is nonabelian and any faithful primitive representation ofG/N has degree |A|.

Given an irreducible G-group A, we define KA as the subset of K containing those normalsubgroups N with the property that the minimal normal subgroups of G/N are equivalentto A. It is clear that

Lemma 24. The set K is the disjoint union of the subsets KA, where A runs over the setof finite irreducible G-groups, up to equivalence.


Now define λ(A) = λ(L), where L = LG(A). Notice that when A is abelian λ(A) = |A|and λ(G/N) = |A| for every N ∈ XG(A), since in this case G/N ∼= L (see Lemma 5);therefore, for n = |A|, Kn ⊇ KA = XG(A). When A is nonabelian we get KA = K1

A ∪ K2A

with K1A = N ∈ KA | G/N ∼= L and K2

A = N ∈ KA | G/N ∼= L2. Notice that K1A =

XG(A) and so λ(G/N) = λ(A) for every N ∈ K1A; also, K2

A is the set of the intersectionsof two different normal subgroups of XG(A) and λ(G/N) = |A| for N ∈ K2

A. Set γA =|CAutA(L/A)| and qA = |EndLA| if A is abelian, qA = 1 otherwise. Since, by Theorems 11and 12, IG(A)/RG(A) ∼= AδG(A) and δG(A) is finite, it can be easily proved that

Lemma 25. Let n = λ(A). If A is abelian, then |Kn ∩KA| = 1 + qA + · · ·+ qδG(A)−1A ; if A

is non abelian, then |Kn ∩ K1A| = δG(A) and |Kn ∩ K2

A| = δG(A)(δG(A)− 1)/2.

Now define:

κabn =∑

A′=1,|A|=n1 + qA + · · ·+ q

δG(A)−1A ;

κ1n=

∑A′=A,λ(A)=n

δG(A);

κ2n=

∑A′=A,|A|=n

(δG(A)

2

).

By Lemma 25 κn(G) = κabn + κ1n + κ2

n. So G is PFG if and only if κabn , κ1n, κ

2n are

polynomially bounded. Now let αn(G) be the number of finite abelian irreducible G-groupsA, with |A| = n and δG(A) > 0.

Lemma 26. κabn is polynomially bounded if and only if αn(G) is polynomially bounded.

Proof. Obviously αn(G) ≤ κabn . We have to prove that if αn(G) is polynomially boundedthen the same is true for κabn . Assume that G can be generated by r elements and let Abe a finite abelian irreducible G-group with |A| = n and δG(A) = 0. By [4] Theorem 18,PL,δG(A)(r) > 0, where L = LG(A). On the other hand

PL,δG(A)(r) = PL,A(r)− (1 + qA + · · ·+ qδG(A)−2A )γA

|A|r .

In particular

(1 + qA + · · ·+ qδG(A)−2A )γA

|A|r < PL,A(r) ≤ 1,

hence

1 + qA + · · ·+ qδG(A)−2A ≤ |A|

r

γA≤ |A|r = nr

and

1 + qA + · · ·+ qδG(A)−1A ≤ (1 + qA)(1 + qA + · · ·+ q

δG(A)−2A ) ≤ (1 + qA)nr = 2nr+1

since qA ≤ n. It follows that κabn ≤ 2αn(G)nr+1.


Now consider the number ξn(G) of finite nonabelian irreducible G-groups A, with |A| = nand δG(A) ≥ 2.

Lemma 27. κ2n(G) is polynomially bounded if and only if ξn(G) is polynomially bounded.

Proof. Obviously ξn(G) ≤ κ2n(G). We have to prove that if ξn(G) is polynomially bounded

then the same is true for κ2n(G). Assume that G can be generated by r elements and let A

be a finite nonabelian irreducible G-group with |A| = n and δG(A) > 1. By [4] Theorem 18,PL,δG(A)(r) > 0, where L = LG(A). On the other hand

PL,δG(A)(r) = PL,A(r)− (δG(A)− 1)γA|A|r .

In particular

(δG(A)− 1)γA|A|r < PL,A(r) ≤ 1,

hence

δG(A)− 1 ≤ |A|r

γA≤ |A|r = nr.

It follows that κ2n(G) ≤ ξn(G)(n+ 1)2r.

Lemma 28. If κ1n(G) is polynomially bounded then ξn(G) and κ2

n(G) are polynomiallybounded.

Proof. If A is nonabelian and |A| = n, then λ(A) ≤ n; so ξn(G) ≤∑m≤n κ1m(G) ≤ nκ1

n(G).

So we conclude:

Theorem 29. G is PFG if and only if αn(G) and κ1n(G) are polynomially bounded.

We conclude with two question:

Question 1. Does there exist a finitely generated profinite group G such that αn(G) is notpolynomially bounded?

Question 2. Let βn(G) be the number of nonabelian irreducible G-groups A, with λ(A) = nand δG(A) = 0. Does there exist a finitely generated profinite group G such that βn(G) isnot polynomially bounded?

We conjecture that both these questions have a negative answer. This would imply:Conjecture.A finitely generated profinite group G is PFG if and only if

ρG(n) = maxδG(A) | A nonabelian, λ(A) = n (4.1)

is polynomially bounded.


References

[1] N. Boston, ‘A probabilistic generalization of the Riemann zeta functions’, Analytic NumberTheory 1 (1996), 155–162.

[2] K. S. Brown, ‘The coset poset and probabilistic zeta function of a finite group’, J. Algebra 225(2000), 989–1012.

[3] F. Dalla Volta and A. Lucchini, ‘Finite groups that need more generators than any properquotient’, J. Austral. Math. Soc., Ser. A 64 (1998), 82–91.

[4] E. Detomi and A. Lucchini, ‘Crowns and factorization of the probabilistic zeta function of afinite group’, J. Algebra to appear.

[5] P. Forster, ‘Chief factors, crowns, and the generalized Jordan-Holder Theorem’, Comm. Alge-bra 16 (1988), 1627–1638.

[6] W. Gaschutz, ‘Praefrattinigruppen’, Arch. Math. 13 (1962), 418–426.[7] P. Hall, ‘The Eulerian functions of a group’, Quart. J. Math. 7 (1936), 134–151.[8] P. Jimenez-Seral and J. Lafuente, ‘On complemented nonabelian chief factors of a finite group’,

Israel J. Math. 106 (1998), 177–188.[9] W. M. Kantor and A. Lubotzky, ‘The probability of generating a finite classical group’, Geom.

Ded. 36 (1990), 67–87.[10] J. Lafuente, ‘Crowns and centralizers of chief factors of finite groups’, Comm. Algebra 13

(1985), 657–668.[11] A. Mann, ‘Positively finitely generated groups’, Forum Math. 8 No. 4 (1996), 429–459.[12] A. Mann and A. Shalev, ‘Simple groups, maximal subgroups and probabilistic aspects of

profinite groups’, Israel J. Math. 96 (1996), 449–46 8.

A. LucchiniDipartimento di MatematicaUniversita di BresciaVia Valotti, 925133 Brescia, [email protected]

E. DetomiDipartimento di MatematicaUniversita di BresciaVia Valotti 925133 Brescia, [email protected]

Current address:Dipartimento di MatematicaUniversita di Padovavia Belzoni, 735131 Padova, [email protected]

THE GALOIS STRUCTURE OF AMBIGUOUS IDEALS IN CYCLICEXTENSIONS OF DEGREE 8

G. GRIFFITH ELDER

Abstract. In cyclic, degree 8 extensions of algebraic number fields N/K, ambiguousideals in N are canonical Z[C8]-modules. Their Z[C8]-structure is determined here.It is described in terms of indecomposable modules and determined by ramificationinvariants. Although infinitely many indecomposable Z[C8]-modules are available (clas-sification by Yakovlev), only 23 appear.

1. Introduction

We are concerned with the interrelationship between two basic objects in algebraicnumber theory: the ring of integers and the Galois group. In particular, we seek to under-stand the effect of the Galois group upon the ring of integers. At the same time, we are alsointerested in the Galois action upon other fractional ideals. So that the action may be simi-lar, we restrict ourselves to ambiguous ideals – those that are mapped to themselves by theGalois group. The setting for our investigation is the family of C8-extensions. This choiceis guided by by a result of E. Noether as well as results in Integral Representation Theory.Noether’s Normal Integral Basis Theorem. A finite Galois extension of number fieldsN/K is said to be at most tamely ramified (TAME) if the factorization of each prime idealPK (of OK) in ON results in exponents (degrees of ramification) that are relatively primeto the ideal PK . A normal integral basis (NIB) is said to exist if there is an element α ∈ ON

(in the ring of integers of N) whose conjugates, σα : σ ∈ Gal(N/K), provide a basis forON over OK (the integers in K).

Noether proved NIB ⇒ TAME; moreover, for local number fields NIB ⇔ TAME, tyingthe Galois module structure of the ring of integers to the arithmetic of the extension [?].This is a nice effect – NIB means that the integers are isomorphic to the group ring,OK [Gal(N/K)]. It is similar to the effect of the Galois group on the field itself (i.e. NormalBasis Theorem). The impact of her result is two-fold: (1) We are encouraged to localize.(2) We are directed away from tamely ramified extensions – toward wildly ramified exten-sions and p-groups (See [?]).Integral Representation Theory (Restricted to p-groups G).Classification of Modules. The number of indecomposable modules over a group ring Z[G]is, in general, infinite. Only Z[Cp] and Z[Cp2 ] are of finite type. Still, among those of infinitetype, there are two whose classifications are somehow manageable. These are the ones of so–called tame type [?]: Z[C2×C2] (classification by L. A. Nazarova [?]) and Z[C8] (classificationby A. V. Yakovlev [?]).Unique Decomposition. The Krull–Schmidt–Azumaya Theorem does not, in general, hold:although a module over a group ring will decompose into indecomposable modules, this

Date: October 6, 2002.

2000 Mathematics Subject Classification. Primary 11S23; Secondary 20C10.

Key words and phrases. Galois Module Structure, Wild Ramification, Integral Representation.

64 G. GRIFFITH ELDER

decomposition may not be unique. Fortunately, it does hold for a few group rings, includingZ[C2 × C2] and Z[C8] [?].Topic. Let G = Gal(N/K). We are led to ask the following natural question: What is theZ[G]-module structure of ambiguous ideals when

• the number theory is ‘bad’ (wild ramification), while• the representation theory is ‘good’ (tame type, K–S–A)?

In other words: What is the Z[G]-module structure of ambiguous ideals in wildly rami-fied C2 × C2 and C8 number field extensions? Previous work solved this for C2 × C2-extensions [?], [?]. So our focus here is on C8-extensions. (Note: This question has alreadybeen addressed for those group rings with ‘very good’ representation theory, those of finitetype. See [?] and [?].)

As with C2 × C2-extensions, the Z[G]-module structure of ambiguous ideals in C8-extensions is completely determined by the structure at its 2-adic completion – our globalquestion reduces to a collection of local ones. We leave it to the reader to fill in the details.(One may follow [?, §2] using [?].)

1.1. Local Question, Answer. Let K0 be a finite extension of the 2-adic numbers Q2 andlet Kn be a wildly ramified, cyclic, degree 2n extension of K0 with G = Gal(Kn/K0). Themaximal ideal Pn in Kn is unique (therefore ambiguous). So every fractional ideal Pi

n isambiguous. We ask: What is the Z2[G]-module structure of Pi

n for n = 1, 2, 3? (Z2 denotesthe 2-adic integers.) The answered is given by the following theorem and the description ofthe modules Ms(i, b1, . . . , bs).

Let T denote the maximal unramified extension of Q2 in K0. Following [?, Ch IV], letG = G−1 ⊇ G0 ⊇ G1 ⊇ · · · denote the ramification filtration. Use subscripts to denote fieldof reference, so Ok denotes the ring of integers of k.

Theorem 1.1. Let Kn/K0 be a cyclic extension of degree 2n and let k ⊆ K0 be an unram-ified extension of Q2. Suppose that |G1| ≤ 8 (i.e. s = 1, 2 or 3) and let b1, . . . , bs be thebreak numbers in the ramification filtration of G1. If Ms(i, b1, . . . , bs) is the Z2[G1]-moduledefined below, then

Pin

∼= Ok[G] ⊗Z2[G1] Ms(i, b1, . . . , bs)[T :k] as left Ok[G]-modules.

1.1.1. Ms(i, b1, · · · , bs). Indecomposable modules are listed in Appendix A and e0 denotesthe absolute ramification index of K0. Following [?] and [?],

M1(i, b1) = (R0 ⊕ R1)(i+b1)/2−i/2 ⊕ Z2[G1]e0−((i+b1)/2−i/2) (1.1)

M2(i, b1, b2) = Ia ⊕

HbA ⊕ GcA ⊕ LdA if b2 + 2b1 > 4e0,HbB ⊕ McB ⊕ LdB if b2 + 2b1 < 4e0.

(1.2)

where a = (i+b2)/4−(i+2b1)/4, bA = e0+i/4−(i+b2)/4, bB = (i+b2+2b1)/4−(i + b2)/4, cA = (i + b2 + 2b1)/4 − e0 − i/4, cB = e0 + i/4 − (i + b2 + 2b1)/4,dA = e0 + (i+ 2b1)/4 − (i+ b2 + 2b1)/4, dB = (i+ 2b1)/4 − i/4.

The description of M3(i, b1, b2, b3) is given by Tables 1 and 2. Note the eight columnsin each table. There are eight cases. Each module that appears in M3(i, b1, b2, b3), exceptfor R3, is listed in the appropriate column of Table 1. The multiplicity of the module isappears in the corresponding spot in Table 2. The multiplicity of R3 follows the tables.

GALOIS STRUCTURE 65

Tab

le1.

AB

CD

EF

GH

HH

HH

I 1I 1

I 1I 1

H1L

H1L

H1L

H1L

H1G

H1G

H2

H2

H1

H1

H1

H1

H1

H1

MH

1,2

H1,2

H1,2

GG

GG

M1

M1

G 4G 4

G 4G 4

G 4G 4

LL

LL

G 3G 3

G 3G 3

L 3L 3

L 3L 3

L 3L 3

G 2G 2

II

IL 2

IL 2

L 2G 1

I 2I 2

I 2I 2

I 2I 2

L 1L 1

Tab

le2. A

BC

DE

FG

Hd

−b

d−b

a−b−e 0

a−b−e 0

b+e 0

−a

b+e 0

−a

b−c

b−c

w+e 0

−a

y+m

−a

w−b

y+m

−e 0

−b

d−y

+e 0

c−y

d+e 0

−d

a−d

−e 0

d−w

d−d

−m

a−w

a+e 0

−c−m

a−b

a−b

a−d

−2e

0d

+2e

0−a

a−d

a−d

d−a

d−a

d−a

d−a

a+e 0

−a

a−d

−e 0

d+e 0

−a

w−m

−a

c−d

y−d

c−d

c−d

c−a

c−a

c−d

−e 0

c−d

−e 0

d+e 0

−c

d+e 0

−c

b−c

b−c

c+e 0

−c

c+e 0

−c

z+b 1

−c

z+b 1

−c

y−d

y+e 0

−d

d+e 0

−b

a−b

b−c−e 0

b−c−e 0

b−c−e 0

c+e 0

−b

b−c−e 0

c+e 0

−b

d+e 0

−b

d+e 0

−a

b−b+e 0

b−b+e 0

b−b+e 0

b−c

a−b

a−c−e 0

c+e 0

−a

c−d


Cases.A. 4e0 − 4b1/3 < b2 (including Stable Ramification, b1 ≥ e0).B. 4e0 − 2b1 < b2 < 4e0 − 4b1/3C. 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 > (4e0 + 4b1)/3D. 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 < (4e0 + 4b1)/3E. b2 < 4e0 − 4b1 and b3 > 8e0 + 4b1 − 2b2F . b2 < 4e0 − 4b1 and 8e0 + 4b1 − 2b2 < b3 < 8e0 + 4b1 − 2b2G. 8e0 − 4b1 − 2b2 < b3 < 8e0 − 2b2H. b3 < 8e0 − 4b1 − 2b2

A graphic representation of these cases appears in §3.2.Constants used in Table 2. a := (i−2b2)/8, a := (i+b3 −2b2)/8, b := (i−2b2 −4b1)/8,b := (i+b3−2b2−4b1)/8, c := (i−4b2)/8, c := (i+b3−4b2)/8, d := (i−4b2−4b1)/8,d := (i + b3 − 4b2 − 4b1)/8, w := (i − 2b2 − 2b1)/8, w := (i + b3 − 2b2 − 2b1)/8,y := (i − 4b2 − 2b1)/8, y := (i + b3 − 4b2 − 2b1)/8, z := (i + b3 − 4b2 − 6b1)/8,m := (b2 − b1)/2.The multiplicity of R3. In Cases A and B it is ((a + b + c + d) − (a + b + c + d) − 3e0)f .In Case C, it is ((a + b + c + d) − (a + b + d) − 2e0 − (z + b1))f . While in Case D it is((a+ b+ c+ d) − (a+ b) − e0 +m− (w + z + b1))f . In Case E, it is (b+ d− a− y − e0)f .In Case F it is ((b+ d+ c) − (a+ y + y) − e0)f . Finally, in Cases G and H, the number ofR3 that appear is (d− a)f .

1.2. Discussion.Cyclic p-Extensions. The Galois module structure of the ring of integers in fully andwildly ramified, cyclic, local extensions of degree pn was studied in [?] and more recentlyin [?]. Both of these papers required a lower bound on the first ramification number b1.In particular, [?] restricted b1 to about half of its possible values, under so-called strongramification. In this paper, by focusing on p = 2 we are able to remove this restriction.Our work sheds light (1) on strong ramification and (2) on the structures that are possibleoutside of it.

(1) Strong ramification for p = 2 means b1 > e0, a small part of Case A. The structureunder strong ramification given by [?, Thm 5.3], when restricted to p = 2, remains validthroughout Case A. What then should Case A be, for odd p?

(2) Suppose that ‘nice’ refers to the structure under strong ramification, indeed underCase A. Does the structure remain relatively ‘nice’ beyond Case A? This depends upona precise definition. Let an indecomposable module be nice if it is made up of distinctirreducible modules. Note only nice modules appear in Case A. But then, as we leave CaseA, the structure turns nasty immediately. At least one of H1,2, H1L and H1G appears inevery Case B through F .Induced Structure. The subfield of Kn fixed by the first ramification group G1 is tameover the base field K0. Miyata generalized Noether’s Theorem proving that each ideal isrelatively projective over G1 [?]. In other words, the ideals are direct summands of modulesthat have been induced from G1 to G [?, §10]. We find, in our situation, that ideals arerelatively free over G1. See [?, Thm 2] for a more general, related result.Extension of Ground Ring. When studying the structure of ideals in an extensionKn/K0 over a group ring, one must choose a ring of coefficients. Does one study ‘fine’structure – over O0[G] where the coefficients are integers in K0. Does one study ‘coarse’structure – over Z2[G]. We study a canonical intermediate structure – over OT [G] where thecoefficients belong to the Witt ring of the residue class field. We determine this structure

GALOIS STRUCTURE 67

by listing generators and relations. Interestingly, the coefficients in these relations alwaysbelong Z2[G]. Therefore OT [G]-structure results, by extension of the ground ring, fromZ2[G]-structure [?, §30B].Realizable Modules. Let SG denote the set of realizable indecomposable Z2[G]-modules:Those indecomposable Z2[G]-modules that appear in the decomposition of some ambiguousideal in an extension N/K with Gal(N/K) ∼= G. Chinburg asked whether SG could beinfinite. In [?], since SC2×C2 is infinite, the answer was found to be yes. We determinehere that although the set of indecomposable Z2[C8]-modules is infinite, SC8 is finite. Thesequence |SC2n |, n = 0, 1, 2, . . . begins

1, 3, 7, 23 . . .

1.3. Organization of Paper. Preliminary results are presented in §2, main results in§3. There are two appendixes. Appendix A lists all necessary indecomposable modules.Appendix B lists bases for our ideals.Preliminary Results: In §2.1 we handle the special case when a ramification break number iseven. In §2.2, we present a strategy for handling odd ramification numbers. To motivate ourwork in §3, we implement this strategy for |G1| = 2 and 4, in §2.2.1 and §2.2.3 respectively.We conclude, in §2.3, with a reduction to totally ramified extensions.Main Results: We begin in §3.1 with a brief outline and discussion. Then, we catalog rami-fication numbers and prove some technical lemmas in §3.2. All this sets the stage for ourwork in §3.3 determining the Galois structure of ideals in fully, though unstably, ramifiedC8-extensions. This is our primary focus. Our work in §3.4 on stably ramified extensions isessentially contained in [?].

2. Preliminary Results

We continue to use the notation of §1.1. Let K0 be a finite extension of Q2 and Kn/K0

be a cyclic extension of degree 2n. Let σ generate G = Gal(Kn/K0) and use subscripts todistinguish among subfields. So Ki denotes the fixed field of 〈σ2i〉, Oi denotes the ring ofintegers of Ki and Pi denotes the maximal ideal of Oi. Let vi be the additive valuation inKi, πi its prime element, so that vi(πi) = 1. Let Tri,j denote the trace from Ki down toKj . Recall the ramification filtration of G. Note G−1 = G0 if and only if Kn/K0 is fullyramified. Also since G is a 2-group and [G1 : G0] is odd, G0 = G1. Furthermore since G iscyclic and Gi/Gi−1 is elementary abelian for i > 1, there are s = log2 |G1| breaks in thefiltration of G1 [?, p 67]. Let b1 < b2 < · · · < bs denote these break numbers. (The breaknumbers of G may include −1 as well.) It is a standard exercise to show that b1, . . . , bs areall either odd or even [?, Ex 3, p 71]. When they are even, we are in an extreme case, calledmaximal ramification. The general case, when they are odd, will be our primary concern.

2.1. Even Ramification Numbers. If b1, . . . , bs are even, we use idempotent elementsof the group algebra, Q2[G], and Ullom’s generalization of Noether’s result [?, Thm 1], todetermine the structure of each ideal. In doing so, we rely upon two observations: (1) Idem-potent elements in Q2[G] that map an ideal into itself, decompose the ideal. (2) Modulesover a principal ideal domain are free.

We illustrate this process in one case, leaving other cases to the reader. Suppose |G| = 8and |G1| = 4. So K3/K0 is only partially ramified and s = 2. From [?, IV §2 Ex 3], b1 = 2e0and b2 = 4e0. Using [?, V §3], one finds that (1/2)(σ4 + 1)Pi

3 ⊆ Pi3. As a result, the

idempotent (σ4 + 1)/2 decomposes the ideal Pi3

∼= Pi/22 ⊕ M2 with (σ4 + 1)M2 = 0.


Meanwhile (1/2)(σ2 + 1)Pi/22 ⊆ P

i/22 . So P

i/22 decomposes as well. This yields Pi

3∼=

Pi/41 ⊕M1 ⊕M2 with (σ4 + 1)M2 = 0 and (σ2 + 1)M1 = 0. Each Mi may be viewed as a

module over OTK[σ]/(σ2i

+1), a principal ideal domain. So Mi is free over OTK[σ]/(σ2i

+1).Ullom’s result provides a normal integral basis for P

i/41 . Counting OT -ranks, we find that

Pi3

∼=OT [σ]

(σ2 − 1)

e0

⊕ OT [σ](σ2 + 1)

e0

⊕ OT [σ](σ4 + 1)

e0

.

2.2. Odd Ramification Numbers. Henceforth the ramification numbers will be odd. Inthis context we will use the following technical result (with Ki/Ki−1).

Lemma 2.1. Let k be a finite extension of Q2 and K/k be a ramified quadratic extension.Let ek be the absolute ramification index of k. Assume that σ generates the Galois groupand that the ramification number, b < 2ek, is odd. Then

(1) vK((σ ± 1)α) = vK(α) + b for vK(α) odd;(2) if τ ∈ k, there is a ρ ∈ K such that (σ + 1)ρ = τ and vK(ρ) = vK(τ) − b;(3) if vK(α) is even and (σ + 1)α = 0, there is a θ ∈ K such that α = (σ − 1)θ and

vK(θ) = vK(α) − b.

Proof. These may be shown using [?, V §3], as in [?, Lem 3.12–14].

Our strategy is based upon the following observations:(1) Under wild ramification, Galois action ‘shifts/increases’ valuation (Lemma 2.1(1)). Soan element may be used to ‘construct’ other elements with distinct valuation.(2) Elements with distinct valuation may be used to construct bases. If the valuation mapvn : Kn → Z is one–to–one on a subsetA ⊆ Kn, while vn(A) is onto i, i+1, . . . , i+vn(2)−1;then A is a basis for Pi

n over the integers in the maximal unramified subfield of Kn. IfKn/K0 is fully ramified, this subfield is T .The strategy is illustrated below. It is: Use Galois Action to Create Bases.

2.2.1. First Ramification Group of Order Two. Suppose that |G1| = 2. To use Observation(1), we pick α ∈ Kn an element with vn(α) = b1 (e.g. α = πb1n ). Let αm := α · πm0 . Sincevn(π0) = 2, vn(αm) = b1 + 2m. Use Lemma 2.1 with Kn/Kn−1. So vn((σ2n−1

+ 1)αm) =2b1 +2m. Since b1 is odd, the valuations of αm and (σ2n−1

+1)αm have opposite parity. Thevaluations for all m lie in one–to–one correspondence with Z. Select those with valuationin i, . . . , i+ vn(2) − 1. Replace πe00 by 2 whenever possible. The result is

B := αm, (σ2n−1+ 1)αm : (i− b1)/2 ≤ m ≤ e0 + i/2 − b1 − 1

∪

(σ2n−1

+ 1)αm, 2αm : i/2 − b1 ≤ m ≤ (i− b1)/2 − 1. (2.1)

Since Kn−1/K0 is unramified, there is a root of unity ζ with Kn−1 = K0(ζ). The maximalunramified extension Q2 in Kn is T (ζ). By Observation (2), B is a basis for Pi

n over OT (ζ).Note that OT (ζ) · αm + OT (ζ) · (σ2n−1

+ 1)αm = OT (ζ) · αm + OT (ζ) · σαm yields the groupring, OT (ζ)[G1], while OT (ζ) · (σ2n−1

+1)αm+OT (ζ) ·2αm = OT (ζ) · (σ2n−1+1)αm+OT (ζ) ·

(σ2n−1 − 1)αm yields the maximal order of OT (ζ)[G1]. Restricting coefficients and countingleads to the Ok[G1]-module structure of Pi

n, and to M1(i, b1) as in (1.1).

GALOIS STRUCTURE 69

Next, we extend B to a basis upon which the action of the whole group can be followed.Since Kn−1/K0 is unramified, there is a normal field basis for Pj

n−1/Pj+1n−1 over O0/P0

(for each j). Of course, [O0/P0 : OT /PT ] = 1. So Pjn−1/P

j+1n−1 has a normal field basis

over OT /PT . For j = b1, this means that there is an element µ ∈ Pb1n−1 and basis

µ, σµ, . . . , σ2n−1−1µ. Using Lemma 2.1(2), there is an α ∈ Kn with vn(α) = b1 such that(σ2n−1

+ 1)α = µ. Then α, σα, . . . , σ2n−1−1α is a normal field basis for Pb1n /P

b1+1n over

OT /PT . Since σj(σ2n−1+ 1)α : j = 0, . . . , 2n−1 − 1 is a basis for Pb1

n−1/Pb1+1n−1 , it is

also a basis for P2b1n /P2b1+1

n over OT /PT . This together with the fact that σjα : j =0, . . . , 2n−1 − 1 is a basis for Pb1

n /Pb1+1n over OT /PT leads to ∪2n−1−1

j=0 σjB being a basisfor Pi

n over OT , and Pin

∼= OT [G] ⊗Z2[G1] M1(i, b1) as OT [G]-modules.

2.2.2. An Application of Nakayama’s Lemma. In the previous section we were able to followthe Galois action from one basis element to another explicitly. This level of detail becomesoverwhelming as we generalize to |G1| = 4, 8. Fortunately, Nakayama’s Lemma allows us topush some of these details into the background.

Lemma 2.2. Let A be a Ok[C2n ]-module (torsion-free over Ok) where C2n = 〈σ〉and k denotes an unramified extension of Q2. Let H denote the subgroup of order 2,AH the submodule fixed by H, and TrHA the image under the trace. Then TrHA/((σ − 1)TrHA + 2AH

)is free over Ok/2Ok. Suppose that B ⊆ A such that TrHB is a

basis for TrHA/((σ − 1)TrHA + 2AH

)then B can be extended to a Ok[C2n ]/〈TrH〉-basis

of A/AH .

Proof. Since A/AH is a module over the principal ideal domain Ok[C2n ]/〈TrH〉, it isfree. So C := A/AH ∼= (Ok[C2n ]/〈TrH〉)a for some exponent a. Now Ok[C2n ]/〈TrH〉 isa local ring with maximal ideal 〈σ − 1〉 dividing 2. Therefore by Nakayama’s Lemmaany collection of elements in A that serves as a Ok/2Ok-basis for C/(σ − 1)C will serveas an Ok[C2n ]/〈TrH〉-basis for C. This leaves us to show that B can be extended to aOk/2Ok-basis for the vector space C/(σ − 1)C = A/(AH + (σ − 1)A). But since TrHB isa basis for TrHA/

((σ − 1)TrHA + 2AH

), the elements of B are linearly independent in

A/(AH + (σ − 1)A) and therefore span a subspace.

2.2.3. First Ramification Group of Order Four. Let |G1| = 4. This case is important becauseit illustrates the utility of Lemma 2.2. (Recall that §2.2.1 and §2.2.3 are included in thispaper to motivate considerations in §3.)

Step 1: Collect |G1| elements whose valuations are a complete set of residues modulo |G1|.We begin with the elements used to determine the structure of ideals in Kn−1 (from §2.2.1),namely αm and (σ+1)αm ∈ Kn−1 (replacing n by n− 1, expressing σ2n−2

as σ). Note thatthe first ramification number of Kn/Kn−2 is the (only) ramification number of Kn−1/Kn−2

(use [?, pg 64 Cor] or switch to upper ramification numbers [?, IV §3]). So vn (αm) =2vn−1 (αm) = 2b1 + 4m and vn((σ + 1)αm) = 4b1 + 4m. We have two elements of evenvaluation. To get elements with odd valuation, we apply Lemma 2.1(2). For each X ∈ Kn−1,Lemma 2.1(2) gives us a preimage X ∈ Kn (under the trace Trn,n−1), a preimage thatsatisfies vn(X) = 2vn−1(X) − b2. So Trn,n−1X = (σ2 + 1)X = X. The integers vn (αm),vn((σ+ 1)αm), vn(αm) = 2b1 − b2 + 4m, vn((σ + 1)αm) = 4b1 − b2 + 4m are a complete setof residues modulo 4.

Step 2: Collect elements with valuation in i, i+1, . . . , i+vn(2)−1. To organize this processwe use Wyman’s catalog of ramification numbers [?]. If b1 ≥ e0, the second ramification


number is uniquely determined, b2 = b1 +2e0. If b1 < e0, then either b2 = 3b1, b2 = 4e0 −b1,or b2 = b1 + 4t for some t with b1 < 2t < 2e0 − b1 [?, Thm 32]. In any case, we havethe bound,

2b1 < b2. (2.2)

Now for a given m, list the infinitely many elements, αm+ke0 , (σ + 1)αm+ke0 , αm+ke0 ,(σ + 1)αm+ke0 , in terms of increasing valuation. Replace αm+ke0 by 2kαm and drop thesubscripts m. So for b2 > 4e0 − 2b1, beginning at α, we have:

· · · −→ α1−→ 1/2(σ + 1)α 2−→ (σ + 1)α 3−→ α

2−→ 2α −→ · · ·

Each increase in valuation, denoted by x−→, is justified as follows: For x = 1, the justificationdepends upon the case either b2 > 4e0 − 2b1 or b2 < 4e0 − 2b1. For x = 2, it is b2 < 4e0. Forx = 3, it is (2.2). If b2 < 4e0 − 2b1, the list is as follows:

· · · −→ α4−→ (σ + 1)α 3−→ α

4−→ (σ + 1)α 1−→ 2α −→ · · ·

Note x = 4 is justified by b1 > 0.Now collect those elements with valuation in i, . . . , i+ vn(2) − 1. This will provide us

with an OT (ζ)-basis for Pin. Begin with the smallest m such that i ≤ vn(αm). Note then

that vn(2(σ + 1)αm) < i + vn(2). Associated with this particular m are four elements ini, . . . , i + vn(2) − 1. They are listed in the first row of the table below. Consider thisinterval to be a ‘window’. As we increase m, new elements appear (e.g. 2X) – appearancecoincides with disappearance (namely of X). Four elements are in ‘view’ always. There arefour ‘views’ (four sets). We list the ‘views’ as rows under the appropriate heading.

D: The OT (ζ)-basis for Pin.

A : b2 < 4e0 − 2b1 B : b2 > 4e0 − 2b1

(1) α (σ + 1)α 2α 2(σ + 1)α α 2α (σ + 1)α 2(σ + 1)α

(2) (σ + 1)α α (σ + 1)α 2α (σ + 1)α α 2α (σ + 1)α

(3) α (σ + 1)α α (σ + 1)α 1/2(σ + 1)α (σ + 1)α α 2α

(4) 1/2(σ + 1)α α (σ + 1)α α α 1/2(σ + 1)α (σ + 1)α α

Should we need to determine the subscripts (associated with a particular ‘view’), we caneasily do so: For example the four elements listed in A(1) and B(1), appear for m withi ≤ vn(αm) and vn(2(σ + 1)αm) ≤ i + 4e0 − 1. In other words, (i − 2b1)/4 ≤ m ≤(i+ b2)/4 − b1 − 1.

Step 3: Identify a basis for the quotient module Pin/P

i/2n−1 , and determine the precise image

of each basis element under the trace Trn,n−1 (in terms of the basis for Pi/2n−1 ). Observe

that Pin/P

i/2n−1 is, in a natural way, free over the principal ideal domain OT (ζ)[G]/〈σ2 +1〉.

We begin by identifying those elements listed in D, the OT (ζ)-basis from Step 2, that canserve as a OT (ζ)[G]/〈σ2 + 1〉-basis. Take D and partition it into two sets. Let D containthose elements X with a bar. Let D0 contain those elements X without a bar. So D is an

GALOIS STRUCTURE 71

OT (ζ)-basis for Pin/P

i/2n−1 , and D0 is an OT (ζ)-basis for P

i/2n−1 . If we knew which elements

from D provide us with OT (ζ)[G]/〈σ2 + 1〉-basis for Pin/P

i/2n−1 we would be done, as it

is easy to express the image (under the trace Trn,n−1) of each element in D in terms ofelements of D0 (there is a one–to–one correspondence).

Before we proceed further, note the following. We may assume without loss of generalitythat for X ∈ D, Trn,n−1X = 0 if and only if X appears together with X (for thesame subscript m) in D. Clearly if X and X appear together, then Trn,n−1X = X = 0.However when 2X and X appear together, after a change of basis, we may assume thatTrn,n−12X = 0. The reason for this is as follows: We can change an element of D by addingan element from D0 and still have a OT (ζ)-basis for Pi

n/Pi/2n−1 . So whenever 2X and X

appear together, replace 2X with 2X −X. Note Trn,n−1(2X −X) = 0. If we perform thischange throughout our basis, but relabel 2X −X as 2X, then we may continue to use thelists, A(1)–A(4) and B(1)–B(4), but assume that Trn,n−12X = 0 if 2X appears togetherwith X.

Our next step will be to provide an OT (ζ)[G]/〈σ2+1〉-basis for Pin/P

i/2n−1 . Consider those

rows with an X such that Trn,n−1X = 0 (namely A(2), A(3), A(4), B(2), B(4)). Let S ⊆ Ddenote the set of left–most X associated with those rows. So, for example, if b2 +2b1 < 4e0,then S is made up of the (σ + 1)αm from A(2), and the αm from A(3) and A(4). Verify thatTrn,n−1S is a OT (ζ)/2OT (ζ)-basis for Trn,n−1P

in/((σ − 1)Trn,n−1P

in + 2P

i/2n−1 ) (observe

that Trn,n−1S generates Trn,n−1Pin/2P

i/2n−1 over OT (ζ)/2OT (ζ)[G]). Now use Lemma 2.2

to extend S to S ′, an OT (ζ)[G]/〈σ2 + 1〉-basis for Pin/P

i/2n−1 . Since Pi

n/Pi/2n−1 has rank e0

over OT (ζ)[G]/〈σ2 + 1〉, we have |S ′| = e0.This OT (ζ)[G]/〈σ2+1〉-basis, S ′, possesses two important properties. First, it contains S.

Second, without loss of generality we may assume that the elements in S ′ − S are killed bythe trace Trn,n−1. These two properties are shared with another set: The set of all left–mostX (an X for every value of m). Clearly the set of all left–most X contains S. Moreover,by an earlier assumption, the compliment of S in the set of all left–most X is mapped tozero under the trace. And so, because the sets have the same cardinality (namely e0), wecan identify them. Without loss of generality, assume that S ′ is the set of all left–most X.This allows us to use the lists, A(1)–A(4) and B(1)–B(4), in the ‘book-keeping’ necessaryfor determining the Galois module structure below.

At this point, we know that Pin/P

i/2n−1 is free over OT (ζ)[G]/〈σ2 + 1〉. Indeed, S ′ (the

set of all left–most X) provides us a OT (ζ)[G]/〈σ2 + 1〉-basis for Pin/P

i/2n−1 . Of course,

the OT (ζ)[G]-structure of Pi/2n−1 is known from §2.2.1 (and can be read off of D0). So a

description of the image of S ′ under σ2 +1 in terms of D0 will determine the Galois modulestructure. See [?, §8]. The Result: For each m associated with A(1) or B(1) we decompose offan OT (ζ)[G1]-summand of OT (ζ) ⊗Z2 I, for A(2) or B(2) we get an OT (ζ) ⊗Z2 H, for A(3) wefind the group ring, OT (ζ)[G1] ∼= OT (ζ) ⊗Z2 G. But, for B(3) we decompose off the maximalorder of OT (ζ)[G1], OT (ζ) ⊗Z2 M. For A(4) and B(4) there is OT (ζ) ⊗Z2 L. All this andcounting determines the OT (ζ)[G1]-module structure of Pi

n from which the Ok[G1]-modulestructure can be inferred. It also determines the module M2(i, b1, b2). To determine theOT [G]-module structure (from which the Ok[G]-module structure can be inferred), we needto take our OT (ζ)-bases for Pi

n and create OT -bases.

2.3. Partially Ramified Extensions. Let Ti denote the maximal unramified extensionof Q2 contained in Ki. So T (ζ) of the previous section can be expressed at Tn, while T = T0.Recall the steps in §2.2.1. We first determined a OTn-basis B for Pi

n, one upon which the


action of G1 could be explicitly followed. Then noting that we can identify G/G1 with theGalois group for Tn/T0, we extended B to an OT0-basis for Pi

n. This time the action of everyelement in the Galois group G could be followed. What were the important ingredients inthis process? It was important that the elements of B lay in one–to–one correspondence,via valuation, with the integers i, . . . , i + vn(2) − 1. Using this fact and the fact that foreach t, Pt

n/Pt+1n had a normal field basis over OT0/PT0 , we were able to make an OT -

basis for Pin, namely B′ = ∪σi∈G/G1σ

iB. At that point we were done. The OT [G]-structurecould simply be read off of this basis. This is not the case when |G1| = 4. Nor is it thecase when |G1| = 8. At this point we still need to change our basis and use Nakayama’sLemma, if only to determine OT [G1]-structure. We leave it to the reader to check that thisprocess of basis change ‘commutes’ with the process of extending our OTn-basis to an OT0-basis. Simply follow the argument using elements of the form σtαm, σt(σ + 1)αm, . . . witht = 0, . . . 2[G:G1]−1 instead of elements of the form αm, (σ+1)αm, . . .. As a consequence, theproblem of determining the OT [G]-module structure reduces to the problem of determiningthe OTn

[G1]-module structure.

3. Fully Ramified Cyclic Extensions of Degree Eight

We consider fully ramified extensions K3/K0 with odd ramification numbers.

3.1. Outline. Our discussion here is focused on the unstably ramified case, b1 < e0. (Thestably ramified case will be addressed separately in §3.4.) Recall Step 1 of §2.2.3 (in referenceto K2/K0). But first note that the first two ramification numbers of K3/K0 are the (only)two ramification numbers of K2/K0 [?, pg 64 Cor]. We began with two elements, namelyα, (σ+1)α in the subfield K1. (The Galois relationship between them was explicit.) Then wecreated α, (σ + 1)α ∈ K2, preimages under the trace Tr2,1. In this section, we will start withthese four elements from K2 and use Lemma 2.1(2) to find further preimages: of α, (σ + 1)α,α, (σ + 1)α under Tr3,2. To avoid confusion (confusion resulting from additional bars denot-ing a preimage under Tr3,2), we relabel. Let α := α and let ρ := (σ + 1)α. So the fourelements in K2 are labeled α, (σ2+1)α, ρ, (σ+1)(σ2+1)α (instead of α, α, (σ + 1)α, (σ+1)αrespectively). The eight resulting elements (four from K2 along with their preimages) lie inone–to–one correspondence with the residues modulo 8.

We would have accomplished all that was accomplished in Step 1 from §2.2.3 if we knewthe Galois relationships among α, (σ2+1)α, ρ, (σ+1)(σ2+1)α explicitly. We need an explicitrelationship between α and ρ. This is accomplished in §3.2.2 through a list of technicalresults – generalizations of Lemma 2.1. Note that ρ is an ‘approximation’ to (σ+1)α – theyhave the same image under the trace Tr2,1. Our results describe their difference, the ‘error’in this ‘approximation’.

As a prerequisite for the technical results of §3.2.2, and in preparation for the analog ofStep 2 from §2.2.3 we use a result of Fontaine to provide a catalog of ramification numbersin §3.2.1. We are then ready for Step 2: First we order the eight elements (that we inheritfrom Step 1) in terms of increasing valuation. This is accomplished in §3.3. There are eightorderings – eight cases. The result is eight different bases, listed as A – H (as opposed tojust two in D from §2.2.3). For the convenience of the reader, they are listed in Appendix B.

We are now ready for the analog of Step 3 from §2.2.3. We are ready to determine thoseelements in each OT -basis that serve as an OT [G]/〈Tr3,2〉-basis, S, for Pi

3/Pi/22 . We will

then be able to describe the image, Tr3,2S, in terms of our OT -basis for Pi/22 (or more to

the point, explicitly in terms of OT [G]-generators for Pi/22 ). To do all this we will need,

as in §2.2.3, to perform certain basis changes. The processes are similar, but there are a

GALOIS STRUCTURE 73

few very important differences. For the convenience of the reader, the results of this stepare summarized in §3.3.1. The steps are then spelled out in §3.3.2 – §3.3.5. The structure ofM3(i, b1, b2, b3) (given in Tables 1 and 2) can then be read off of the bases in Appendix B.Note however, that we still need to determine the structure under b1 ≥ e0 (part of Case A).This situation is addressed in § 3.3.4.

3.2. Preliminary Results. We catalog the ramification triples and generalize Lemma 2.1,describing the difference ρ− (σ + 1)α.

3.2.1. Ramification Triples. There is stability and instability.

Theorem 3.1 ([?, Prop 4.3]).Stability:

b1 ≥ e0 ⇒ b2 = b1 + 2e0, and b1 + b2 ≥ 2e0 ⇒ b3 = b2 + 4e0.

Instability:

b1 < e0 ⇒ 3b1 ≤ b2 ≤ 4e0 − b1, b1 + b2 < 2e0 ⇒ 3b2 + 2b1 ≤ b3 ≤ 8e0 − b2 − 2b1.

In particular, when b1 < e0, either b2 = 3b1, b2 = 4e0 − b1, or b2 = b1 + 4t for b1 <2t < 2e0 − b1, while if b1 + b2 < 2e0, then either b2 = 3b2 + 2b1, b2 = 8e0 − b2 − 2b1, orb3 = 8s− b2 + 2b1 for b2 < 2s < 2e0 − b1.

Plot these ramification triples (b1, b2, b3) in 3, and project this plot to the first twocoordinates, (x, y, z) → (x, y, 0), thus creating Figure 1 (next page). This projection ispartly a line: for b1 ≥ e0, each point (b1, b2) is restricted to b2 = b1 + 2e0. It is partly atriangular region: for b1 < e0, each point (b1, b2) is bound between the lines b2 = 3b1 andb2 = 4e0 − b1. The significance of the regions A,B,C, . . . will be explained later. Note thatfor points, (b1, b2), above the line b2 = −b1 + 2e0, the plot of the (b1, b2, b3) in 3 will be aplane – b3 is uniquely determined.

In Figure 2 we have plotted a slice, at a particular value of b1, through our plot oframification triples in 3. Part of this slice is a line – when b3 is uniquely determined. Thusthe line from (2e0 − b1, 6e0 − b1) to (4e0 − b1, 8e0 − b1). Indeed, as drawn, Figure 2 implicitlyassumes that the slice was taken at b1 for b1 < e0/2. Otherwise there would be no triangularregion. Observe that in Figure 1, the lines b2 = 2e0 − b1 and b2 = 3b1 intersect at b1 = e0/2.If b1 ≥ e0/2, the third ramification number is uniquely determined by b2. The triangularregion bound by the lines b2 = 3b1, b3 = 3b2 + 2b1 and b3 = 8e0 − b2 − 2b1 exists onlyfor b1 < e0/2.

Because the ramification numbers are odd, the triangular part of Figure 1 can be parti-tioned as follows:

A. 4e0 − 4b1/3 < b2B. 4e0 − 2b1 < b2 < 4e0 − 4b1/3C. 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 > (4e0 + 4b1)/3D. 4e0 − 4b1 < b2 < 4e0 − 2b1 and b2 < (4e0 + 4b1)/3E. 2e0 − b1 ≤ b2 < 4e0 − 4b1 and b2 > (4e0 + 4b1)/3F . 2e0 − b1 ≤ b2 < 4e0 − 4b1 and b2 < (4e0 + 4b1)/3


Assuming that b1 < e0/2, there is a triangular region in Figure 2. This can be partitionedinto the following cases:

E. 8e0 + 4b1 − 2b2 < b3F . 8e0 − 2b2 < b3 < 8e0 + 4b1 − 2b2G. 8e0 − 4b1 − 2b2 < b3 < 8e0 − 2b2H. b3 < 8e0 − 4b1 − 2b2

Note that if b1 > 8e0/17, region G is empty; if b1 > 8e0/21, region H is empty; ifb1 > 8e0/28, region E is empty. So as drawn, we have assumed that b1 < 2e0/7. If howeverthe slice were taken for a value 8e0/17 < b1 < 8e0/16, note that the triangular regionwould consist of only one case, namely F . The relationship between E, F and E, F will beexplained in §3.3.

3.2.2. Technical Lemmas. The difference ρ− (σ + 1)α depends upon ramification.Unstable Ramification. Assume that b1 < e0. These results may be thought of as conse-quences of indirect ‘routes’ from α to ρ. For example, we may begin with α ∈ K2, create(σ2 + 1)α, then (σ + 1)(σ2 + 1)α and let ρ be the inverse image of (σ + 1)(σ2 + 1)α underTr2,1. This results in an expression for the ρ− (σ + 1)α.

Lemma 3.2. If b2 ≡ b1 mod 4 (equivalently 3b1 < b2 < 4e0 − b1), let t = (b2 − b1)/4. Thereare elements αm ∈ K2 with v2(αm) = b2 + 4m, such that

ρm = (σ + 1)αm + (σ2 ± 1)αm−t

has valuation v2(ρm) = b2 + 2b1 + 4m. The ‘+’ or ‘−’ depends on our needs.

Proof. Let α ∈ K2 with valuation, v2(α) ≡ b2 mod 4. Using Lemma 2.1, v2((σ + 1)α) =v2(α) + b1, v2((σ2 + 1)α) = v2(α) + b2. Since (σ2 + 1)α ∈ K1 and v1((σ2 + 1)α) = (v2(α) +b2)/2 ≡ b2 mod 2, v1((σ + 1)(σ2 + 1)α) = (v2(α) + b2)/2 + b1. Using Lemma 2.1(2), thereis a ρ ∈ K2 with v2(ρ) = v2(α) + 2b1 such that (σ2 + 1)ρ = (σ + 1)(σ2 + 1)α. Since(σ2+1) [ρ− (σ + 1)α] = 0. Using Lem 2.1(3), there is a θ ∈ K2 with v2(θ) = (v2(α)−b2)+b1and ρ = (σ+1)α+(σ2−1)θ. Since b1 < e0, v2(2θ) > v2(ρ). We may replace ρ by ρ′ := ρ+2θ(they have the same valuation), and get ρ′ = (σ + 1)α+ (σ2 + 1)θ. Once αm is chosen, welet αm−t := θ.

Lemma 3.3. If b2 ≡ −b1 mod 4 (equivalently b2 = 3b1 or b2 = 4e0−b1), let s := (b2+b1)/4.There are elements αm ∈ K2 with v2(αm) = b2 + 4m, such that

ρm = (σ + 1)αm + (σ + 1)(σ2 + 1)αm−s

has valuation, v2(ρm) = 2b2 − b1 + 4m. Note if b2 = 3b1, v2(ρm) = b2 + 2b1 + 4m.

GALOIS STRUCTURE 75

Proof. There is a τ ∈ K0 with v0(τ) = (b2 − b1)/2. Using Lemma 2.1(2), let ρ ∈ K2 withv2(ρ) = b2 − 2b1 such that (σ2 + 1)ρ = τ . Clearly (σ2 + 1) · (σ − 1)ρ = 0, so there is aθ ∈ K2 with v2(θ) = −b1 such that (σ − 1)ρ = (σ2 − 1)θ. Since (σ − 1) · [ρ− (σ + 1)θ] = 0,τ ′ := ρ− (σ + 1)θ is a unit in K0. Let ρ′ = ρ/τ ′ and θ′ = θ/τ ′, so 1 = ρ′ − (σ + 1)θ′. Nowlet β = (σ + 1)(σ2 + 1)θ′. Clearly (σ2 + 1)θ′ ∈ K1 and v1((σ2 + 1)θ′) = (b2 − b1)/2 is odd.Therefore v2(β) = b2 + b1. Replacing 1 with the expression, (σ + 1)(σ2 + 1)(θ′/β), yields

ρ′ = (σ + 1)θ′ + (σ + 1)(σ2 + 1)(θ′/β). (3.1)

By choosing τ ∈ K0 with other valuations, the result follows.

Unfortunately, if b2 = 4e0 − b1 then s = e0 (valuation can not distinguish between αm/2and αm−s). To avoid this confusion, we include the following.

Lemma 3.4. Let b2 = 4e0 − b1. There are αm ∈ K2 with v2(αm) = b2 + 4m, so

ρm := (σ + 1)αm − 12(σ + 1)(σ2 + 1)αm +

12(σ + 1)(σ2 + 1)αm+e0−b1

has valuation, v2(ρm) = 2b2 − b1 + 4m.

Proof. From (3.1) we have ρ′ = (σ + 1)θ′ + (σ + 1)(σ2 + 1)(θ′/β). Apply (σ2 + 1)/β toboth sides. So (σ2 + 1)(ρ′/β) = 1 + 2/β. Since v2((σ2 + 1)ρ′) = 8e0 − 4b1 and v2(β) = 4e0,then v0(1 + 2/β) = e0 − b1. Replace θ′/β with (1/2) · [−θ′ + θ′(1 + 2/β)], and distribute(σ + 1)(σ2 + 1).

Remark 3.5. Note (σ−1)ρm = (σ2 −1)αm and (σ2 +1)ρm = (σ+1)(σ2 +1)αm+e0−b1 , usingLemma 3.4. Apparently, ρm is ‘torn’ between αm and αm+e0−b1 . We chose to emphasizeρm’s tie to αm. If we relabel ρm−e0+b1 as ρm (keep the αm the same), Lemma 3.4 reads

ρm := (σ + 1)αm−e0+b1 − 12(σ + 1)(σ2 + 1)αm−e0+b1 +

12(σ + 1)(σ2 + 1)αm

has valuation, v2(ρm) = b2 + 2b1 + 4m – thus tying ρm to (1/2)(σ + 1)(σ2 + 1)αm. Thisvaluation of ρm is as in Lemmas 3.2 and 3.3 (for b2 = 3b1).

Stably Ramified Extensions. Assume that b1 ≥ e0. The results may be seen as direct routesfrom α to ρ. We create ρ immediately from (σ+1)α ∈ K2. For discussion and generalization,see [?].

Lemma 3.6. Let b1 > e0. For every odd integer, a, there are elements α, ρ ∈ K2 withv2(α) = a, v2(ρ) = a+ (b2 − b1). such that

(σ + 1)α− ρ = µ ∈ K1,

with v2(µ) = v2(α) + b1. Furthermore µ ∈ K0 for v2(µ) = v2(α) + b1 ≡ 0 mod 4.

Proof. Since v2((σ+1)α) = v2(α)+ b1 is even, we may express (σ+1)α as a sum µ+ρ withµ ∈ K1, ρ ∈ K2, v2(µ) = v2(α)+b1 and odd v2(ρ). Apply (σ−1). So (σ2 −1)α = (σ−1)µ+(σ−1)ρ. Since b2 = b1 +2e0 < 3b1, v2((σ2 −1)α) = v2(α)+b2 < v2(α)+3b1 ≤ v2((σ−1)µ).So v2((σ2 − 1)α) = v2((σ − 1)ρ) and v2(ρ) = v2(α) + (b2 − b1). If v2(µ) ≡ 0 mod 4, we may


choose α so that µ ∈ K0. Pick a µ∗ ∈ K0 with v2(µ∗) = v2(µ). Relabel α as α0. Chooseαi ∈ K2 with v2(αi) = v2(α0) + 2i. As before, generate µi and ρi with αi = µi + ρi. Clearlyµ∗ =

∑∞i=0 aiµi for some units ai ∈ K0. Let α∗ =

∑∞i=0 aiαi and ρ∗ =

∑∞i=0 aiρi.

Lemma 3.7. Let b1 = e0 be odd. For every odd integer, a, there are elements α, ρ ∈ K2

with v2(α) = a, v2(ρ) = a+ (b2 − b1) such that

(σ − 1)α− ρ = µ1 ∈ K1 if a ≡ e0 mod 4,

(σ + 1)α− ρ = µ0 ∈ K0 if a ≡ 3e0 mod 4.

with v2(µi) = v2(α) + b1.

Proof. Let τ ∈ K0 be a unit. From Lemma 2.1(2), there is a ρ ∈ K2 with v2(ρ) = −b2and (σ2 + 1)ρ = τ . So (σ2 + 1) · (σ − 1)ρ = 0. Use Lemma 2.1(3) to find θ ∈ K2 withv2(θ) = b1 − 2b2 and (σ2 − 1)θ = (σ− 1)ρ. For a ≡ e0 mod 4, we may assume that α = ρπm0for some m. Let µ1 = (σ2 + 1)θπm0 ∈ K1 and ρ = −2θπm0 ∈ K2. The statement follows. Fora ≡ 3e0 mod 4, (σ2 − 1)θ = (σ − 1)ρ can be interpreted to mean that ρ − (σ + 1)θ ∈ K0.Multiplying by an appropriate power of π0, we let α = θπm0 , µ0 = −(ρ− (σ+ 1)θ)πm0 ∈ K0

and ρ = ρπm0 ∈ K2.

3.3. The Galois module structure under unstable ramification. Assume b1 < e0.First we determine the OT -bases in Appendix B. From Lemmas 3.2, 3.3, 3.4 we have αm, ρm,(σ2+1)αm, (σ+1)(σ2+1)αm ∈ K2, with valuations (measured in v2) for every residue classmodulo 4. Recall v2(αm) = b2+4m, v2((σ2+1)αm) = 2b2+4m, v2((σ+1)(σ2+1)αm) = 2b2+2b1+4m and v2(ρm) = 8e0−3b1+4m if b2 = 4e0−b1, otherwise v2(ρm) = b2+2b1+4m. UsingLemma 2.1(2) we determine elements αm, ρm, (σ2 + 1)αm, (σ + 1)(σ2 + 1)αm ∈ K3, with(σ4 + 1)X = X and v3(X) = 2v2(X) − b3. These eight elements have valuations (measuredin v3) in one–to–one correspondence with the residue classes modulo 8. By varying m, it ispossible to choose those with valuation i ≤ v3(x) < 8e0 + i.

To organize this process, we list these elements in terms of increasing valuation. Thereare eight orderings – eight cases. In each case X (or X), an increase in valuation is denotedby an arrow, −→, and justified by an inequality assigned a number. Numbers above anarrow apply to X. Numbers below the arrow apply to X. As we see below, the orderingof the elements in E is the same as in E (also in F as in F ). This explains the use ofsimilar notation.

A. ρ1−→ 2ρ 2−→ (σ2 + 1)α 1−→ 2(σ2 + 1)α 1−→ 2α 1−→

4α 6−→ (σ + 1)(σ2 + 1)α 1−→ 2(σ + 1)(σ2 + 1)α 4−→ 2ρ

In Case A, the valuation of ρm depends upon whether or not b2 = 4e0 − b1. If b2 = 4e0 − b1,0 < b1 justifies 2 while b1 < 2e0 justifies 4. All other increases, including 2 and 4 forb2 = 4e0 − b1, are justified by the inequalities listed below.

In Cases B through H, there is only one valuation of ρm.

B. ρ1−→ 2ρ 2−→ (σ2 + 1)α 1−→ 2(σ2 + 1)α 4−→ 2α 5−→

(σ + 1)(σ2 + 1)α 6′−→ 4α 5−→ 2(σ + 1)(σ2 + 1)α 4−→ 2ρ

GALOIS STRUCTURE 77

C. ρ1−→ 2ρ 2−→ (σ2 + 1)α 1−→ 2(σ2 + 1)α 7−→ (σ + 1)(σ2 + 1)α 5′

−→

2α 7−→ 2(σ + 1)(σ2 + 1)α 5′−→ 4α 7−→ 2ρ

D. ρ9−→ (σ2 + 1)α 2′

−→ 2ρ 9−→ 2(σ2 + 1)α 7−→ (σ + 1)(σ2 + 1)α 5′−→

2α 7−→ 2(σ + 1)(σ2 + 1)α 5′−→ 4α 7−→ 2ρ

E = E. ρ7′

−→11

2α 8−→8

2ρ 2−→13

(σ2 + 1)α 8−→8

(σ + 1)(σ2 + 1)α 7′−→11

2(σ2 + 1)α 8−→8

2(σ + 1)(σ2 + 1)α 7′−→14

2α 8−→8

2ρ

F = F . ρ7′

−→11

2α 10−→12

(σ2 + 1)α 2′−→13′

2ρ 10−→12

(σ + 1)(σ2 + 1)α 7′−→11

2(σ2 + 1)α 8−→8

2(σ + 1)(σ2 + 1)α 7′−→14

2α 8−→8

2ρ

G. ρ9−→ (σ2 + 1)α 12′

−→ 2α 15−→ (σ + 1)(σ2 + 1)α 12′−→ 2ρ 9−→ 2(σ2 + 1)α 8−→

2(σ + 1)(σ2 + 1)α 14−→ 2α 8−→ 2ρ

H. ρ9−→ (σ2 + 1)α 8−→ (σ + 1)(σ2 + 1)α 15′

−→ 2α 8−→ 2ρ 9−→ 2(σ2 + 1)α 8−→2(σ + 1)(σ2 + 1)α 14−→ 2α 8−→ 2ρ

Numbered Inequalities: (1) b1 < 2e0, b2 < 4e0, b3 < 8e0. (2) 3b2 > 4e0 + 4b1. (2′) 3b2 <4e0+4b1. (3) 4e0−4b1 < 3b2 (true for A–F since b2 ≥ 2e0−b1). (4) 2b2 < b3. (5) 4e0−2b1 <b2. (5′) 4e0 − 2b1 > b2. (6) 4e0 − 4b1/3 < b2. (6′) 4e0 − 4b1/3 > b2. (7) 4e0 − 4b1 < b2. (7′)4e0 − 4b1 > b2. (8) b1 > 0. (9) b2 > 2b1. (10) b2 > 4e0/3 (true for A–F , since b2 ≥ 3e0/2).(11) Since b2 > 2b1 and b3 ≤ 8e0 − 2b1 − b2, b3 < 8e0 − 4b1. (12) 8e0 − 2b2 < b3. (12′)8e0 − 2b2 > b3. (13) 8e0 + 4b1 − 2b2 < b3. (13′) 8e0 + 4b1 − 2b2 > b3. (14) Since b2 > 2b1and b3 ≥ 3b2 + 2b1, b3 > 2b1 + 4b2. (15) 8e0 − 4b1 − 2b2 < b3. (15′) 8e0 − 4b1 − 2b2 > b3.

We leave it to the reader to verify Appendix B.

3.3.1. Summary: Results of Basis Changes and Nakayama’s Lemma.Basis Changes. Except in four rows,

C(2), D(2), E(2), F (2), (3.2)

we find we may change the OT -bases in Appendix B so that the Galois action upon eachbasis is as if ρ and ρ had been everywhere replaced by (σ + 1)α and (σ + 1)α. In the fourexceptional cases there are nontrivial Galois relationships among the basis elements. Thisis explained in §3.3.5.Nakayama’s Lemma. We find, without loss of generality, that the set S of ‘left–most’elements X (as in S ′ of §2.2.3) from each basis in Appendix B will serve as a OT [G]/〈Tr3,2〉-basis for Pi

3/Pi/22 , except that S contains both (σ + 1)(σ2 + 1)α and 2α in B(3),

C(3), D(3).At this point, the reader can skip the verification of these assertions, ignore Cases C

through F , replace ρ with (σ + 1)α, and lift the Galois module structure off of the baseslisted in Appendix B. See [?, §8] The result of the readers effort will be the statement ofour main result in every case except those associated with (3.2).


3.3.2. Trivial Difference. The elements αm, ρm (or ρm, 2αm) from each basis in Appendix Bprovide a OT -basis for P

i/22 /P

i/41 . We can change ρm by an element in P

i/41 and still

have a OT -basis. So when ρm−(σ+1)αm ∈ Pi/41 , the difference between ρm and (σ+1)αm

is trivial.Since v2((σ + 1)α) = v2(ρ − (σ + 1)α), checking ρm − (σ + 1)αm ∈ P

i/41 is equiv-

alent to checking v3((σ + 1)α) ≥ i. In Case A, because b2 + b1 ≤ 4e0 we find thatv3((1/2) · (σ + 1)(σ2 + 1)αm) ≤ v3((σ + 1)αm). Therefore, in A(3) through A(8), we mayreplace ρm by (σ+ 1)αm. We refrain from doing so in A(8) as it may hamper our ability todetermine the effect of Tr3,2 on ρ. We will return to this issue in §3.3.4. In Case B, becauseb2 > 4e0 − 2b1 we find v3(2α) < v3((σ+ 1)α). We may replace ρ in B(3) through B(8). Forsimilar reasons, we refrain in B(8). In Cases C and D, b3 > 2b2+2b1 (since b3 = b2+4e0 andb2 < 4e0 − 2b1). As a consequence, v3((σ + 1)(σ2 + 1)α) < v3((σ + 1)α). We may replace ρin C(3) through C(8), and in D(3) through D(6). In Cases E through H, we clearly havev3(α) < v3((σ+1)α). We may replace ρ in E(1) or E(3) – E(8), F (1) or F (3) – F (8), G(1)or G(3) – G(8), H(1) or H(3) – H(8). We replace ρ everywhere that we may, except thatwe refrain for

A(8), B(8), C(8), D(7), D(8), E(8), F (7), F (8), G(6), G(7), H(6). (3.3)

Now we consider the difference between ρ and (σ + 1)α and replace ρ with (σ + 1)α (2ρwith (σ + 1)2α) in

E(1), F (1), G(1), G(8), H(1), H(7), H(8). (3.4)

Since (σ4 + 1) · [ρ − (σ + 1)α] = 0, we may use Lem 2.1(2) and find an element ω ∈ K3

with v3(ω) = 2b2 + b1 − 2b3 so that (σ4 − 1)ω = ρ − (σ + 1)α. As long as b3 < 8e0 − 3b1,which holds in Cases E through H, we have v3(ρ) = v3(ρ+ 2ω). On the basis of valuation,we may replace ρ with ρ + 2ω and still have a basis (i.e. Observation (2)). Now since(ρ+2ω)− (σ+1)α = (σ4 +1)ω ∈ K2, we may replace (ρ+2ω) with (σ+1)α and still havea basis. All we need is v3((σ + 1)α) ≥ i. But this clearly holds since v3(α) ≥ i.

3.3.3. Nakayama’s Lemma and an OT [G]/〈Tr3,2〉-basis for Pi3/P

i/22 . The collection of

X in our bases provide an OT -basis for Pi3/P

i/22 . As in §2.2.3, whenever X and (1/2)·

X appear in the same row, we may replace X with X − (1/2) ·X and still have a OT -basis.Since Tr3,2(X − (1/2) · X) = 0, we relabel and assume, without loss of generality, thatfor these X’s, Tr3,2X = 0. Let T=0 denote this set (trace zero). Let T=0 denote the set ofX’s with X in the same row. For each such X ∈ T=0, Tr3,2X ≡ 0 mod 2. This is the setof trace not zero. Note that Tr3,2T=0 is an OT /2OT -basis for Tr3,2Pi

3/2Pi/22 . Following

§2.2.3, we select from T=0 a set S (notation as in §2.2.3) such that Tr3,2S is a OT /2OT -basisfor Tr3,2Pi

3/((σ − 1)Tr3,2Pi3 + 2P

i/22 ). It turns out that just as in §2.2.3, S is the set of

left-most X for which X appears in the same row, except that S contains both X’s in T=0

from B(3), C(3), D(3).Note that σ acts trivially (modulo 2) upon (σ + 1)(σ2 + 1)α and 2α in B(3), C(3) and

D(3). These elements are linearly independent over OT /2OT [G]. Since both contribute tothe OT /2OT -basis for Tr3,2Pi

3/2Pi/22 , both (σ + 1)(σ2 + 1)α and 2α are in S. When a

row contributes exactly one X to T=0, the phrase ‘left–most’ is unnecessary. Indeed σ actstrivially (modulo 2) on the loneX = Tr3,2X, and sinceX is needed for the OT /2OT -basis forTr3,2Pi

3/2Pi/22 , X must appear in S. Note this is the only situation to consider in Case A.

GALOIS STRUCTURE 79

In the other cases, we need to show that each X, corresponding to the left–most X ofT=0, generates over OT /2OT [G] all other elements in the same row (in Tr3,2T=0). This iseasy to see for rows E(1), E(5), F (1), F (5), G(1), G(5), G(8) and H(1), H(5), H(7), H(8).More work is required for rows D(7), F (7), G(6), G(7), H(6). Note that ρ − (σ + 1)αm =(σ2 + 1)αm−t or (σ + 1)(σ2 + 1)αm−s depending upon b2 > 3b1 or b2 = 3b1, respectively. Ifρ−(σ+1)αm = (σ+1)(σ2+1)αm−s, then (σ−1)ρ = (σ2+1)α−2α ≡ (σ2+1)α mod 2P

i/22 .

So ρ generates (σ2 + 1)α. If ρ − (σ + 1)αm = (σ2 + 1)αm−t the analysis is a little moreinvolved. Note (σ − 1)ρ− (σ2 + 1)α ≡ (σ − 1)(σ2 + 1)αm−t mod 2P

i/22 . For m associated

with D(7), F (7), G(6), G(7), H(6), check that m− t lies in D(3), F (4), G(4), H(4) or later.In any case (σ + 1)(σ2 + 1)αm−t ∈ Pi

3. So ρm and another X, namely (σ + 1)(σ2 + 1)αm−t,combine together to generate (σ2 + 1)αm.

Apply Lemma 2.2 and extend S to an OT [G]/〈Tr3,2〉-basis for Pi3/P

i/22 . Except in

Cases B, C, D (where a row contributes more than one element), we may assume that thisbasis is the set of left–most elements X, one from each row.

3.3.4. Essentially Trivial Difference. In §3.3.2 we did not replace ρ by (σ + 1)α in rowsA(1), A(2), B(1), B(2), C(1), C(2), D(1), D(2), E(2), F (2), G(2), H(2). It was not clearthat the difference ρ− (σ + 1)α lay in P

i/41 . Neither did we replace ρ by (σ + 1)α in the

rows listed in (3.3). In this section we remedy this situation. We show, except in four cases,C(2), D(2), E(2), F (2), we may change our basis so that the Galois action is as if ρ hadbeen replaced by (σ + 1)α (ρ by (σ + 1)α).

We begin with Case A, explaining why the difference between ρ and (σ+1)α is essentiallytrivial and then determine the Galois module structure (to illustrate the process). ConsiderA(1), A(2) and A(8). Recall there are three expressions for ρm corresponding to 3b1 < b2 <4e0 − b1, b2 = 3b1, and b2 = 4e0 − b1. Suppose 3b1 < b2 < 4e0 − b1, and ρm = (σ + 1)αm +(σ2 − 1)αm−t. Consider ρm in A(8). Since b1 + b2 < 4e0, v3(ρm) ≤ v3(2αm−t). So for m inA(8), m− t is in A(4) or later. In any case, (σ− 1)αm−t = (σ+ 1)αm−t − 2αm−t ∈ Pi

3 and(1/2)(σ−1)(σ2+1)αm−t = (1/2)(σ+1)(σ2+1)αm−t−(σ2+1)αm−t ∈ Pi

3 (i.e. these elementsare available). We replace αm with x = αm+(σ−1)αm−t− (1/2)(σ−1)(σ2 +1)αm−t. Note(σ + 1)x = ρ and (σ2 + 1)x = (σ2 + 1)αm. The Galois action on x and ρm is the same asthe Galois action on αm and (σ+1)αm. It is as if ρm had been replaced by (σ+1)αm andρm by (σ + 1)αm. Now consider A(1) and A(2), ρm = (σ + 1)αm + (1/2)(σ2 − 1)αm−t+e0 .Since v2(ρm) < v2((1/2)(σ + 1)(σ2 + 1)αm−t+e0), for m in A(1) or A(2), m− t+ e0 lies inA(3) or later. In any case, (1/2)(σ − 1)(σ2 − 1)αm−t+e0 is available. So in A(1) and A(2),we replace 2αm by 2αm − (1/2)(σ − 1)(σ2 − 1)αm−t+e0 . The effect of this replacement onthe Galois action is, again, the same as if we replaced ρm by (σ + 1)αm.

Now suppose b2 = 3b1 and ρm = (σ+1)αm+(σ+1)(σ2 +1)αm−s. Note s = b1. Startingwith the smallest m such that i ≤ v3(ρm) we replace αm by αm + (1/2)(σ + 1)αm+e0−b1so long as m + e0 − b1 is associated with A(8). If i ≤ v3(ρm−b1), we replace αm byαm + (σ2 + 1)αm−b1 . In any case, we can systematically replace αm by x = αm +(1/2)(σ2 + 1)αm+e0−b1 or αm + (σ2 + 1)αm−b1(1/2)(σ2 + 1)αm by (1/2)(σ2 + 1)x and(1/2)(σ+1)(σ2+1)αm by (1/2)(σ+1)(σ2+1)x. The Galois action after this change of basisis as if ρm = (σ + 1)αm and ρm = (σ + 1)αm. Consider A(1) and A(2). Note (σ − 1)ρm =(σ−1) ·(σ+1)αm. Moreover, for m associated with these two cases, (σ+1)(σ2+1)αm+e0−b1and (σ2 + 1)αm+e0−b1 are available elsewhere in our basis. So we replace (σ2 + 1)αm by(σ2 +1)(αm+αm+e0−b1) and (σ+1)(σ2 +1)αm by (σ+1)(σ2 +1)(αm+αm+e0−b1). Note form associated with A(2), m+e0 − b1 is associated with A(3) or later. We achieve the desiredeffect by replacing (σ + 1)(σ2 + 1)αm with (σ + 1)(σ2 + 1)αm + (σ + 1)(σ2 + 1)αm+e0−b1 .


This leaves b2 = 4e0−b1. Because this case is more complicated (recall Remark 3.5: ρm is‘torn’ between αm and αm+e0−b1), we first determine the Galois module structure for b2 <4e0−b1. Eachm in A(1) results in an OT⊗Z2 (R3⊕H);m in A(2) in an OT⊗Z2H2;m in A(3)in an OT ⊗Z2 (R3 ⊕ M); m in A(4) in an OT ⊗Z2 M1; m in A(5) in an OT ⊗Z2 (R3 ⊕ L);m in A(6) in an OT ⊗Z2 L3; m in A(7) in an OT ⊗Z2 (R3 ⊕ I); m in A(8) in an OT ⊗Z2 I2.Counting the number of m associated with each A(j) yields the first column of Table 2.

Now consider b2 = 4e0 −b1. Because v2(ρm) = 2b2 −b1 +4m, the number of m associatedwith A(1) and A(7) are different. The number for A(7) is e0 − b1 too low, while A(1) ise0 − b1 too high. We seem to be missing e0 − b1 of OT ⊗Z2 I and have e0 − b1 too many ofOT ⊗Z2 H. Let us look at this more carefully. Note ρm in A(8) maps (via Tr3,2) to

ρm = (σ + 1)(αm − (1/2)(σ2 + 1)αm) +

(1/2)(σ + 1)(σ2 + 1)αm+e0−b1(σ + 1)(σ2 + 1)αm−b1

So ρm maps into the OT -module spanned by αm − (1/2)(σ2 + 1)αm and (σ + 1)(αm −(1/2)(σ2+1)αm) along with either (1/2)(σ2+1)αm+e0−b1 and (1/2)(σ+1)(σ2+1)αm+e0−b1or (σ2 +1)αm−b1 and (σ+1)(σ2 +1)αm−b1 . In any case, the elements (1/2)(σ2 +1)αm and(1/2)(σ+1)(σ2 +1)αm for (i+b3 −4b2 +2b1)/8 ≤ m ≤ (i+b3 −4b2 +2b1)/8+e0 −b1 −1are not associated with a ρm in A(8). The ρm in A(1) map to (σ2 − 1)αm (under (σ − 1))and so (σ+1)(σ2+1)αm+e0−b1 (under (σ2+1)) yielding a H, unless m+e0−b1 is associatedwith A(2). In fact, there are e0 − b1 ρm that map into A(2) under (σ2 + 1). For each m inA(2) we have (σ4 + 1)(σ + 1)(σ2 + 1)αm = (σ2 + 1)ρm−e0+b1 = (σ+ 1)(σ2 + 1)αm, yieldinga copy of H2. But for the last e0 − b1 elements ρm in A(2), namely those m such thatm+ e0 − b1 is in A(3) we may replace ρm by ρm − (1/2)(σ + 1)(σ2 + 1)αm+e0−b1 . For eachof these m we have the OT [G]-submodule spanned by ρm − (1/2)(σ + 1)(σ2 + 1)αm+e0−b1and (σ2 − 1)αm. These e0 − b1 together with the elements left out of a module in A(8) yielda e0 − b1 copies of I, precisely making up the counts.

Cases B – H: In the remaining cases, we only have two situations: b2 = 3b1 and 3b1 <b2 < 4e0 − b1. Consider 3b1 < b2 < 4e0 − b1 first, and ρm = (σ+1)αm+(σ2 ±1)αm−t wherewe may choose between ± as we like. We are concerned with the image of the trace, Tr3,2,in particular Tr3,2ρm = (σ + 1)αm + (σ2 + 1)αm−t, for ρm appearing in B(8), C(8), D(7),D(8), E(8), F (7), F (8), G(6), G(7), and H(6). Note if (σ2 + 1)αm−t ∈ Pi

3, we may replaceρm with ρm − (σ2 + 1)αm−t. So if (σ2 + 1)αm−t appears in B(6), C(6), D(6), E(5), F (5),G(5), H(5) or later we may replace ρm with (σ + 1)αm and ρm with ρm − (σ2 + 1)αm−t.The later replacement exhibits the same Galois action as a replacement of ρm by (σ+1)αm.Without loss of generality we will call it a replacement of ρm by (σ + 1)αm.

Since b2 ≤ 4e0 − b1, v3(2αm) ≥ v3(ρm). What happens when (σ2 + 1)αm−t appears inB(3) – B(5), C(3) – C(5), D(3) – D(5), E(4), F (6), G(4), H(6)? In this case (σ−1)αm−t =(σ + 1)αm−t − 2αm−t ∈ Pi

3. In B(8), C(8), D(7), D(8), E(8), F (8), G(6), G(7), H(6), wereplace αm with αm+(σ−1)αm−t, and (σ2 +1)αm with (σ2 +1)αm+(σ−1)(σ2 +1)αm−t.Note ρm = (σ + 1) · [αm + (σ − 1)αm−t]. The Galois action upon these basis elements:Tr3,2ρm = ρm = (σ + 1) · [αm + (σ − 1)αm−t], (σ2 + 1) · [αm + (σ − 1)αm−t] = (σ2 + 1)αm + (σ − 1)(σ2 + 1)αm−t, and (σ + 1) · [(σ2 + 1)αm + (σ − 1)(σ2 + 1)αm−t] = (σ2 + 1)ρm = (σ + 1)(σ2 + 1)αm, is similar to the Galois action upon: (σ + 1)αm, αm, (σ + 1)αm,(σ2 + 1)αm, (σ + 1)(σ2 + 1)αm. We may assume (σ + 1)αm and (σ + 1)αm appear insteadof ρm and ρm.

Now consider the appearance of ρ in B(1), B(2), C(1), D(1), G(2), H(2). Suppose ρm =(σ+1)αm+(1/2)·(σ2−1)αm+e0−t. One may check v3(ρm) ≤ v3((1/2)(σ+1)(σ2+1)αm+e0−t

GALOIS STRUCTURE 81

and v3(2ρm+e0−t) ≤ v3(4αm). So (1/2)(σ+1)(σ2+1)αm+e0−t appears in B(4) – B(7), C(6) –C(8) or D(6) – D(8). Note in these sets of elements, ρm+e0−t has already been replaced by(σ + 1)αm+e0−t. Importantly, (1/2)(σ − 1)(σ2 + 1)αm+e0−t along with (σ − 1)αm+e0−t areavailable to us. We replace 2αm with 2αm−(1/2)(σ−1)(σ2+1)αm+e0−t+(σ−1)αm+e0−t =2αm− (1/2)(σ−1)(σ2 −1)αm+e0−t in B(1), B(2), C(1) and D(1). The effect of this changeof basis is the same as if we replaced ρm by (σ + 1)αm.

Now consider G(2) and H(2). Again ρm = (σ+1)αm+(1/2)(σ2−1)αm+e0−t. In G and H,b3 ≤ 8e0 − 2b2. As a result, v3(ρm) ≤ v3((σ − 1)αm+e0−t). Note we refer to (σ − 1)αm+e0−tand not (σ − 1)αm+e0−t. The valuation of the first is b1 more than the valuation of thesecond. As one may check v3(ρm) ≤ v3((1/2)(σ+1)(σ2+1)αm+e0−t), so (1/2)(σ+1)(σ2+1)αm+e0−t appears in G(7), G(8) or H(8). If (σ + 1)(σ2 + 1)α appeared in G(1) or H(1),(σ2 + 1)αm−t would be available and so we would replace ρm with ρm − (σ2 + 1)αm−t. If(1/2)(σ+1)(σ2 +1)αm+e0−t appears in G(7), then we may assume (σ − 1)αm+e0−t appearsthere instead of ρm+e0−t, because v3((σ2 + 1)αm+e0−2t) = v3((σ − 1)αm+e0−2t) ≥ i, and wewould have replaced ρm+e0−t previously in our discussion with ρm+e0−t−(σ2 + 1)αm+e0−2t.We may now replace 2αm with 2αm− (σ − 1)αm+e0−t. We replace (σ2 +1)αm with (σ2 +1)αm + (1/2)(σ − 1)(σ2)αm+e0−t. We may assume without loss of generality that (σ + 1)αmappears in G(2) and H(2) instead of ρm.

Now we work with Cases B through H under the assumption b2 = 3b1. So ρm = (σ+1) ·[αm+(σ2 +1)αm−b1 ]. First note if (σ+1)(σ2 +1)αm−b1 appears in B(2), C(3), D(3), E(4),F (4), G(4), H(4), or later we may replace ρm in B(8), C(8), D(7), D(8), E(7), F (7), G(5),G(6), H(5) with ρm − (σ + 1)(σ2 + 1)αm. Suppose (σ2 + 1)αm−b1 appears elsewhere. In B,these elements can appear in B(1), B(2), or as (1/2)·(σ2+1)αm+e0−b1 elsewhere in B(8). Incases C throughH, since b1 < 4e0/5, v3(ρm) ≤ v3(ρm−b1). So (σ2+1)αm−b1 appears in C(1),C(2), D(1), D(2), E(2), E(3), F (2), F (3), G(2), G(3), H(2), H(3). In these cases, we mayeither replace αm with αm+(1/2)·(σ2+1)αm+e0−b1 or αm+(σ2+1)αm+−b1 . If for example,we replace αm with αm+(σ2 +1)αm−b1 , (σ2 +1)αm with (σ2 +1)αm+2(σ2 +1)αm−b1 , and(σ+ 1)(σ2 + 1)αm with (σ+ 1)(σ2 + 1)αm + 2(σ+ 1)(σ2 + 1)αm−b1 , then the Galois actionon this new basis is the same as if (σ+ 1)αm and (σ+ 1)αm appear instead of ρm and ρm.

We now concern ourselves with B(1), B(2), C(1), D(1), G(2) and H(2). Checkv3((σ2 + 1) αm+e0−b1) ≥ v3(ρm). We replace (σ2+1)αm with (σ2+1)αm+(σ2+1)αm+e0−b1 ,and (σ + 1)(σ2 + 1)αm with (σ + 1)(σ2 + 1)αm + (σ + 1)(σ2 + 1)αm+e0−b1 . In B(2),v3((σ + 1)(σ2 + 1)αm+e0−b1) ≥ v3((σ + 1)(σ2 + 1)αm), we replace (σ + 1)(σ2 + 1)αm with(σ + 1)(σ2 + 1)αm + (σ + 1)(σ2 + 1)αm+e0−b1 . All this has the same effect upon the Galoisaction as a replacement of ρm by (σ + 1)αm.

3.3.5. Non-Trivial Difference. We consider ρ in C(2), D(2), E(2), F (2).First consider the case b2 = 3b1 where ρm = (σ+ 1)αm + (1/2)(σ+ 1)(σ2 + 1)αm+e0−b1 .

Note C and E do not intersect the line b2 = 3b1. We focus on D(2), F (2). In D withb2 = 3b1, we have b1 < 4e0/5. So v3(2αm) ≤ v3(αm+e0−b1). Since v3(2(σ + 1)(σ2 + 1)αm) ≤v3((σ+ 1)(σ2 + 1)αm+e0−b1), for m associated with D(2), (σ+ 1)(σ2 + 1)αm+e0−b1 appearsin D(4), or (1/2)(σ + 1)(σ2 + 1)αm+e0−b1 appears in D(5) or later. If (1/2)(σ + 1)(σ2 + 1)αm+e0−b1 is available, we may replace ρm with (σ + 1)αm. The Galois action when m is inD(2) and m + e0 − b1 is in D(4) is our primary concern. But first consider F (or F ) withb2 = 3b1. Note then b3 ≤ 8e0 + 2b2 − 8b1. So v3(ρm) ≤ v3((σ + 1)(σ2 + 1)αm+e0−b1). Sinceb3 ≤ 8e0+2b2−8b1, v3(αm) ≤ v3(2(σ2 + 1)αm+e0−b1). So form associated with F (2), (σ+1)(σ2 +1)αm+e0−b1 appears in F (4), or in F (5) or later. If m+e0 − b1 is associated with F (5)or later, we have (σ2 + 1)αm+e0−b1 available. We replace 2αm with 2αm+(σ2 + 1)αm+e0−b1 .


We replace (σ2 + 1)αm and (σ + 1)(σ2 + 1)αm with (σ2 + 1)αm + (σ2 + 1)αm+e0−b1 and(σ + 1)(σ2 + 1)αm + (σ + 1)(σ2 + 1)αm+e0−b1 . The effect of these changes upon the Galoisaction is the same as the replacement of ρm by (σ + 1)αm. This leaves the situation whenm belongs to D(2), F (2) while m + e0 − b1 belongs to D(4), F (4). In both of these cases,we replace (σ + 1)(σ2 + 1)αm+e0−b1 with (σ + 1)(σ2 + 1)αm+e0−b1 +(σ+1)2αm−ρm. Thisnew basis element has trace, Tr3,2, zero. For each such pair (m,m + e0 − t) we get a copyof H1G ⊕ R3.

Let us now turn to the case where 3b1 < b2 < 4e0−b1 and ρm = (σ+1)αm+(1/2)(σ2+1)αm+e0−t. Consider cases C and E. Because v3(2α) ≤ v3((σ2 + 1)α), if m appears in C(2),then m+e0 − t appears in C(6) or later. Since v3((σ2 +1)αm+e0−t > v3(2(σ + 1)(σ2 + 1)α),not every m+ e0 − t is in C(6) when m is in C(2). Since v3(ρ) ≤ v3((1/2)(σ2 + 1)α), if mappears in E(2), then m+e0−t appears in E(6) or later. Since v3((σ2+1)αm+e0−t > v3(2α),some m+ e0 − t spill over into C(7). Consequently, whenever a pair (m,m+ e0 − t) has min C(2), E(2) while m+ e0 − t is in C(6), E(6) we get a copy of H1L ⊕ R3.

Consider cases D and F (including F ). Consider D first. Since v3(2αm) <v3((σ2 + 1)αm+e0−t), for m in D(2), m+e0 −t lands in D(6) or later. Note since v3(2αm) >v3(ρm+e0−t), some m+ e0 − t land in D(6). Since v3((σ2 + 1)αm+e0−t) > v3(2(σ2 + 1)αm),the collection of m + e0 − t overlap into D(8). When m + e0 − t is in D(8), the element(1/2)(σ2 + 1)αm+e0−t is available and we replace ρm by ρm − (1/2)(σ2 + 1)αm+e0−t =(σ+1)αm. For each pair (m,m+ e0 − t) such that m is associated with D(2) and m+ e0 − tis associated with D(6), we get a copy of H1L ⊕ R3. What we are principally concernedwith is what happens when for m in D(2), m + e0 − t is in D(7). In this case, becauseρm+e0−t = (σ + 1)αm+e0−t + (σ2 + 1)αm+e0−2t, there is some new interaction to consider.

Suppose m is in D(2), while m + e0 − t is in D(7). Since v3(ρm+e0−t) ≤ v3(αm+e0−2t)and v3(2(σ + 1)(σ2 + 1)αm) ≤ v3((σ + 1)(σ2 + 1)αm+e0−2t), for m in D(2) and m+ e0 − tin D(7), we find m + e0 − 2t is associated with D(4), or D(5) or later. Consider m inD(2), m + e0 − t in D(7), and m + e0 − 2t in D(4). Perform change of basis: Replace2αm with 2αm + 2αm+e0−t − 2αm+e0−t, ρm with ρm − αm+e0−t, (σ2 + 1)αm with (σ2 + 1)αm+(σ2+1)αm+e0−2t+1/2(σ−1)(σ2+1)αm+e0−t, and (σ+1)(σ2+1)αm with (σ+1)(σ2+1)αm + (σ + 1)(σ2 + 1)αm+e0−2t. The effect of these base changes upon the Galois actionis the same as if we were to replace ρm with (σ + 1)αm − (1/2)(σ + 1)(σ2 + 1)αm+e0−2t.Notice the similarity between this expression and the expression for ρm used when b2 = 3b1.Consequently, this scenario results in copies of H1G ⊕ R3. (Note if b2 = 3b1, then 2t = b1.)

In the alternative situation, whenm is inD(2),m+e0−t inD(7), andm+e0−2t is inD(5)or later, we perform the same basis changes. Except, since the element (1/2)(σ+1)(σ2 +1)αm+e0−2t is available, we replace ρm with ρm − αm+e0−t + (1/2)(σ + 1)(σ2 + 1)αm+e0−2t.The effect of this alternative basis change upon the Galois action is the same as a simplereplacement of ρm with (σ + 1)αm. We now turn our attention to Cases F and F . Since0 < 2b1, v3(ρm) < v3((1/2)(σ+1)(σ2+1)αm+e0−t. So for m associated with F (2), m+e0−tis associated with F (6) or later. We leave it to the reader to check that m+ e0 − t lands inF (6) or F (7). If m+ e0 − t is associated with F (7), then m+ e0 − 2t lands in F (4) or F (5).In any case, all this is analogous to D.

3.4. The Galois module structure under stable ramification. For p = 2, stableramification b1 ≥ e0 is nearly strong ramification b1 > (1/2) · pe0/(p − 1), (the conditionsdiffer only when e0 is odd – K0 tame over Q2). In [?], the structure of the ring of integers wasdetermined under strong ramification for any prime p. We revisit that argument extendingit to ambiguous ideals and the case b1 = e0.

GALOIS STRUCTURE 83

Following §2.1, Pi/22 /P

i/41

∼=(OT [σ]/〈σ2 + 1〉

)e0 . So e0 elements generate Pi/22 /P

i/41

over OT [G]. Use Lemmas 3.6, 3.7 to select elements, α, with odd valuation a such thati/2 ≤ a ≤ i/2 + 2e0 − 1. Each of these e0 elements gives rise (via the action of (σ ± 1))to another element, ρ in K2, with odd valuation, a + (b2 − b1) = a + 2e0. These α alongwith their Galois translates, ρ ≡ (σ ± 1)α mod P

i/41 , have valuations in one–to–one

correspondence (via v2) with the odd integers in i/2, . . . , 4e0 + i/2 − 1, and as a resultserve as a OT -basis for P

i/22 /P

i/41 . The α provide a OT [G]/〈Tr2,1〉-basis.

We need this basis for Pi/22 /P

i/41 to be compatible with our OT -basis for P

i/41

(as determined as in §2.2.1), as well as our OT [G]/〈Tr3,2〉-basis for Pi3/P

i/22 . First we

consider compatibility with Pi/41 . The OT -basis for P

i/41 consists of pairs: either ((σ +

1)η, η) or ((σ+1)η, 2η) ∈ K0×K1 where v1(η) is odd. Because of Lemma 2.1 each coordinateuniquely determines the other. Now consider pairs where the valuation v3 of both elementsis bound between i and 8e0 + i − 1. For example, pairs of the form ((σ + 1)η, η) appearfor i/4 ≤ v1(η) ≤ 2e0 + i/4 − b1 − 1, while pairs of the form ((σ + 1)η, 2η) appear fori/4 − b1 ≤ v1(η) ≤ i/4 − 1. The coordinates of all pairs provides us with an OT basisfor P

i/41 . Each α with v2((σ2 + 1)α) ≤ 4e0 + i/2 − 1 determines (via (σ2 + 1)α ∈ K1)

a pair of elements in the OT -basis for Pi/41 . If v1((σ2 + 1)α) is odd, then α determines

a pair of the form ((σ + 1)η, 2η). If even, it determines a pair of the form ((σ + 1)η, η). Ingeneral for α with v2((σ2 + 1)α) ≥ 4e0 + i/2, v2(1/2(σ2 + 1)α) ≥ i/2. So 1/2(σ2 + 1)αis available and we may replace α in by α − 1/2(σ2 + 1)α and still have a basis. Note(σ2+1)

(α− 1/2(σ2 + 1)α

)= 0. So we can assume, without loss of generality, (σ2+1)α = 0.

This posses no complication, unless (σ ± 1)α = µ + ρ with ρ in the image of Tr3,2Pi3. In

other words, v2(ρ) ≥ (b3 + i+ 1)/2. (Note for α with v2((σ2 + 1)α) ≤ 4e0 + i/2 − 1 and(σ±1)α = µ+ρ, we have v2(ρ) < (b3 + i+1)/2.) For these α (actually α−1/2(σ2 +1)α),µ (actually µ − (σ ± 1)1/2(σ2 + 1)α) will determine a pair ((σ + 1)η, 2η) or ((σ + 1)η, η)in our OT -basis for P

i/41 . We need simply to show µ and µ − (σ ± 1)1/2(σ2 + 1)α have

the same properties. We leave it to the reader to do this (use Lemma 3.6 and 3.7 to showthat the valuations are the same, that µ− (σ± 1)1/2(σ2 + 1)α ∈ K0 if and only if µ ∈ K0).The only issue that remains is whether there can be any conflict between a pair of basiselements for P

i/41 determined directly, via (σ2 + 1)α, and a pair determined indirectly via

µ = (σ ± 1)α − ρ. Note any element in the image of the trace, Tr2,1, has valuation thatis larger than the valuation of every µ ∈ K1 that arises from the expression for a Galoistranslate ρ = (σ ± 1)α− µ.

We select our OT [G]-basis for Pi3/P

i/22 now. There is one element X in our OT -basis

for Pi/22 for each valuation v2 in

(i+ b3 + 1)/2, . . . , 4e0 + i/2 − 1. (3.5)

The reader may check for v2(X) even, X = (σ2 + 1)α for some α in our OT [G]-basis forP

i/22 /P

i/41 . For v2(X) odd, since i/2+(b2−b1) < (i+b3+1)/2, X = ρ = (σ±1)α−µ

also for some α. Use Lem 2.1 to create elements X ∈ Pi3 such that Tr3,2X = X and

v3(X) = v3(X) − b3. Note the elements (σ2 + 1)α and µ (from each case) have expressionsin terms of our OT -basis for P

i/41 . These expressions depend solely upon the valuations

of (σ2 + 1)α and µ.Before we move on to our result, we should say something about our basis for Pi

3/Pi/22 .

Since OT [σ]/〈σ4 + 1〉 is a principal ideal domain, Pi3/P

i/22 is free over OT [σ]/〈σ4 + 1〉 of


rank e0. Given elements of K2 with valuation v2 listed in (3.5) we may use Lem 2.1(2) tofind elements, ρ ∈ Pi

3, whose images under the trace, Tr3,2, lie one–to–one correspondence(via valuation) with (3.5). Refer to this set of elements in Pi

3 as S. One can check b1 + (i+b3 + 1)/2 > 4e0 + i/2. Therefore (σ − 1)Tr3,2Pi

3 ⊆ 2Pi/22 . Since Tr3,2S is an OT -basis

for Tr3,2Pi3 ⊆ 2P

i/22 and σ acts trivially upon Tr3,2Pi

3 ⊆ 2Pi/22 we may use Lemma 2.2

and extend S to an OT [G]/〈σ4 + 1〉-basis for Pi3/P

i/22 .

At this point we may put the preceding discussion together with our work in §2.2.3 (thatdetermines the structure of P

i/22 ) and determine the Galois module structure of Pi

3. Weneed to express the image of S under the trace, Tr3,2, in terms of our OT [G]-basis for P

i/22 .

This is the same as a determination of the expression (in terms of Galois generators ofP

i/2q2 ) for each valuation in (3.5). First note under stable ramification, b2 > 4e0−2b1 so the

structure of Pi/22 is determined by the basis listed as Case B in §2.2.3. However it is more

convenient for us to use the basis listed as Case A in Appendix B. To translate between thetwo bases, note in the elements α, (σ + 1)α, α, (σ+ 1)α from §2.2.3 are referred to as α, ρ,(σ2+1)α, (σ+1)(σ2+1)α in §3.1 and then in Appendix B. So row B(1) in §2.2.3 correspondswith a pair of rows A(7) and A(8) in Appendix B. Moreover B(2) corresponds to rows A(1)and A(2), B(3) corresponds to A(3) and A(4), and B(4) corresponds to A(5) and A(6).

There are four types of expression with valuation listed in (3.5). If the valuation a satisfiesa− (b2 − 2b1) ≡ 0 mod 4 then a is the valuation of a Galois translate ρ where the differencebetween (σ ± 1)α and ρ is an element (σ + 1)µ ∈ K0 where µ is in the basis for P

i/41 .

Note each such a corresponds with the appearance of I2 in the OT [G] decomposition ofPi

3. Counting such a one finds the same count as in A(8). Note therefore A(7) counts thenumber of I that are not mapped to under the trace, Tr3,2, from Pi

3.Each valuation a satisfying a ≡ 0 mod 4 is the valuation of (σ2 +1)α = (σ+1)µ for some

α in the basis for Pi/22 and µ in the basis for P

i/41 P

i/80 . So each such a, corresponds with

the appearance of an H2. A count of such a equals the count in A(2). Note A(1) counts thenumber of H not interacted with. Each valuation a satisfying a−(b2 −2b1) ≡ 2 mod 4 is thevaluation of a Galois translate ρ where the difference between (σ± 1)α and ρ is an element2µ ∈ P

i/41 where (σ + 1)µ is in the basis for P

i/80 . Each such a, therefore corresponds with

the appearance of an M1. The count of such a is the same as the count for A(4). Thenumber of M that appear in Pi

3 is the same as the count for A(3). Finally each valuationa satisfying a ≡ 2 mod 4 is the valuation of (σ2 + 1)α = 2µ for some α in the basis forP

i/22 . Also (σ+1)µ is in the basis for P

i/80 , so each such a, therefore corresponds with the

appearance of an L3. The count of such a is the same as the count for A(6). Again, A(5)counts the number of L in Pi

3.Note the structure of Pi

3 under stable ramification is consistent with the structure of Pi3

under unstable ramification so long as b2 > 4e0 − 4b1/3.

Appendix A. The Modules

In this section we introduce twenty–three indecomposable Z2[C8]-modules. It is left tothe interested reader to translate our notation into Yakovlev’s [?].

Irreducibles: Four of the Z2[C8]-modules are irreducible: R0, R1, R2, and R3 where Rn :=Z2[ζ2n ], ζ2n denotes a primitive 2n root of unity, and σ the generator of C8 acts via multi-plication by ζ2n .

The other nineteen modules are ‘compounds’. They are organized according to fixedpart – those fixed by σ2 are listed first, followed by those fixed by σ4, etc.

GALOIS STRUCTURE 85

Z2[C2]-modules: Besides the two irreducibles R0, R1, the group ring Z2[σ]/〈σ2〉 is the onlyother indecomposable module that is fixed by σ2.

Notation for ‘compounds’: The group ring, Z2[σ]/〈σ2〉, is made up of two irreducibles. Tomake the relationships between irreducibles and their ‘compounds’ explicit, we will usediagrams like

R1 → 1 ∈ R0

(instead of Z2[σ]/〈σ2〉). These diagrams are to be interpreted as follows: The number ofZ2[σ]-generators is the number of irreducible modules that appear in the diagram. Forexample, R1 → 1 ∈ R0 means two generators. Let us call them c and d. (Think : c generatesR1 while d generates R0.) Relations determine the module. If there is no ‘arrow’ leavingan irreducible Ri, then the trace Φ2i(σ) maps the generator to zero. So Φ20(σ)d = 0. NoteΦ2i(x) denotes the cyclotomic polynomial and x8 − 1 = Φ20(x) · Φ21(x) · Φ22(x) · Φ23(x). Ifthere is an ‘arrow’ leaving an irreducible Ri (pointing to an element), then the trace Φ2i(σ)maps the generator to that element. In this case Φ21(σ)c = 1 · d.

Z2[C4]-modules: There are three indecomposable modules fixed by σ4 (yet not fixed by σ2).Notation for two other decomposable modules is included as it will be needed to describecertain modules later (those not fixed by σ4). For three (of these five), the submodule fixedby σ2 is the group ring Z2[σ]/〈σ2〉 (note how their diagams include R1 → 1 ∈ R0):

(G) : R2 → 1 ∈ R1 → 1 ∈ R0, (H) :R2

R1 → 1 ∈ R0

, (I) : R2 ⊕ (R1 → 1 ∈ R0).

Denote the three generators by b, c, d. (Think: generating R2,R1,R0, respectively.) Recall(σ − 1)d = 0 while (σ + 1)c = d. In G, we have Φ22(σ)b = 1 · c. So G is the group ringZ2[σ]/〈σ4〉. In H, we have Φ22(σ)b = 1 · d. While in I, Φ22(σ)b = 0.

For two (of these five), the submodule fixed by σ2 is the maximal order of Z2[σ]/〈σ2〉(note how R1 ⊕ R0 appears):

(L) : R2 →

1 ∈ R1⊕1 ∈ R0

, (M) : R2 ⊕ R1 ⊕ R0.

Denote the three generators by b, c, d where (σ − 1)d = 0 and (σ + 1)c = 0. In L, we haveΦ22(σ)b = 1 ·c+1 ·d. In M, we have Φ22(σ)b = 0. So M is the maximal order of Z2[σ]/〈σ4〉.

Z2[C8]-modules: The remaining fifteen indecomposable modules can now be listed. They arecollected according to submodule fixed by σ4.Fixed part G.

(G1) : R3 → 1 ∈ R2 → 1 ∈ R1 → 1 ∈ R0 (G3) :R3

R2 → 1 ∈ R1 → 1 ∈ R0

(G2) : R3 → λ ∈ R2 → 1 ∈ R1 → 1 ∈ R0 (G4) :R3

R2 → 1 ∈ R1 → 1 ∈ R0

Call the generators a, b, c, d, where the Z2[σ]-relations among b, c, d are as in G. In G1, wehave Φ23(σ)a = 1 · b. So G1 is the group ring Z2[σ]. In G2, we have Φ23(σ)a = λ · b whereλ = σ − 1. In G3, Φ23(σ)a = 1 · c. In G4, Φ23(σ)a = 1 · d.


Fixed part H.

(H1) : R3 →λ ∈ R2⊕ 1 ∈ R1 → 1 ∈ R0

(H2) :

R3

R2 → 1 ∈ R0

R1

Call the generators a, b, c, d, where the Z2[σ]-relationships among b, c, d are as in H. In H1,Φ23(σ)a = λ · 1 · b+ 1 · c. In H2, Φ23(σ)a = d.Fixed part I.

(I1) : R3 →

1 ∈ R2⊕1 ∈ R1 → 1 ∈ R0

(I2) : R3 →R1 →

1 ∈ R2⊕1 ∈ R0

Each module is generated by a, b, c, d, where the Z2[σ]-relationships among b, c, d are as inI. In I1, Φ23(σ)a = 1 · b+ 1 · c. In I2, Φ23(σ)a = 1 · b+ 1 · d.Fixed part L or M.

(L1) : R3 → 1 ∈ R2 →

1 ∈ R1⊕1 ∈ R0

(L3) :

R3

R2 →

1 ∈ R1⊕1 ∈ R0

(L2) : R3 →λ ∈ R2 →

1 ∈ R1⊕1 ∈ R0

(M1) : R3 →

λ ∈ R2⊕1 ∈ R1⊕1 ∈ R0

The generators are a, b, c, d, where the Z2[σ]-relationships among b, c, d are as in L or Mrespectively. In L1, Φ23(σ)a = b. In L2, Φ23(σ)a = λ · b. In L3, Φ23(σ)a = 1 · c + 1 · d. InM1, Φ23(σ)a = 1 · b+ 1 · c+ 1 · d.Hybrids of H1. The next three modules result from the linking of an H1 with either anotherR3, or with a G, or with a L.

(H1,2) :

R3

R3 →

1 ∈ R1 → 1 ∈ R0⊕ λ ∈ R2

This module is generated by a1, a2, b, c, d with the Z2[σ]-relationships among b, c, d as in H,while Φ23(σ)a1 = λ · b+ 1 · c and Φ23(σ)a2 = d. If Φ23(σ)a1 = 0, H2 would decompose off.If Φ23(σ)a2 = 0, H1 would decompose off. It is a mixture of H1 and H2, hence the name.

(H1G) :R3 →

1 ∈ R1 → 1 ∈ R0⊕λ ∈ R2 →⊕

R2 → 1 ∈ R1 → 1 ∈ R0

GALOIS STRUCTURE 87

This module is generated by a1, b1, c1, d1 and b2, c2, d2. The Z2[σ]-relationships amongb2, c2, d2 are as in G. The Z2[σ]-relationships among a1, c1, d1 are as in H1 with (σ2 +1)b1 =1 · d1 + 1 · d2.

(H1L) :

R3 →

1 ∈ R1 → 1 ∈ R0⊕λ ∈ R2 →⊕

R2 →

1 ∈ R0⊕1 ∈ R1

This module is generated by a1, b1, c1, d1 and b2, c2, d2. The Z2[σ]-relationships amongb2, c2, d2 are as in L. The Z2[σ]-relationships among a1, c1, d1 are as in H1 with (σ2 +1)b1 =1 · d1 + (1 · c2 + 1 · d2).

Appendix B. The Bases by Case, A through H

From §3.4, we inherit sequences of elements ordered in terms of increasing valu-ation (for Case A, we have . . . ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, 2α, 4α, (σ + 1)(σ2 + 1)α,2(σ + 1)(σ2 + 1)α, 2ρ, . . .). Following §2.2.3, we are interested in those elements ‘in view’(i.e. with valuation in i, i+ 1, . . . , i+ v3(2) − 1). As we vary m the ‘view’ changes. Indeed,for each case, there are eight views (eight sets). They are listed below. Recall from §2.2.3it is easy to determine the subscripts m associated with a particular ‘view’. For example,the elements in A(2) appear for i ≤ v3

((σ + 1)(σ2 + 1)α

)and v3

((σ + 1)(σ2 + 1)α

)≤

8e0 + i− 1. In other words, (i+ b3 − 4b1 − 4b2)/8 ≤ m ≤ (i+ 8e0 − 4b1 − 4b2)/8 − 1.

Case A

(1) ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, 2α, 4α, (σ + 1)(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α

(2) (σ + 1)(σ2 + 1)α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, 2α, 4α, (σ + 1)(σ2 + 1)α

(3)12(σ + 1)(σ2 + 1)α, (σ + 1)(σ2 + 1)α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, 2α, 4α

(4) 2α,12(σ + 1)(σ2 + 1)α, (σ + 1)(σ2 + 1)α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, 2α

(5) α, 2α,12(σ + 1)(σ2 + 1)α, (σ + 1)(σ2 + 1)α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α

(6) (σ2 + 1)α, α, 2α,12(σ + 1)(σ2 + 1)α, (σ + 1)(σ2 + 1)α, ρ, 2ρ, (σ2 + 1)α

(7)12(σ2 + 1)α, (σ2 + 1)α, α, 2α,

12(σ + 1)(σ2 + 1)α, (σ + 1)(σ2 + 1)α, ρ, 2ρ

(8) ρ,12(σ2 + 1)α, (σ2 + 1)α, α, 2α,

12(σ + 1)(σ2 + 1)α, (σ + 1)(σ2 + 1)α, ρ


Case B

(1) ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, 4α, 2(σ + 1)(σ2 + 1)α

(2) (σ + 1)(σ2 + 1)α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, 4α

(3) 2α, (σ + 1)(σ2 + 1)α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α

(4)12(σ + 1)(σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, 2α

(5) α,12(σ + 1)(σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α

(6) (σ2 + 1)α, α,12(σ + 1)(σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, ρ, 2ρ, (σ2 + 1)α

(7)12(σ2 + 1)α, (σ2 + 1)α, α,

12(σ + 1)(σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, ρ, 2ρ

(8) ρ,12(σ2 + 1)α, (σ2 + 1)α, α,

12(σ + 1)(σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, ρ

Case C

(1) ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α, 2(σ + 1)(σ2 + 1)α, 4α

(2) 2α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α, 2(σ + 1)(σ2 + 1)α

(3) (σ + 1)(σ2 + 1)α, 2α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α

(4) α, (σ + 1)(σ2 + 1)α, 2α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α, (σ + 1)(σ2 + 1)α

(5)12(σ + 1)(σ2 + 1)α, α, (σ + 1)(σ2 + 1)α, 2α, ρ, 2ρ, (σ2 + 1)α, 2(σ2 + 1)α

(6) (σ2 + 1)α,12(σ + 1)(σ2 + 1)α, α, (σ + 1)(σ2 + 1)α, 2α, ρ, 2ρ, (σ2 + 1)α

(7)12(σ2 + 1)α, (σ2 + 1)α,

12(σ + 1)(σ2 + 1)α, α, (σ + 1)(σ2 + 1)α, 2α, ρ, 2ρ

(8) ρ,12(σ2 + 1)α, (σ2 + 1)α,

12(σ + 1)(σ2 + 1)α, α, (σ + 1)(σ2 + 1)α, 2α, ρ

Case D

(1) ρ, (σ2 + 1)α, 2ρ, 2(σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α, 2(σ + 1)(σ2 + 1)α, 4α

(2) 2α, ρ, (σ2 + 1)α, 2ρ, 2(σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α, 2(σ + 1)(σ2 + 1)α

(3) (σ + 1)(σ2 + 1)α, 2α, ρ, (σ2 + 1)α, 2ρ, 2(σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α

(4) α, (σ + 1)(σ2 + 1)α, 2α, ρ, (σ2 + 1)α, 2ρ, 2(σ2 + 1)α, (σ + 1)(σ2 + 1)α

(5)12(σ + 1)(σ2 + 1)α, α, (σ + 1)(σ2 + 1)α, 2α, ρ, (σ2 + 1)α, 2ρ, 2(σ2 + 1)α

(6) (σ2 + 1)α,12(σ + 1)(σ2 + 1)α, α, (σ + 1)(σ2 + 1)α, 2α, ρ, (σ2 + 1)α, 2ρ

(7) ρ, (σ2 + 1)α,12(σ + 1)(σ2 + 1)α, α, (σ + 1)(σ2 + 1)α, 2α, ρ, (σ2 + 1)α

(8)12(σ2 + 1)α, ρ, (σ2 + 1)α,

12(σ + 1)(σ2 + 1)α, α, (σ + 1)(σ2 + 1)α, 2α, ρ

Case E

(1) 2α, 2ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α, 2α, 2ρ

(2) ρ, 2α, 2ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α, 2α

(3) α, ρ, 2α, 2ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α

GALOIS STRUCTURE 89

(4) (σ + 1)(σ2 + 1)α, α, ρ, 2α, 2ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2(σ2 + 1)α

(5) (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, 2α, 2ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α

(6)12(σ + 1)(σ2 + 1)α, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, 2α, 2ρ, (σ2 + 1)α

(7)12(σ2 + 1)α,

12(σ + 1)(σ2 + 1)α, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, 2α, 2ρ

(8) ρ,12(σ2 + 1)α,

12(σ + 1)(σ2 + 1)α, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, 2α

Case F

(1) 2α, (σ2 + 1)α, 2ρ, (σ + 1)(σ2 + 1)α, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α, 2α, 2ρ

(2) ρ, 2α, (σ2 + 1)α, 2ρ, (σ + 1)(σ2 + 1)α, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α, 2α

(3) α, ρ, 2α, (σ2 + 1)α, 2ρ, (σ + 1)(σ2 + 1)α, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α

(4) (σ + 1)(σ2 + 1)α, α, ρ, 2α, (σ2 + 1)α, 2ρ, (σ + 1)(σ2 + 1)α, 2(σ2 + 1)α

(5) (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, 2α, (σ2 + 1)α, 2ρ, (σ + 1)(σ2 + 1)α

(6)12(σ + 1)(σ2 + 1)α, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, 2α, (σ2 + 1)α, 2ρ

(7) ρ,12(σ + 1)(σ2 + 1)α, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, 2α, (σ2 + 1)α

(8)12(σ2 + 1)α, ρ,

12(σ + 1)(σ2 + 1)α, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, 2α

Case G

(1) (σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, 2ρ, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α, 2α, 2ρ

(2) ρ, (σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, 2ρ, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α, 2α

(3) α, ρ, (σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, 2ρ, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α

(4) (σ + 1)(σ2 + 1)α, α, ρ, (σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, 2ρ, 2(σ2 + 1)α

(5) (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, (σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α, 2ρ

(6) ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, (σ2 + 1)α, 2α, (σ + 1)(σ2 + 1)α

(7)12(σ + 1)(σ2 + 1)α, ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, (σ2 + 1)α, 2α

(8) α,12(σ + 1)(σ2 + 1)α, ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, (σ2 + 1)α

Case H

(1) (σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α, 2ρ, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α, 2α, 2ρ

(2) ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α, 2ρ, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α, 2α

(3) α, ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α, 2ρ, 2(σ2 + 1)α, 2(σ + 1)(σ2 + 1)α

(4) (σ + 1)(σ2 + 1)α, α, ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α, 2ρ, 2(σ2 + 1)α

(5) (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α, 2ρ

(6) ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, 2α

(7) α, ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α

(8)12(σ + 1)(σ2 + 1)α, α, ρ, (σ2 + 1)α, (σ + 1)(σ2 + 1)α, α, ρ, (σ2 + 1)α

Department of Mathematics, University of Nebraska at Omaha, Omaha, Nebraska 68132-0243


AN INTRODUCTION TO NONCOMMUTATIVE DEFORMATIONSOF MODULES

EIVIND ERIKSEN

Abstract. Let k be an algebraically closed (commutative) field, let A be an associativek-algebra, and let M = M1, . . . , Mp be a finite family of left A-modules. We studythe simultaneous formal deformations of this family, described by the noncommutativedeformation functor DefM : ap → Sets introduced in Laudal [8]. In particular, we provethat this deformation functor has a pro-representing hull, and describe how to calculatethis hull using the cohomology groups Extn

A(Mi, Mj) and their matric Massey products.

Introduction

In this paper, I shall give an elementary introduction to the noncommutative deformationtheory for modules, due to Laudal. This theory, which generalizes the classical deforma-tion theory for modules, was introduced by Laudal in [8]. Earlier versions of this materialappeared in the preprints Laudal [3], [4], [5], [6], [7].

This noncommutative deformation theory has several applications: In the paperLaudal [8], Laudal used it to construct algebras with a prescribed set of simple modules,and also to study the moduli space of iterated extensions of modules. In the preprintLaudal [7], he also showed that this theory is a useful tool in the study of algebras, and inestablishing a noncommutative algebraic geometry.

These applications are an important part of the motivation for the noncommutativedeformation theory. But we shall not go into the details of these applications in this elemen-tary introduction. Instead, we refer to the papers and preprints of Laudal mentioned abovefor applications and further developments of the theory.

Throughout this paper, we shall fix the following notations: Let k be an algebraicallyclosed (commutative) field, let A be an associative k-algebra, and let M = M1, . . . ,Mpbe a finite family of left A-modules. Notice that this notation differs from Laudal’s: WhileLaudal considers families of right modules in all his paper, I consider families of left modules.Of course, the difference is only in the appearance — the resulting theories are obvi-ously equivalent.

We shall present a noncommutative deformation functor DefM : ap → Sets, whichdescribes the simultaneous formal deformations of the family M of left A-modules. Further-more, we shall prove that this deformation functor has a pro-representable hull (H, ξ) whenthe family M satisfy a certain finiteness condition. We shall also describe a method forfinding the pro-representable hull explicitly.

In section 1, we describe the category ap. It is a full sub-category of the category Ap

of p-pointed k-algebras. The objects of Ap are the k-algebras R equipped with k-algebrahomomorphisms kp → R→ kp, such that the composition kp → kp is the identify. For any

This research has been supported by a Marie Curie Fellowship of the European Communityprogramme “Improving Human Research Potential and the Socio-economic Knowledge Base” under contract

number HPMF-CT-2000-01099.

NONCOMMUTATIVE DEFORMATIONS OF MODULES 91

such object, R = (Rij) is a k-algebra of p × p matrices. The radical of this object is theideal I(R) = ker(R → kp) ⊆ R. The category ap is the full sub-category of Ap consistingof objects such that R is Artinian and complete in the I(R)-adic topology.

In section 2, we describe the noncommutative deformation functor associated to thefamily M of left A-modules,

DefM : ap → Sets

It is constructed in the following way: Let R be an object of ap, and consider the vectorspace MR = (Mi ⊗k Rij), equipped with the natural right R-module structure induced bythe multiplication in R. A deformation of M to R consists of the following data:

• A left A-module structure on MR making MR a left A⊗k Rop-module,• Isomorphisms ηi : MR ⊗R ki →Mi of left A-modules for 1 ≤ i ≤ p.

The set of equivalence classes of such deformations is denoted DefM(R), and this definesthe covariant functor DefM. Notice that the fact that

MR∼= (Mi ⊗k Rij)

as right R-modules replaces the flatness condition in classical deformation theory. If p = 1and R is commutative, the above condition is of course equivalent to the flatness condition,so the noncommutative deformation functor generalizes the classical one.

In section 3, we look at noncommutative deformations from the point of view of resolu-tions. Let R be any object of ap. An M-free module over R is a left A⊗k Rop-module F ofthe form

F = (Li ⊗Rij),

where L1, . . . , Lp are free left A-modules. M-free complexes and M-free resolutions aredefined similarly. Let us fix a free resolution of Mi the form

0←Mi ← L0,i ← · · · ← Lm,i ← · · ·

for 1 ≤ i ≤ p. We prove that there is a bijective correspondence between deformations ofM to R and complexes of M-free modules over R of the form

(L0,i ⊗k Rij)← · · · ← (Lm,i ⊗k Rij)← · · ·

In fact, each such complex of M-free modules is an M-free resolution of the correspondingdeformation MR of M to R.

In section 4, we recall some general facts about pointed functors and their representability.In section 5, we consider the special case of the noncommutative deformation functor DefM.From this point in the text, we assume that the family M satisfy the finiteness condition

(FC) dimk ExtnA(Mi,Mj) is finite for 1 ≤ i, j ≤ p, n = 1, 2.

When this condition holds, we define T1,T2 to be the formal matrix rings (in the senseof section 1) given by the families of k-vector spaces Vij = ExtnA(Mj ,Mi)∗ for n = 1, 2.

92 EIVIND ERIKSEN

Assuming condition (FC), we show the following theorem of Laudal, which generalizes thecorresponding theorem for the classical deformation functor:

Theorem 0.1. There exists an obstruction morphism o : T2 → T1, such that H = T1⊗T2kp

is a pro-representable hull for the noncommutative deformation functor DefM : ap → Sets.

In the rest of the paper, we show how to construct the hull H explicitly, which can beaccomplished by using matric Massey products. In section 6, we introduce the immediatelydefined matric Massey products. In section 7, we define the matric Massey products ingeneral, and show that the hull H of the noncommutative deformation functor DefM isdetermined by the vector spaces ExtnA(Mi,Mj) for n = 1, 2 and 1 ≤ i, j ≤ p and theirmatric Massey products. We also describe a general method for calculating the hull H inconcrete terms.

In appendix A, we describe the Yoneda and Hochschild representations of the cohomologygroups ExtnA(Mi,Mj). In this paper, we have chosen to express the matric Massey productsusing the Yoneda representation and M-free resolutions. It is also possible to express thematric Massey products using the Hochschild representation, see for instance Laudal [8].

1. Categories of pointed algebras

Let p be a fixed natural number, and consider the ring kp. This ring has a natural k-algebra structure given by the map α → (α, . . . , α) for α ∈ k. Let pri : kp → kp be the i’thprojection, and consider the ideal ki = pri(kp) ⊆ kp as a kp-module for 1 ≤ i ≤ p. Clearly,kp is an Artinian k-algebra and k1, . . . , kp is the full set of isomorphism classes of simplekp-modules, each of them of dimension 1 over k. This simple example will serve as a modelfor the p-pointed algebras that we shall consider in this section.

A p-pointed k-algebra is a triple (R, f, g), where R is an associative ring and f : kp →R, g : R → kp are ring homomorphisms such that g f = id. A morphism u : (R, f, g) →(R′, f ′, g′) of p-pointed k-algebras is a ring homomorphism u : R→ R′ such that the naturaldiagrams commute (that is, such that u f = f ′ and g′ u = g). We shall denote thecategory of p-pointed k-algebras by Ap. Notice that if (R, f, g) is an object of Ap, then fis injective and g is surjective, and we shall identify kp with its image in R. We often writeR for the object (R, f, g) to simplify notation.

Let (R, f, g) be an object in Ap. We define the radical of R to be I(R) = ker(g), whichis an ideal in R. Furthermore, we denote by J(R) the Jacobson radical of R

J(R) = x ∈ R : xM = 0 for all simple left R-modules M,

which is also an ideal in R. We shall write I, J for the radicals I(R), J(R) when there is nodanger of confusion. Notice that the Jacobson radical J depends only on the ring R, whilethe radical I depends on the structural morphism g as well.

For all objects R in Ap, we have an inclusion J(R) ⊆ I(R): We have J(kp) = 0 sincekp is semi-simple, and g(J(R)) ⊆ J(kp) = 0 since g : R → kp is a surjection. In general,we know that R and R/J(R) have the same simple left modules. So if we consider ki asa left R-module via the morphism g : R → kp for 1 ≤ i ≤ p, we see that k1, . . . , kpis contained in the set of isomorphism classes of simple left R-modules, and the equalityJ(R) = I(R) holds if and only if k1, . . . , kp is the full set of isomorphism classes of simpleleft R-modules. Equivalently, the equality I(R) = J(R) holds if and only if there are exactlyp isomorphism classes of simple left R-modules.


It is therefore clear that the equality I(R) = J(R) does not hold in general: It is easyto find examples where R has ‘too many’ simple modules. For instance, consider R =k[x]/(x − x2) with the natural k-algebra structure f : k → R and let g : R → k be givenby x → 0. Then R is an object of A1, but J(R) = I(R) because R has two non-isomorphicsimple left R-modules (given by x → 0 and x → 1).

Let ei be the idempotent (0, 0, . . . , 1, . . . , 0) ∈ kp for 1 ≤ i ≤ p. Notice that eiej = 0 ifi = j, and that e1 + · · · + ep = 1. For any object R in Ap, we identify e1, . . . , ep withidempotents in R via the inclusion kp → R. Denote by Rij the k-linear sub-space eiRej ⊆ R.We immediately see, using the properties of the idempotents, that the following relationshold for 1 ≤ i, j, l,m ≤ p:

(1) RijRlm ⊆ δjlRim,(2) Rij ∩Rlm = 0 if (i, j) = (l,m),(3)

∑Rij = R.

In particular, we have that R = ⊕Rij , so every element r ∈ R may be written in matrixform r = (rij) with rij ∈ Rij for 1 ≤ i, j ≤ p. Furthermore, elements of R multiply asmatrices when we write them in this form. It is therefore reasonable to call an object R inAp a matrix ring, and to write it R = (Rij). Notice that Rii is an associative ring (withidentity ei), and that Rij is a (unitary) Rii − Rjj bimodule for 1 ≤ i, j ≤ p. For any idealK ⊆ R, we see that eiKej = K ∩ Rij , and we shall denote this k-linear subspace Kij for1 ≤ i, j ≤ p. Since K = ⊕Kij , we write K = (Kij).

Let R be an object of Ap, so R = (Rij) is a matrix ring in the above sense. The followingstandard result gives useful information on when R is an Artinian or Noetherian ring:

Proposition 1.1. Let R = (Rij) be an object in Ap. Then R is Noetherian (Artinian) ifand only if the following conditions hold:

i) Rii is Noetherian (Artinian) for 1 ≤ i ≤ p,ii) Rij is a Noetherian (Artinian) left Rii-module and a Noetherian (Artinian) right Rjj-

module for 1 ≤ i = j ≤ p.We recall that a finitely generated, associative k-algebra is not necessarily Noetherian.

That is, Hilbert’s basis theorem does not hold for associative rings. For a counter-example,let R = kx1, . . . , xn be the free associative k-algebra on n generators. It is well-knownthat R is Noetherian only if n = 1. However, we know from the Hopkins-Levitzki theoremthat an associative Artinian ring is Noetherian.

A k-algebra R of finite dimension as vector space over k is Artinian. This is clear, sinceevery one-sided ideal is a vector space over k of finite dimension. We have a conversestatement under the following conditions:

Lemma 1.2. Let R be an object of Ap. If R is Artinian and I(R) is nilpotent, then R hasfinite dimension as a vector space over k.

Proof. We write I = I(R). Since R is Artinian and therefore Noetherian, Im is finitely gener-ated as a left R-module for allm. Consequently, Im/Im+1 is a finitely generated R/I-modulefor all m, and hence has finite k-dimension. But In = 0 for some n, so Im has finite k-dimension for all m ≥ 0. In particular, R has finite dimension as a vector space over k.

We define the category ap to be the full sub-category of Ap consisting of objects R in Ap

such that R is Artinian and I(R) = J(R). The condition I(R) = J(R) might equivalentlybe replaced by the condition that I(R) is a nilpotent ideal, since the Jacobson radical is the

94 EIVIND ERIKSEN

largest nilpotent ideal in an Artinian ring. So by lemma 1.2, all objects R in ap have finitek-dimension. Since R is Artinian, the condition that I(R) is nilpotent is also equivalent to∩ I(R)n = 0. Finally, there is a geometric interpretation of the condition I(R) = J(R): Bythe comment earlier in this section, I(R) = J(R) if and only if k1, . . . , kp is the full setof isomorphism classes of simple left R-modules (or equivalently, that the number of suchisomorphism classes is exactly p).

Lemma 1.3. Let R be an associative ring. Then there exists morphisms f : kp → R andg : R → kp making (R, f, g) an object of ap if and only if R is an Artinian k-algebra withexactly p isomorphism classes of simple left R-modules, each of them of dimension 1 over k.

Proof. One implication follows from the comments above. For the other, assume that Ris Artinian with the prescribed isomorphism classes of simple left R-modules. This definesa morphism g : R → kp. Clearly, I = ker(g) = J(R) by the comments above. So it isenough to lift the idempotents e1, . . . , ep of kp to idempotents r1, . . . , rp in R such thatr1 + · · · + rp = 1 and rirj = 0 when i = j. But R is Artinian and therefore I = J(R) isnilpotent, so this is clearly possible.

Let R be an object in Ap with radical I = I(R). Then the I-adic filtration definesa topology on R compatible with the ring operations, and we shall always consider R atopological ring in this way. We say that the topology on R is Hausdorff (or separated) ifand only if ∩In = 0.

For all objects R in Ap, there is an I-adic completion R of R and a canonical morphismR→ R in Ap. The I-adic completion R is defined by the projective limit

R = lim← R/In,

and the morphism R → R is the natural one induced by this projective limit. Notice thatthe kernel of this morphism is ∩In. We say that R is complete (or separated complete) ifthe natural morphism R→ R is an isomorphism in Ap. In particular, this implies that themorphism is injective, so R is Hausdorff (or separated). This gives a new characterizationof the category ap:

Lemma 1.4. The category ap is the full sub-category of Ap consisting of objects such thatR is Artinian and I-adic complete.

We define the pro-category ap of ap to be the full sub-category of Ap consisting of objectssuch that R is complete and R/I(R)n belongs to ap for all n ≥ 1. It is clear that we havean inclusion of (full) sub-catgories ap ⊆ ap.

Let R be an object in ap with radical I = I(R). To fix notation, we write grn(R) =In/In+1 for n ≥ 0 (with I0 = R). We also write grR = ⊕ grn(R), this is the graded ringassociated to the I-adic filtration of R. The tangent space of R is defined to be the k-linearspace dual to gr1(R),

tR = Homk(I/I2, k) = (I/I2)∗,

which is clearly of finite dimension over k. In particular, we have (tR)∗ ∼= I/I2.Let u : R → S be a morphism in ap. As usual, we consider R and S with the I-adic

filtrations, where I is I(R) and I(S) respectively. Since u preserves these filtrations, itinduces a morphism of graded rings gr(u) : grR→ grS. This morphism is homogeneous ofdegree 0, so u also induces morphisms of k-vector spaces grn(u) : grn(R) → grn(S) for all


n ≥ 0. In particular, we have a morphism of k-vector spaces gr1(u) : gr1(R)→ gr1(S), anda dual morphism tu : tS → tR.

Proposition 1.5. Let u : R→ S be a morphism in ap. Then u is a surjection if and onlyif gr1(u) is a surjection. Furthermore, u is injective if gr(u) is injective.

Proof. If u is surjective, then clearly gr1(u) is also surjective. To prove the other implication,let us consider the map gr(u) : gr(R) → gr(S). Since grS is generated by the elements ingr1 S as an algebra, it follows that if gr1(u) is surjective, then gr(u) is also surjective. FromBourbaki [1], chapter III, §2, no. 8, corollary 1 and 2, we have that u is surjective (injective)if gr(u) is surjective (injective), and the result follows.

Let n be any natural number. We define the category ap(n) to be the full sub-categoryof ap consisting of objects R in ap such that I(R)n = 0. Notice that ap(n) ⊆ ap(n + 1)for all n ≥ 1. Furthermore, each object R in ap belongs to a sub-category ap(n) for someinteger n.

Let u : R → S be a morphism in ap, and denote by K = ker(u) the kernel of u. We saythat u is a small morphism if we have I(R) · K = K · I(R) = 0. We prove the followingimportant fact about small surjections:

Lemma 1.6. Let u : R → S be a surjection in ap. Then u can be factored into a finitenumber of small surjections.

Proof. Let I = I(R), then InK = 0 for some n ≥ 0. Consider the surjection uq : R/IqK →R/Iq−1K for 1 ≤ q ≤ n. Clearly I(R/IqK) ker(uq) = 0 for all q. Moreover, u1 · · · un = uwhen u1 : R/IK → R/K is considered as a morphism onto S ∼= R/K. It is therefore enoughto prove the lemma for a surjection u : R→ S with IK = 0. In this situation, KIn = 0 forsome n ≥ 0. Now consider the surjection vq : R/KIq → R/KIq−1 for 1 ≤ q ≤ n. Clearly,vq is a small surjection for all q. Moreover, u = v1 · · · vn when v1 : R/KI → R/K isconsidered as a morphism onto S ∼= R/K. It follows that u can be factorized in a finitenumber of small surjections in ap.

We conclude this section with an important family of examples: Let Vij be a finitedimensional k-vector space for 1 ≤ i, j ≤ p, with dimk Vij = dij . Let furthermore rij(l) :1 ≤ l ≤ dij be a basis of Vij for 1 ≤ i, j ≤ p (or simply rij if dij = 1). We define thefree matrix ring R = R(Vij) defined by the vector spaces Vij in the following way: Wesay that a monomial in R of type (i, j) and degree n is an expression of the form

ri0i1(l1)ri1i2(l2) . . . rin−1in(ln)

with i0 = i, in = j. To these, we add the monomials ei for 1 ≤ i ≤ p, which we considerto be of type (i, i) and degree 0. We define R to be the k-linear space generated by allmonomials in R, with the obvious multiplication: If M is a monomial of type (i, j), and M ′

is a monomial of type (l,m), then MM ′ = 0 if j = l, and MM ′ is the monomial obtained byjuxtapositioning M and M ′ (possibly after having erased unnecessary ei’s) if j = l. We seethat (R, f, g) is an object of the category Ap, where f, g are the obvious maps kp → R→ kp.In fact, Rij is the k-linear subspace generated by monomials in R of type (i, j), and theideal I = I(R) is the k-linear subspace generated by all monomials of positive degree.

We denote by R = R(Vij) the completion of R = R(Vij), and call this the formalmatrix ring defined by the vector spaces Vij . Explicitly, every element in Rij is an infinitek-linear sum of monomials in R of type (i, j). Let I = I(R), then we have that Rn = R/In ∼=

96 EIVIND ERIKSEN

R/I(R)n belongs to ap for n ≥ 1: Clearly, Rn has finite dimension as k-vector space, so Rnis Artinian, and I(Rn) = I/In, so the radical is nilpotent. Since R clearly is complete, itfollows that R belongs to ap.

Notice that neither the free matrix ring R nor the formal matrix ring R is Noetherian ingeneral. For a counter-example, it is enough to consider the case when p = 1 and d11 = 2, orthe case when p = 2 and d11 = d12 = d21 = 1, d22 = 0. In the first case, R ∼= kx, y, whichwe know is not Noetherian. In the second case, we have that R11 = kr11, r12r21 ∼= kx, y,which again is not Noetherian. So by proposition 1.1, R is not Noetherian in this case either.A similar argument shows that R is not Noetherian in any of the two cases.

2. Noncommutative deformations of modules

We recall that k is an algebraically closed (commutative) field, A is an associativek-algebra, and M = M1, . . . ,Mp is a finite family of left A-modules. In this section, weshall define the noncommutative deformation functor

DefM : ap → Sets

describing the simultaneous formal deformations of the family M.Let R be an object of ap. A lifting of the family M of left A-modules to R is a left

A⊗k Rop-module MR, together with isomorphisms ηi : MR ⊗R ki →Mi of left A-modulesfor 1 ≤ i ≤ p, such that MR

∼= (Mi ⊗k Rij) as right R-modules. We remark that a leftA⊗kRop-module is the same as an A-R bimodule such that the left and right k-vector spacestructures coincide. Furthermore, the notation (Mi ⊗k Rij) refers to the k-vector space

(Mi ⊗k Rij) = ⊕i,j

(Mi ⊗k Rij)

with the natural right R-module structure coming from the multiplication in R. The condi-tion that MR

∼= (Mi ⊗k Rij) as right R-modules generalizes the flatness condition incommutative deformation theory.

Let M ′R,M′′R be two liftings of M to R. We say that these two liftings are equivalent if

there exists an isomorphism τ : M ′R →M ′′R of left A⊗k Rop-modules such that the naturaldiagrams commute (that is, such that η′′i (τ ⊗R ki) = η′i for 1 ≤ i ≤ p). We let DefM(R)denote the set of equivalence classes of liftings of M to R, and we refer to these equivalenceclasses as deformations of M to R. We shall often denote a deformation represented by(MR, ηi) by MR to simplify notation.

Let u : R → S be a morphism in ap, and let MR be a lifting of M to R, representingan element in DefM(R). We define MS = MR ⊗R S, which has a natural structure asa left A ⊗k Sop-module. Since u is a morphism in ap, we have natural isomorphisms ofleft A-modules

(MR ⊗R S)⊗S ki ∼= MR ⊗R ki,

inducing isomorphisms of left A-modules ρi : MS ⊗S ki → Mi via ηi for 1 ≤ i ≤ p. Astraight-forward calculation shows that MS together with the isomorphisms ρi for 1 ≤ i ≤ pconstitutes a lifting of M to S, and furthermore that the equivalence class of this lifting isindependent upon the representative of the equivalence class of MR. Hence, we obtain a mapDefM(u) : DefM(R)→ DefM(S), and we see that DefM : ap → Sets is a covariant functor.


LetR = (Rij) be an object in ap. We shall describe how one, in principle, could attempt tocalculate DefM(R) explicitly: We may assume that every element of DefM(R) is representedby a lifting MR, such that MR = (Mi ⊗k Rij) considered as a right R-module. In orderto describe this lifting completely, it is enough to describe the left action of A on MR.Furthermore, it is enough to describe this action on elements of the form mi ⊗ ei withmi ∈Mi, since we have

a(mi ⊗ rij) = (a(mi ⊗ ei))rijfor all a ∈ A, mi ∈ Mi, rij ∈ Rij . For a fixed a ∈ A, mi ∈ Mi, assume that a(mi ⊗ ei) =∑

(m′j ⊗ r′jl) with m′j ∈ Mj , r′jl ∈ Rjl. Then multiplication by ei on the right givesthe equality

a(mi ⊗ ei) =∑j

(m′j ⊗ r′ji),

and the isomorphism ηi gives a further restriction on the left action of A, expressed bythe formula

(1) a(mi ⊗ ei) = (ami)⊗ ei +∑j

m′j ⊗ r′ji,

where a ∈ A, mi ∈Mi, m′j ∈Mj , r

′ji ∈ I(R)ji. Consequently, the set DefM(R) consists of

all possible choices of left A-actions on elements of the form mi⊗ ei, fulfilling condition (1)and the associativity condition, up to equivalence.

Let R be any object in ap. Then the formula a(mi ⊗ ei) = (ami) ⊗ ei for a ∈ A, mi ∈Mi defines a left A-module structure on (Mi ⊗ Rij) compatible with the right R-modulestructure. Hence, there exists a trivial lifting MR to R for all objects R in ap, and DefM(R)is non-empty. Notice that in the case R = kp, we have I = I(R) = 0, so this trivial liftingis the only one possible. Consequently, we have DefM(kp) = ∗, where ∗ denotes theequivalence class of the trivial lifting.

Let u : R → S be a morphism in ap, and let MS ∈ DefM(S) be a given deforma-tion. We say that a deformation MR ∈ DefM(R) is a lifting of MS or is lying over MS ifDefM(u)(MR) = MS . Given any object R in ap and a deformation MR ∈ DefM(R), we seethat MR is a lifting of the trivial deformation ∗ in DefM(kp) in the above sense via thestructural morphism g : R→ kp. Hence, our notation is consistent.

For another example, consider the test algebras R(α, β) for 1 ≤ α, β ≤ p, constructed inthe following way: Let R be the free matrix algebra defined by the k-vector spaces Vij withdimensions dα,β = 1 and dij = 0 when (i, j) = (α, β). We define R(α, β) = R/I(R)2, whichis an object in ap(2) by construction. We know that any lifting of M to R(α, β) is definedby a left A-action

a(mβ ⊗ eβ) = (amβ)⊗ eβ + ψ(a)(mβ)⊗ εα,βfor all a ∈ A, mβ ∈ Mβ , where ψ : A × Mβ → Mα is a k-bilinear map and εα,β isthe class of rα,β . Clearly, we must have a(mi ⊗ ei) = (ami) ⊗ ei for all a ∈ A,mi ∈ Mi

when i = β. Moreover, ψ defines an associative A-module structure if and only if ψ ∈Derk(A,Homk(Mβ ,Mα)). In this case, we shall denote the corresponding lifting by M(ψ) ∈DefM(R(α, β)). Given two derivations ψ,ψ′, we see that M(ψ) and M(ψ′) are equivalentliftings if and only if there is a φ ∈ Homk(Mβ ,Mα) such that

(ψ − ψ′)(a)(mβ) = aφ(mβ)− φ(amβ)

98 EIVIND ERIKSEN

for all a ∈ A, mβ ∈Mβ .

Lemma 2.1. There is a bijective correspondence DefM(R(α, β)) ∼= Ext1A(Mβ ,Mα) for 1 ≤α, β ≤ p.Proof. From the definition of Hochschild cohomology (see appendix A), we see thatψ → M(ψ) induces a bijective correspondence between HH1(A,Homk(Mβ ,Mα)) andDefM(R(α, β)). Moreover, HH1(A,Homk(Mβ ,Mα)) ∼= Ext1A(Mβ ,Mα) by proposition A.3.

3. M-free resolutions and noncommutative deformations

We recall that k is an algebraically closed (commutative) field, A is an associative k-algebra, and M = M1, . . . ,Mp is a finite family of left A-modules. In this section, we shalldefine M-free resolutions and relate them to noncommutative deformations of modules. Inparticular, we shall show that M-free resolutions are useful computational tools in order tostudy the deformation functor DefM.

Let R be any object of ap. An M-free module over R is a left A ⊗k Rop-module F ofthe form

F = (Li ⊗k Rij),where L1, . . . , Lp are free left A-modules, and the left A-module structure on F is the trivialone. In other words, F is the trivial lifting of a family L1, . . . , Lp of free left A-modulesto R.

Although an M-free module over R is not free considered as a left A ⊗k Rop-module, itbehaves as a free module when interpreted as a module of matrices in the correct way:

Lemma 3.1. Let u : R → S be a surjection in ap, and consider a left A ⊗k Rop-moduleMR = (Mi ⊗k Rij) and a left A ⊗k Sop-module MS = (Mi ⊗k Sij) such that the naturalmap v : MR → MS induced by u is left A-linear. If FS is any M-free module over S givenby the free left A-modules L1, . . . , Lp and fS : FS → MS is any left A ⊗k Sop-linear map,then there exists a left A⊗k Rop-linear map fR : FR →MR making the diagram

MR

v

FR

(id⊗u)

fR

MS FSfS

commutative, where FR is the M-free module over R given by the free left A-modulesL1, . . . , Lp.

Proof. Clearly, the map fS is determined by its values on Li ⊗ ei, and therefore by thecorresponding left A-linear maps Lj → ⊕(Mi ⊗k Sij). Since each left A-module Lj isprojective, we can lift these maps to left A-linear maps Lj → ⊕(Mi⊗kRij), and these mapsdetermine fR.

Let R be any object of ap, and let MR = (Mi ⊗k Rij) ∈ DefM(R) be a lifting of M toR. An M-free resolution of MR is an exact sequence of left A⊗k Rop-linear maps

0←MR ← FR0 ← FR1 ← · · · ← FRm ← · · ·


where FRm is an M-free module over R for m ≥ 0. So we have FRm = (Lm,i ⊗k Rij) whereLm,i are free left A-modules for 1 ≤ i ≤ p, m ≥ 0. We shall denote the differentials bydRm : FRm+1 → FRm for m ≥ 0.

We fix a k-linear basis rij(l) : 1 ≤ l ≤ dimk Rij of Rij for 1 ≤ i, j ≤ p such that eiis contained in the basis of Rii for 1 ≤ i ≤ p. Consider the differential dRm in the M-freeresolution of MR above. Clearly, we can write dRm uniquely in the form

(2) dRm =∑i,j,l

α(rij(l))m ⊗ rij(l)

for all m ≥ 0, where α(rij(l))m : Lm+1,j → Lm,i is a homomorphism of left A-modules for1 ≤ i, j ≤ p, 1 ≤ l ≤ dimk Rij . In particular, the M-free resolution of MR defines a familyof 1-cochains α(rij(l)) ∈ Hom1(L∗j , L∗i), indexed by a k-linear basis for R.

From now on, we fix a free resolution (L∗i, d∗i) of Mi considered as left A-module for1 ≤ i ≤ p. These free resolutions correspond to an M-free resolution (F∗, d∗) of the trivialdeformation (Mi ⊗k (kp)ij) ∈ DefM(kp). In fact, the M-free resolution (F∗, d∗) is given byFm = (Lm,i ⊗k (kp)ij) and dm =

∑dm,i ⊗ ei for m ≥ 0. We have therefore fixed an M-free

resolution (F∗, d∗) of the trivial lifting of M to kp.Let R be any object of ap. We say that a complex (FR∗ , d

R∗ ) of M-free modules FRm =

(Lm,i ⊗k Rij) over R is a lifting of the complex (F∗, d∗) if the following diagram commutes

FR0

v0

FR1dR0

v1

FR2dR1

v2

. . .

F0 F1d0

F2d1

. . .

where vm : FRm → Fm are the natural maps induced by R→ kp.

Lemma 3.2. Let R be any object of ap, and let (FR∗ , dR∗ ) be a lifting of the complex (F∗, d∗).

Then we have:

(1) Hm(FR∗ , dR∗ ) = 0 for all m ≥ 0,

(2) H0(FR∗ , dR∗ ) is a lifting of the family M to R.

Proof. Clearly, the lemma holds for R = kp. We shall consider a small surjection u : R→ Sin ap and liftings of complexes (FU∗ , d

U∗ ) of (F∗, d∗) to U for U = R,S such that the following

diagram commutes:

FR0

v0

FR1dR0

v1

FR2dR1

v2

. . .

FS0 FS1dS0

FS2dS1

. . .

In this situation, we shall prove that if the conclusion of the lemma holds for S, it holds forR as well. This is clearly enough to prove the lemma.

Let K = ker(u), then we clearly have ker(vm) = (Fm,i ⊗k Kij) with the trivial leftA-action for all m ≥ 0. We denote this kernel by FKm , then (FK∗ , d

K∗ ) is a complex of

100 EIVIND ERIKSEN

left A ⊗k Rop-modules, where dK∗ is the restriction of dR∗ . Moreover, it is clear that vm issurjective for m ≥ 0. Define MU = H0(FU∗ , d

U∗ ) for U = R,S, let v : MR → MS be the

induced map, and denote the kernel by MK = ker(v). Then clearly v is surjective, and wehave the following commutative diagram of complexes:

0 0 0 0

0 MK

i

FK0ρK

i0

FK1dK0

i1

FK2dK1

i2

. . .

0 MR

v

FR0ρR

v0

FR1dR0

v1

FR2dR1

v2

. . .

0 MS FS0ρS

FS1dS0

FS2dS1

. . .

0 0 0 0

Clearly all columns are exact, so the diagram gives a short exact sequence of complexes.By assumption, the bottom row is exact and MS = (Mi ⊗k Sij) is a lifting of M to S. Letus first show that Hm(FK∗ , d

K∗ ) = 0 for m ≥ 1: This follows since the complex is a lifting

of (F∗, d∗) and because I(R)K = 0 (since u : R → S is small). The long exact sequence ofcohomologies of the complexes above now implies that Hm(FR∗ , d

R∗ ) = 0 for all m ≥ 1 and

that we have a short exact sequence

0→ H0(FK∗ , dK∗ )→MR →MS = (Mi ⊗k Sij)→ 0,

of left A-modules, so in particular MK∼= H0(FK∗ , d

K∗ ). But since I(R)K = 0, it follows

that H0(FK∗ , dK∗ ) ∼= (H0(L∗,i, d∗,i) ⊗k Kij) = (Mi ⊗k Kij) with the trivial left A-module

structure. It follows that MR∼= (Mi ⊗k Rij) considered as a k-vector space, and therefore

MR is a lifting of M to R.

Lemma 3.3. Let R be any object of ap, and let MR be a lifting of M to R. Then thereexists an M-free resolution of MR which lifts the complex (F∗, d∗) to R.

Proof. Clearly, the lemma holds for R = kp. We shall consider a small surjection u : R→ Sin ap, deformations MU ∈ DefM(U) for U = R,S such that MR lifts MS to R, and anM-free resolution (FS∗ , d

S∗ ) of MS which lifts the complex (F∗, d∗) to S. In this situation,

we shall prove that there exists an M-free resolution (FR∗ , dR∗ ) of MR compatible with the

M-free resolution of MS . This is clearly enough to prove the lemma.Let FRm = (Lm,i ⊗k Rij) for all m ≥ 0. Moreover, we write FKm = (Lm,i ⊗k Kij) for all

m ≥ 0, where K = ker(u). To complete the proof, we have to find the differentials dRm form ≥ 0 and the augmentation map ρR: By lemma 3.1, we can find a homomorphism ρR :FR0 → MR lifting ρS . Denote by ρK : FK0 → MK its restriction, where MK = ker(MR →MS). Since u is small, ρK is surjective, and this implies that the induced map ker(ρR) →ker(ρS) is surjective. By lemma 3.1, we can find a homomorphism dR0 : FR1 → FR0 lifting dS0such that ρRdR0 = 0. Let dK0 be the restriction of dR0 , then clearly ker(ρK) = Im(dK0 ) since


u is small. An easy induction argument shows that we can construct a complex (FR∗ , dR∗ )

lifting the complex (FS∗ , dS∗ ) in such a way that the restriction (FK∗ , d

K∗ ) is a resolution of

MK . By the proof of lemma 3.2, it follows that Hm(FR∗ , dR∗ ) = 0 for m ≥ 1 and that there

is an exact sequence

0→MK → H0(FR∗ , dR∗ )→MS → 0.

This implies that MR = H0(FR∗ , dR∗ ), and (FR∗ , d

R∗ ) is the required M-free resolution of MR

compatible with the given M-free resolution of MS .

Proposition 3.4. Let u : R→ S be a surjection in ap, and consider a deformation MS ∈DefM(S) and any M-free resolution (FS∗ , d

S∗ ) of MS which lifts the complex (F∗, d∗) to S.

There is a bijective correspondence between the set of liftings

MR ∈ DefM(R) : DefM(u)(MR) = MS

and the set of M-free complexes (FR∗ , dR∗ ) which lift the resolution (FS∗ , d

S∗ ) to R, up to

equivalence.

Proof. For a small surjection, this follows from lemma 3.2 and lemma 3.3. But any surjectionin ap is a composition of small surjections.

Let R be any object in ap. In section 2, we described how to, in principle, calculateDefM(R) by considering the possible left A-module structures on the right R-module (Mi⊗kRij). The M-free resolutions give us another way of viewing deformations in DefM(R): Byproposition 3.4, we can view DefM(R) as the set of liftings of the complex (F∗, d∗) to R,up to equivalence. Using equation 2, each lifting of complexes corresponds to a family of1-cochains α(rij(l)) ∈ Hom1(L∗j , L∗i), parametrized by a k-basis for R. We leave it as anexercise for the reader to use this approach to calculate DefM(R) in the case R = Rα,β —this will give a new proof of lemma 2.1 via the Yoneda representation of Ext1A(Mβ ,Mα).

4. Pro-representing hulls of pointed functors

We say that a covariant functor F : ap → Sets is pointed if F(kp) = ∗. In this section,we shall consider pointed functors defined on the category ap, and study their representabil-ity. Of course, the motivation for this is the fact that DefM is such a pointed functor.

Let R be any object of ap, and consider the functor hR : ap → Sets given by hR(S) =Mor(R,S) for all objects S in ap. The notation Mor(R,S) denotes the set of morphismsfrom R to S in the pro-category ap. Then hR is clearly a pointed functor defined on ap.

We say that a pointed functor F : ap → Sets is representable is F is isomorphic to hRfor some object R in ap, and pro-representable if F is isomorphic to hR for some objectR in ap. However, it is well-known that deformation functors seldom are representable oreven pro-representable. So a weaker notion is required, and we shall define the notion of apro-representing hull of a pointed functor on ap. We start by introducing some notation:

Any pointed functor F : ap → Sets has an extension to a functor F : ap → Sets definedon the pro-category ap. This extension is defined by the formula

F(R) = lim← F(R/In)

102 EIVIND ERIKSEN

for any object R in ap with I = I(R). Clearly, any pointed functor F : ap → Sets also hasa restriction to the sub-category ap(n) ⊆ ap for all n ≥ 1. We shall denote this restrictionby Fn : ap(n)→ Sets.

Lemma 4.1. Let R be an object in ap, and let F : ap → Sets be a pointed functor. Thenthere is a natural isomorphism of sets α : F(R)→ Mor(hR,F).

Proof. Let ξ ∈ F(R), then ξ = (ξn) with ξn ∈ F(R/In) for all n ≥ 1. For any object S inap, we construct a map of sets α(ξ)S : Mor(R,S) → F(S): Let u : R → S be a morphismin ap, then u(I(R)) ⊆ I(S), and I(S) is nilpotent since S is in ar, so there exists n ≥ 1such that u factorizes through un : R/I(R)n → S. We define α(ξ)S(u) = F(un)(ξn), anda straight-forward calculation shows that this expression is independent upon the choiceof n, and gives rise to a natural transformation of functors. Conversely, let φ : hR → Fbe a natural transformation of functors on ap. Then we define ξn ∈ F(R/I(R)n) to beξn = φR/I(R)n(R → R/I(R)n), where R → R/I(R)n is the natural morphism. Again, astraight-forward calculation shows that ξ = (ξn) defines an element in F(R), and that thismap of sets defines an inverse to α.

There is also a version of lemma 4.1 for the category ap(n): For an object R in ap(n),and a pointed functor F : ap(n) → Sets, there is a natural isomorphism of sets αn :F(R)→ Mor(hR,F). The construction of this isomorphism is similar to the construction inlemma 4.1.

We recall that a morphism φ : F → G of pointed functors F,G : ap → Sets is smoothif the following condition holds: For all surjective morphisms u : R → S in ap, the naturalmap of sets

(3) F(R)→ F(S) ×G(S)

G(R),

given by x → (F(u)(x), φR(x)) for all x ∈ F(R), is a surjection. Clearly, it is enough tocheck this for small surjections in ap. Also notice that any morphism φ : F→ G of functorsnaturally extends to a morphism φ : F → G of functors on ap, and if φ is a smoothmorphism, then φR : F(R)→ G(R) is surjective for all objects R in ap.

Similarly, we say that a morphism φ : F→ G of functors F,G : ap(n)→ Sets on ap(n) issmooth if the map of sets (3) is surjective for all surjective morphisms u : R→ S in ap(n).Clearly, a morphism φ : F → G of functors on ap is smooth if and only if the restrictionφn : Fn → Gn is smooth for all n ≥ 1.

Let F be a pointed functor on ap. A pro-couple for F is a pair (R, ξ), where R is anobject in ap and ξ ∈ F(R). A morphism u : (R, ξ) → (R′, ξ′) of pro-couples is a morphismu : R → R′ in ap such that F(u)(ξ) = ξ′. If (R, ξ) is a pro-couple for F such that R is alsoan object of ap, then it is called a couple for F.

We say that a pro-couple (R, ξ) pro-represents F if α(ξ) : hR → F is an isomorphism offunctors on ap. If (R, ξ) pro-represents F and (R, ξ) is also a couple for F, then we say that(R, ξ) represents F. It is clear that if the couple (R, ξ) represents F, then (R, ξ) is uniqueup to a unique isomorphism of couples.

Similarly, let F be a pointed functor on ap(n). A couple for F is a pair (R, ξ), where Ris an object of ap(n) and ξ ∈ F(R). We say that the couple (R, ξ) represents F if and onlyif αn(ξ) is an isomorphism of functors defined on ap(n). It is clear that if this is the case,the couple (R, ξ) is unique up to a unique isomorphism of couples.


Let F be a functor on ap, and let (R, ξ) be a pro-couple for F. For all n ≥ 1, let (Rn, ξn)be given by Rn = R/I(R)n and ξn = F(un)(ξ), where un : R→ Rn is the natural surjection.Then (Rn, ξn) is a couple for the restriction Fn : ap(n) → Sets of F for all n ≥ 1. Noticethat αn(ξn) is the restriction of the morphism α(ξ) to ap(n) for all n ≥ 1. Consequently,(R, ξ) pro-represents F if and only if (Rn, ξn) represents Fn for all n ≥ 1. In particular, itfollows that if (R, ξ) pro-represents F, then (R, ξ) is unique up to a unique isomorphismof pro-couples.

Let F : ap → Sets be a pointed functor on ap. A pro-representing hull of F is a pro-couple(R, ξ) of F such that the following conditions hold:

(1) α(ξ) : hR → F is a smooth morphism of functors on ap(2) α2(ξ2) : hR2 → F2 is an isomorphism of functors on ap(2)

To simplify notation, we sometimes call the pro-representing hull (R, ξ) a hull of F.

Proposition 4.2. Let F : ap → Sets be a pointed functor on ap, and assume that(R, ξ), (R′, ξ′) are pro-representing hulls of F. Then there exists an isomorphism of pro-couples u : (R, ξ)→ (R′, ξ′).

Proof. Let φ = α(ξ), φ′ = α(ξ′). Since φ, φ′ are smooth morphisms, we have that φR′ andφ′R are surjective. So we can find morphisms u : (R, ξ) → (R′, ξ′) and v : (R′, ξ′) → (R, ξ)of pro-couples of F. The restriction to ap(2) gives us morphisms u2 : (R2, ξ2) → (R′2, ξ

′2)

and v2 : (R′2, ξ′2) → (R2, ξ2). But both (R2, ξ2) and (R′2, ξ

′2) represent F2, so u2 and v2 are

inverses. In particular, gr1(u2) and gr1(v2) are inverses, and (v u)2 = v2 u2 = id. Fromthe proof of proposition 1.5, we see that gr(v u) is surjective. This means that grn(v u) isa surjective endomorphims of a finite dimensional k-vector space for all n ≥ 1, so gr(v u)is an isomorphism. By proposition 1.5, v u is an isomorphism as well, and the same holdsfor u v by a symmetric argument. It follows that u and v are isomorphisms.

So if there exists a pro-representing hull of a pointed functor F, we know that it is unique,and we shall denote it by (H, ξ). Notice that (H, ξ) is only unique up to non-canonicalisomorphism. By abuse of language, we shall sometimes omit ξ from the notation, and saythat H is the hull of F.

5. Hulls of noncommutative deformation functors

We recall that k is an algebraically closed (commutative) field, A is an associativek-algebra, and M = M1, . . . ,Mp is a finite family of left A-modules. In this section, weprove that if the family M satisfy the finiteness condition (FC), then there exists a hullH = H(M) of the noncommutative deformation functor DefM. The proof follows Laudal[8], and the essential point is the following obstruction calculus:

Proposition 5.1. Let u : R → S be a small surjective morphism in ap with kernelK = ker(u), and let MS ∈ DefM(S) be a deformation. Then there exists a canonicalobstruction

o(u,MS) ∈ (Ext2A(Mj ,Mi)⊗k Kij),

such that o(u,MS) = 0 if and only if there exists a deformation MR ∈ DefM(R) liftingMS. If this is the case, the set of deformations in DefM(R) lifting MS is a torsor under thek-vector space (Ext1A(Mj ,Mi)⊗k Kij).

104 EIVIND ERIKSEN

Proof. We recall from section 2 that up to equivalence, we may assume that MS has thefollowing form: MS = (Mi ⊗k Sij) with right S-module structure given by the multiplica-tion in S, and with left A-module structure given by k-linear homomorphisms ai : Mi →⊕(Mj⊗kSji) for all a ∈ A. Via the natural projections, the map ai gives rise to k-linear mapsaji : Mi →Mj ⊗k Sji for a ∈ A, 1 ≤ i, j ≤ p. Since u is surjective, we may choose k-linearmaps L(a)ji : Mi →Mj ⊗k Rji such that (id⊗ u) L(a)ji = aji for a ∈ A, 1 ≤ i, j ≤ p. Let

L(a) = (L(a)ij) ∈ (Homk(Mj ,Mi ⊗k Rij)),

this defines a k-linear left action of A on MR = (Mi⊗k Rij), lifting the left A-modulestructure on MS . We let Q′ = (Homk(Mj ,Mi ⊗k Rij)), and remark that this is an asso-ciative k-algebra in a natural way: We compose the k-linear morphisms in Q′ by using themultiplication in R.

For a, b ∈ A, consider the expression L(ab)−L(a)L(b) ∈ Q′. By the associativity of the leftA-module structure on MS , we see that L(ab)−L(a)L(b) ∈ Q, where Q = (Homk(Mj ,Mi⊗kKij)) ⊆ Q′. Furthermore, we notice that Q ⊆ Q′ is an ideal, and Q has a natural structureas an A-A bimodule via L, since K2 = 0. We define ψ ∈ Homk(A ⊗k A,Q) to be givenby ψ(a, b) = L(ab) − L(a)L(b) for all a, b ∈ A. A straight-forward calculation shows thatψ is a 2-cocycle in HC∗(A,Q), so ψ gives rise to an element o(u,MS) ∈ HH2(A,Q) — seeappendix A for the definition of the Hochschild complex and its cohomology. Since K2 = 0,it follows that if L′ is another k-linear lifting of the left A-module structure on MS , thenthe A-A bimodule structures of Q given by L and L′ coincide. Therefore, HH∗(A,Q) isindependent upon the choice of L, and a straight-forward calculation shows that the sameholds for the obstruction o(u,MS).

We remark that there exists a deformation MR ∈ DefM(R) lifting MS if and only if thereexists some k-linear lifting L′ : A → Q′ of the left A-module structure of MS such thatL′(ab) = L′(a)L′(b) for all a, b ∈ A. Let τ = L′ − L, then τ : A→ Q is a k-linear map, anda straight-forward calculation shows that L′(ab) = L′(a)L′(b) if and only if the relation

L(ab)− L(a)L(b) = L(a)τ(b)− τ(ab) + τ(a)L(b) + τ(a)τ(b)

holds. Since K2 = 0, the last term vanishes. The fact that the above relation holds forall a, b ∈ A is therefore equivalent to the fact that o(u,MS) = 0 in HH2(A,Q). So wehave established that there exists a canonical obstruction o(u,MS) ∈ HH2(A,Q) such thato(u,MS) = 0 if and only if there is a lifting of MS to R.

Assume that L : A → Q′ is such that L(ab) = L(a)L(b) for all a, b ∈ A, that is, suchthat it defines a deformation MR lying over MS . For any other k-linear lifting L′ : A→ Q′

of the left A-module structure on MS , we may consider the difference τ = L′ − L : A→ Q.A straight-forward calculation shows that τ is a 1-cocycle in HC∗(A,Q) if and only ifL′(ab) = L′(a)L′(b) for all a, b ∈ A, that is, if and only if L′ defines a left A-module structureon MR. Furthermore, we have that L and L′ give rise to equivalent deformations if and onlyif τ is a 1-coboundary: It is clear that any equivalence between the left A-module structuresof MR = (Mi⊗kRij) given by L and L′ has the form id+ψ, where ψ ∈ Q. Furthermore, themap id+ψ : MR →MR (with the left A-module structure from L′ and L respectively) is aleft A-module homomorphism if and only if L(r)(id+ψ) = (id+ψ)L′(r) holds for all a ∈ A,and this last condition is equivalent with the fact that τ = d(ψ), so that τ is a 1-coboundary.If τ is a 1-boundary in HC∗(A,Q), it is also clear that id+ψ defines an equivalence betweenthe two deformations given by L and L′. Therefore, the set of deformations MR lying overMS is a torsor under the k-vector space HH1(A,Q).


To end the proof, we have to show that there are isomorphisms of k-vector spacesHHn(A,Q) ∼= (ExtnA(Mj ,Mi)⊗k Kij) for n = 1, 2: Since L(a) is a lifting to MR of the leftmultiplication of a on MS (satisfying equation 1), L(a) satisfies equation 1 as well. Thatis, we have L(a)ji(mi) − δij(ami) ⊗ ei ∈ Mj ⊗k Iji for all a ∈ A, mi ∈ Mi, 1 ≤ i, j ≤ p.Since K2 = 0, this means that the A-A bimodule structure of Q defined via L coincideswith the following natural one: Since Mi,Mj ⊗k Kji are left A-modules, we have thatQij = Homk(Mj ,Mi ⊗k Kij) and Q = ⊕Qij has natural A-A bimodule structures. Clearly,we have

HHn(A,Q) ∼= ⊕i,j

HHn(A,Qij) = (HHn(A,Qij)).

By appendix A, proposition A.3, we have that HHn(A,Qij) ∼= ExtnA(Mj ,Mi ⊗k Kij) forn ≥ 0. Moreover, ExtnA(Mj ,Mi ⊗k Kij) ∼= ExtnA(Mj ,Mi) ⊗k Kij since Kij is a k-vectorspace of finite dimension. This completes the proof of the proposition.

We remark that it is easy to find an alternative proof of proposition 5.1 using resolu-tions and the Yoneda representation of ExtnA(Mi,Mj). This is straight-forward, but makesessential use of proposition 3.4.

Also notice that the obstruction calculus is functorial in the following sense: Let u : R→S and u′ : R′ → S′ be two small surjections in ap, and write K = ker(u) and K ′ = ker(u′).Assume that v : R → R′ and w : S → S′ are morphisms such that u′ v = w u. Thenv(K) ⊆ K ′, and the map v induces a k-linear map of obstruction spaces

(Ext2A(Mj ,Mi)⊗k Kij)→ (Ext2A(Mj ,Mi)⊗k K ′ij).

If MS is a deformation of M to S and MS′ = DefM(w)(MS) is the corresponding deforma-tion to S′, then this map of obstruction spaces maps o(u,MS) to o(u′,MS′). This followsfrom the proof of proposition 5.1.

Let us start the construction of the pro-representing hull (H, ξ) of DefM, using theobstruction calculus for DefM given above. From now on, we shall assume that the familyM satisfy the finiteness condition

(FC) dimk ExtnA(Mi,Mj)is finite for 1 ≤ i, j ≤ p, n = 1, 2.

We fix the following notation: Let xij(l) : 1 ≤ l ≤ dij be a basis for Ext1A(Mj ,Mi)∗

and let yij(l) : 1 ≤ l ≤ rij be a basis for Ext2A(Mj ,Mi)∗ for 1 ≤ i, j ≤ p, with dij =dimk Ext1A(Mj ,Mi) and rij = dimk Ext2A(Mj ,Mi). Moreover, we consider the formalmatrix rings in ap corresponding to these vector spaces, and denote them by T1 = R

(Ext1A(Mj ,Mi)∗) and T2 = R(Ext2A(Mj ,Mi)∗).First, let us show that DefM restricted to ap(2) is representable: We define H2

to be the object H2 = T12 = T1/I(T1)2 in ap(2). For all objects R in ap(2), we

get Mor(H2, R) ∼= (Homk(Ext1A(Mj ,Mi)∗, I(R)ij)) ∼= (Ext1A(Mj ,Mi) ⊗k I(R)ij), andDefM(R) ∼= (Ext1A(Mj ,Mi) ⊗k I(R)ij) by proposition 5.1 applied to the small surjectionR → kp. The isomorphisms we obtain in this way are compatible, so they induce anisomorphism φ2 : hH2 → DefM of functors on ap(2). From the version of lemma 4.1 forthe category ap(2), we see that there is a unique deformation ξ2 ∈ DefM(H2) such thatα2(ξ2) = φ2. By definition, (H2, ξ2) represents the deformation functor DefM restrictedto ap(2).

106 EIVIND ERIKSEN

Let us also give an explicit description of the deformation ξ2: We have H2 = T12, so let us

denote by εij(l) the image of xij(l) in H2 for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij . In this notation, ξ2 isrepresented by the right H2-module (Mi⊗k (H2)ij), with left A-module structure defined by

a(mj ⊗ ej) = amj ⊗ ej +∑i,l

ψlij(a)(mj)⊗ εij(l)

for all a ∈ A, mj ∈Mj , 1 ≤ j ≤ p, where ψlij ∈ Derk(A,Homk(Mj ,Mi)) is a representativeof xij(l)∗ ∈ Ext1A(Mj ,Mi) via Hochschild cohomology.

There is also an alternative description of ξ2 using M-free resolutions and the Yonedarepresentation of Ext1A(Mi,Mj): Let α(εij(l)) ∈ Hom1(L∗j , L∗i) be a 1-cocycle representingxij(l)∗ ∈ Ext1A(Mj ,Mi) for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij . Then by construction, the formula

dH2m =

∑i

dm,i ⊗ ei +∑i,j,l

α(εij(l))m ⊗ εij(l)

defines a differential which lifts the complex (F∗, d∗) to H2. By proposition 3.4, the liftedcomplex is in fact an M-free resolution of some deformation of M toH2, and this deformationis ξ2 ∈ DefM(H2).

Theorem 5.2. Assume that dimk ExtnA(Mi,Mj) is finite for 1 ≤ i, j ≤ p, n = 1, 2. Thenthere exists a morphism o : T2 → T1 in ap such that H(M) = T1⊗T2kp is a pro-representinghull for DefM.

Proof. For simplicity, let us write I for the ideal I = I(T1), and for all n ≥ 1, let us writeT1n for the quotient T1

n = T1/In, and tn : T1n+1 → T1

n for the natural morphism. Fromthe paragraphs preceding this theorem, we know that (H2, ξ2) represents DefM restrictedto ap(2). Let o2 : T2 → T1

2 be the trivial morphism given by o2(I(T2)) = 0 and let a2 = I2,then H2 = T1/a2

∼= T12 ⊗T2 kp. Using o2 and ξ2 as a starting point, we shall construct on

and ξn for n ≥ 3 by an inductive process. So let n ≥ 2, and assume that the morphismon : T2 → T1

n and the deformation ξn ∈ DefM(Hn) is given, with Hn = T1n ⊗T2 kp. We

shall also assume that tn−1 on = on−1 and that ξn is a lifting of ξn−1.Let us now construct the morphism on+1 : T2 → T1

n+1: We let a′n be the ideal inT1n generated by on(I(T2)). Then a′n = an/I

n for an ideal an ⊆ T1 with In ⊆ an, andHn∼= T1/an. Let bn = Ian + anI, then we obtain the following commutative diagram:

T2

on

T1n+1 T1/bn

T1n Hn = T1/an,

Observe that T1/bn → T1/an is a small surjection. So by proposition 5.1, there is anobstruction o′n+1 = o(T1/bn → Hn, ξn) for lifting ξn to T1/bn, and we have

o′n+1 ∈ (Ext2A(Mj ,Mi)⊗k (an/bn)ij) ∼= (Homk(gr1(T2)ij , (an/bn)ij)).


Consequently, we obtain a morphism o′n+1 : T2 → T1/bn. Let a′′n+1 be the ideal in T1/bngenerated by o′n+1(I(T

2)). Then a′′n+1 = an+1/bn for an ideal an+1 ⊆ T1 with bn ⊆ an+1 ⊆an. We define Hn+1 = T1/an+1 and obtain the following commutative diagram:

T2

on

o′n+1

T1n+1 T1/bn Hn+1 = T1/an+1

T1n Hn = T1/an

By the choice of an+1, the obstruction for lifting ξn to Hn+1 is zero. We can therefore finda lifting ξn+1 ∈ DefM(Hn+1) of ξn to Hn+1.

The next step of the construction is to find a morphism on+1 : T2 → T1n+1 which

commutes with o′n+1 and on: We know that tn−1 on = on−1, which means that an−1 =In−1 + an. For simplicity, let us write O(K) = (Homk(gr1(T

2)ij ,Kij)) for any ideal K ⊆T1. Consider the following commutative diagram of k-vector spaces, in which the columnsare exact:

0 0

O(bn/In+1)jn

O(bn−1/In)

O(an/In+1)

rn+1

knO(an−1/I

n)

rn

O(an/bn)ln

O(an−1/bn−1)

0 0

We may consider consider on as an element in O(an−1/In), since an ⊆ an−1. On the other

hand, o′n+1 ∈ O(an/bn). Let o′n = rn(on), then the natural map T1/bn → T1/bn−1 maps theobstruction o′n+1 to the obstruction o′n by the second remark following proposition 5.1. Thisimplies that o′n+1 commutes with o′n, so ln(o′n+1) = o′n = rn(on). But we have on(I(T2)) ⊆an, so we can find an element on+1 ∈ O(an/In+1) such that kn(on+1) = on. Since an−1 =an+In−1, jn is surjective. Elementary diagram chasing using the snake lemma implies thatwe can find on+1 ∈ O(an/In+1) such that rn+1(on+1) = o′n+1 and kn(on+1) = on. It followsthat the obstruction on+1 defines a morphism on+1 : T2 → T1

n+1 compatible with on suchthat T1

n+1 ⊗T2 kp ∼= Hn+1.By induction, it follows that we can find a morphism on : T2 → T1

n and a deformationξn ∈ DefM(Hn), with Hn = T1

n ⊗T2 kp, for all n ≥ 1. From the construction, we see thattn−1 on = on−1 for all n ≥ 2, so we obtain a morphism o : T2 → T1 by the universalproperty of the projective limit. Moreover, the induced morphisms hn : Hn+1 → Hn are suchthat ξn+1 ∈ DefM(Hn+1) is a lifting of ξn ∈ DefM(Hn) to Hn+1. Notice that I(Hn)n = 0

108 EIVIND ERIKSEN

and that Hn/I(Hn)n−1 ∼= Hn−1 for all n ≥ 2. It follows that H/I(H)n = Hn for all n ≥ 1,so H is an object of the pro-category ap. Let ξ = (ξn), then clearly ξ ∈ DefM(H), so (H, ξ)is a pro-couple for DefM. It remains to show that (H, ξ) is a pro-representable hull for DefM.

It is clearly enough to show that (Hn, ξn) is a pro-representing hull for DefM restricted toap(n) for all n ≥ 3. So let φn = αn(ξn) be the morphism of functors on ap(n) correspondingto ξn. We shall prove that φn is a smooth morphism. So let u : R → S be a small surjec-tion in ap(n), and assume that MR ∈ DefM(R) and v ∈ Mor(Hn, S) are given such thatDefM(u)(MR) = DefM(v)(ξn) = MS . Let us consider the following commutative diagram:

T1/bn

T1 Hn+1 R

u

Hn v S

Let v′ : T1 → R be any morphism making the diagram commutative. Then v′(an) ⊆ K,where K = ker(u), so v′(bn) = 0. But the induced map T1/bn → R maps the obstructiono′n+1 to o(u,MS), and we know that o(u,MS) = 0. So we have v′(an+1) = 0, and v′ inducesa morphism v′ : Hn+1 → R making the diagram commutative. Since v′(I(Hn+1)n) = 0,we may consider v′ a map from Hn+1/I(Hn+1)n ∼= Hn. So we have constructed a mapv′ ∈ Mor(Hn, R) such that u v′ = v. Let M ′R = DefM(v′)(ξn), then M ′R is a lifting of MS

to R. By proposition 5.1, the difference between MR and M ′R is given by an element

d ∈ (Ext1A(Mj ,Mi)⊗k Kij) = (Homk(gr1(T1)ij ,Kij)).

Let v′′ : T1 → R be the morphism given by v′′(xij(l)) = v′(xij(l))+d(xij(l)) for 1 ≤ i, j ≤ p,1 ≤ l ≤ dij . Since an+1 ⊆ I(T1)2, we have

v′′(an+1) ⊆ v′(an+1) + I(R)K +KI(R) +K2.

But u is small, so v′′(an+1) = 0 and v′′ induces a morphism v′′ : Hn → R. Clearly,u v′′ = u v′ = v, and DefM(v′′)(ξn) = MR by construction. It follows that φn is smoothfor all n ≥ 3.

We remark that the conclusion of the theorem still holds if we relax the finiteness condi-tion (FC). If we only assume that

dimk Ext1A(Mi,Mj)is finite for 1 ≤ i, j ≤ p,

then the object T2 is in Ap, but not necessarily in ap. However, the rest of the proof is stillvalid as stated, so the finiteness condition on Ext2A(Mi,Mj) is clearly not essential.

In general, it is possible to generalize theorem 5.2 to the case when ExtnA(Mi,Mj) hascountable dimension as a vector space over k for 1 ≤ i, j ≤ p, n = 1, 2, see Laudal [2].However, we shall always assume (FC) in this paper.


Assume that M satisfy (FC). If Ext2A(Mi,Mj) = 0 for 1 ≤ i, j ≤ p, we say that thedeformation functor DefM is unobstructed. For instance, DefM is unobstructed for any finitefamily M of left A-modules satisfying (FC) if A is left hereditary (that is, the left globalhomological dimension of A is at most 1). If DefM is unobstructed, H = T1 is the hullof DefM.

In general, DefM can be obstructed, and there is no simple formula for the hullH of DefMif this is the case. However, there exists an algorithm for calculating the hull H using matricMassey products. In the next sections, we shall introduce the matric Massey products andexplain how the hull can be calculated when M satisfy (FC).

6. Immediately defined matric Massey products

We recall that k is an algebraically closed (commutative) field, A is an associativek-algebra, and M = M1, . . . ,Mp is a finite family of left A-modules. From now on, wealso assume that the family M satisfy the finiteness condition (FC). In this section, weshall define the immediately defined matric Massey products and their defining system, andshow how to calculate these products using matrices.

Let us fix a monomial X ∈ I(T1) of type (i, j) and degree n ≥ 2. Then we can write Xuniquely in the form

X = xi0i1(l1)xi1i2(l2) . . . xin−1in(ln),

where (i0, in) = (i, j). Let X ′ be another monomial in T1. We shall say that X ′ divides Xif there exist monomials X(l),X(r) ∈ T1 such that X = X(l)X ′X(r), and write X ′ | X ifthis is the case.

Consider the set of monomials X ′ ∈ I(T1) : X ′ | X, and denote by J(X) the ideal inT1 generated by these monomials. We define R(X) = T1/J(X) and S(X) = R(X)/(X) =T1/(J(X),X). Then the natural map

π(X) : R(X)→ S(X)

is a small surjection in ap, and it has a 1-dimensional kernel which is generated by themonomial X. We write I(X) = I(S(X)) and S(X)n = S(X)/I(X)n for all n ≥ 1.

Let us consider the set B(X) = (i, j, l) : 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij , xij(l) | X, anddenote by vij(l) the image of xij(l) in S(X)2 for all (i, j, l) ∈ B(X). Then the set

vij(l) : (i, j, l) ∈ B(X)

is a natural k-basis for I(X)/I(X)2.Assume that a morphism φ(X) : H → S(X) is given, and denote the composition of

φ(X) with the natural morphism S(X) → S(X)2 by φ(X)2 : H → S(X)2. This morphismcan be written uniquely in the form

φ(X)2 =∑

(i,j,l)∈B(X)

αij(l)⊗ vij(l),

where αij(l) ∈ Ext1A(Mj ,Mi) for all (i, j, l) ∈ B(X).

110 EIVIND ERIKSEN

Conversely, consider a family αij(l) ∈ Ext1A(Mj ,Mi) : (i, j, l) ∈ B(X) of extensionsindexed by B(X), corresponding to a morphism φ(X)2 : H → S(X)2 given by φ(X)2 =∑αij(l)⊗ vij(l). If there exists a lifting of φ(X)2 to a morphism φ(X) : H → S(X), we say

that the matric Massey product

〈α;X〉 = 〈αi0,i1(l1), αi1,i2(l2), . . . , αin−1,in(ln)〉

is defined, and that φ(X) is a defining system for this matric Massey product. If thisis the case, we denote the deformation induced by the defining system φ(X) by MX ∈DefM(S(X)), and by proposition 5.1, the obstruction for lifting MX to R(X) is an element

o(π(X),MX) ∈ (Ext2A(Mj ,Mi)⊗k K(X)ij) ∼= Ext2A(Mj ,Mi),

where K(X) = ker(π(X)) ∼= kX. In general, this element depends upon the deformationMX , and therefore on the defining system φ(X). We define the value of the matric Masseyproduct to be

〈α;X〉 = 〈αi0,i1(l1), αi1,i2(l2), . . . , αin−1,in(ln)〉 = o(π(X),MX).

Consequently, the value of the matric Massey product 〈α;X〉 will in general depend uponthe chosen defining system.

Let us fix the monomial X. Then the matric Massey product α → 〈α;X〉 is a noteverywhere defined k-linear map

Ext1A(Mi1 ,Mi0)⊗k · · · ⊗k Ext1A(Min ,Min−1) Ext2A(Min ,Mi0).

In fact, this map is defined for α if and only if the morphism φ(X)2 : H → S(X)2 corre-sponding to α can be lifted to a morphism φ(X) : H → S(X). Moreover, even when thismap is defined for α, it is not necessarily uniquely defined: In general, its value 〈α;X〉depends upon the chosen lifting φ(X), the defining system. The matric Massey products〈α;X〉 defined above are called the immediately defined matric Massey products.

We remark that if X is a monomial of degree n = 2, then the situation is much simpler:We have S(X) = S(X)2, so the matric Massey product 〈α;X〉 is uniquely defined for anyfamily of extensions αij(l) : (i, j, l) ∈ B(X). In fact, the matric Massey product is justthe usual cup product in this case.

Let us fix a monomial X ∈ I(T1) of degree n ≥ 2. Then there exists a natural family ofextensions indexed by B(X) given by αij(l) = xij(l)∗,

xij(l)∗ ∈ Ext1A(Mj ,Mi) : (i, j, l) ∈ B(X).

The matric Massey products of these extensions are the ones that we shall use for theconstruction of the hull H of DefM in the next section. We therefore introduce the notation

〈x∗;X〉 = 〈xi0i1(l1)∗, xi1i2(l2)∗, . . . , xin−1in(ln)∗〉

for their immediately defined matric Massey products.


The matric Massey products are called matric because these products (and their definingsystems) can be described completely in terms of linear algebra and matrices. We shall endthis section by giving such a description.

Let αij(l) ∈ Ext1A(Mj ,Mi) : (i, j, l) ∈ B(X) be a family of extensions indexed byB(X), and consider the corresponding matric Massey product

(4) 〈α;X〉 = 〈αi0i1(l1), αi1i2(l2), . . . , αin−1in(ln)〉.

We assume that there exists a defining system φ(X) : H → S(X) for this matric Masseyproduct. Then φ(X) induces a deformation MX ∈ DefM(S(X)). We notice that the matricMassey product (4) only depends upon this deformation. By abuse of language, we shalltherefore let the notion defining system refer to the deformationMX as well as the morphismφ(X) : H → S(X) which induces MX .

We know that any deformation MX ∈ DefM(S(X)) can be described by a complex whichlifts (F∗, d∗) to S(X). Such a complex is given by differentials of the form

dS(X)m : (Lm+1,i ⊗k S(X)ij)→ (Lm,i ⊗k S(X)ij).

We write v(X ′) for the image of X ′ in S(X) whenever X ′ is a monomial in T1, and defineB(X) = X ′ ∈ I(T1) : X ′ is a monomial such that X ′ | X ∪ e1, . . . , ep. Then the set

v(X ′) : X ′ ∈ B(X)

is a natural k-basis for S(X), and B(X) contains xij(l) : (i, j, l) ∈ B(X) and e1, . . . , epas subsets. Let us write B(X)ij = B(X) ∩ S(X)ij for 1 ≤ i, j ≤ p. With this notation, theabove differentials have the form

dS(X)m =

∑1≤i≤p

dm,i ⊗ ei +∑

X′∈B(X)

α(X ′)m ⊗ v(X ′),

where α(X ′) ∈ Hom1A(L∗j , L∗i) is a 1-cochain whenever X ′ ∈ B(X ′)ij .

Let dS(X)m be arbitrary maps between M-free modules over S(X) defined by a family

of 1-cochains α(X ′) : X ′ ∈ B(X) as above. These maps lifts the complex (F∗, d∗) ifα(ei) = d∗i for 1 ≤ i ≤ p. Moreover, these maps are differentials if and only if the followingcondition holds: For all monomials Z ∈ B(X) and for all integers m ≥ 0, we have

(5)∑

X′,X′′∈B(X)X′X′′=Z

α(X ′)m α(X ′′)m+1 =∑

X′,X′′∈B(X)X′X′′=Z

α(X ′′)m+1α(X ′)m = 0.

In the first sum, the symbol denotes composition of maps. We recall that each of the mapsinvolved can be considered as right multiplication by a matrix. In the second summation, weidentify the maps with such matrices, and re-write the composition of maps as multiplicationof the corresponding matrices.

Assume that these conditions hold. Then the family α(X ′) : X ′ ∈ B(X) of 1-cochainsdefines a lifting of complexes of (L∗, d∗) to S(X) given by the differentials dS(X) as above,

112 EIVIND ERIKSEN

and this lifting corresponds to a deformation MX ∈ DefM(S(X)). The deformation MX

is a defining system for the matric Massey product (4) if and only if α(X ′) is a 1-cocyclewhich represents αij(l) ∈ Ext1A(Mj ,Mi) whenever X ′ = xij(l) for some (i, j, l) ∈ B(X).In this case, we shall refer to the family of 1-cochains α(X ′) : X ′ ∈ B(X) as a definingsystem for the matric Massey product (4).

Finally, assume that the family of 1-cochains α(X ′) : X ′ ∈ B(X) is a defining systemof the matric Massey product (4). Then the value of this matric Massey product is given by

(6) 〈α;X〉m =∑

X′,X′′∈B(X)X′X′′=X

α(X ′′)m+1α(X ′)m

for all m ≥ 0, where the multiplication denotes matrix multiplication of the correspondingmatrices.

Proposition 6.1. Let αij(l) ∈ Ext1A(Mj ,Mi) : (i, j, l) ∈ B(X) be a family of extensions.A defining system for the matric Massey product

〈α;X〉 = 〈αi0i1(l1), . . . , αin−1in(ln)〉

corresponds to a family α(X ′) ∈ Hom1A(L∗j , L∗i) : 1 ≤ i, j ≤ p, X ′ ∈ B(X)ij of

1-cochains satisfying the following conditions:

• α(ei) = d∗i for 1 ≤ i ≤ p,• α(X ′) is a 1-cocycle representing αij(l) whenever X ′ = xij(l) for some (i, j, l) ∈ B(X),• For all Z ∈ B(X) and for all m ≥ 0, we have

∑X′,X′′∈B(X)X′X′′=Z

α(X ′′)m+1 α(X ′)m = 0.

Moreover, given such a family of 1-cochains, the matric Massey product 〈α;X〉 is representedby the 2-cocyle given by

〈α;X〉m =∑

X′,X′′∈B(X)X′X′′=X

α(X ′′)m+1 α(X ′)m

for all m ≥ 0.

Hence we have described the immediately defined matric Massey products and theirdefining systems in terms of linear algebra and matrices, as we set out to do. We remark thatthe description given in proposition 6.1 is extremely useful for doing concrete calculationswith matric Massey products, and even for implementing such computations on computers.It also justifies the name matric.


7. Calculating hulls using matric Massey products

We recall that k is an algebraically closed (commutative) field, A is an associativek-algebra, and M = M1, . . . ,Mp is a finite family of left A-modules. We also assume thatthe family M satisfy the finiteness condition (FC). In this section, we show how to calculatethe hull H of the deformation functor DefM using matric Massey products.

By theorem 5.2, there exists an obstruction morphism o : T2 → T1 in ap such thatH = T1⊗T2kp is a hull for the deformation functor DefM. We shall write I = I(T1) andfij(l) = o(yij(l)) for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Then fij(l) is a formal power series in I2

ij byconstruction. Let us define a ⊆ T1 to be the ideal generated by fij(l) : 1 ≤ i, j ≤ p, 1 ≤l ≤ rij. Then a ⊆ I2, and we have

H = T1⊗T2kp ∼= T1/a.

We shall use the matric Massey products from section 6 to calculate the coefficients of thepower series fij(l). Clearly, this is sufficient to determine the hull H.

Let us fix an integer N ≥ 2 such that a ⊆ IN . This is always possible, since a ⊆ I2. Sofij(l) ∈ IN for all fij(l), and we can write fij(l) in the form

fij(l) =∑|X|=N

alij(X) ·X +∑|X|>N

alij(X) ·X

for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij , with alij(X) ∈ k for all monomials X ∈ IN . As usual, we usethe notation |X| to denote the degree of the monomial X.

Let 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij and let n ≥ N . Then we agree to write fij(l)n for thetruncated power series

fij(l)n =n∑

|X|=Nalij(X) ·X.

Moreover, let an+1 = In+1 + (fn) for all n ≥ N , where (fn) ⊆ T1 is the ideal generatedby fij(l)n : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij, and let an = In for 2 ≤ n ≤ N . We writeHn = H/I(H)n as usual, then Hn = T1/an for all n ≥ 2, in accordance with the notationin the proof of theorem 5.2.

Recall that H2 = T12 and that ξ2 ∈ DefM(H2) denotes the universal deformation with

the property that the couple (H2, ξ2) represents DefM restricted to ap(2). We have assumedthat a ⊆ IN , and this means that there exists a lifting of ξ2 to HN = T1/aN = T1

N . Let usproceed to find such a lifting MN ∈ DefM(HN ) explicitly.

We choose to describe the deformation MN in terms of M-free resolutions. Let us defineB(N − 1) to be the set of all monomials in T1 of degree at most N − 1. Then X : X ∈B(N − 1) is a monomial basis of HN , and any M-free resolution of MN can be describedby a family α(X) : X ∈ B(N − 1) of 1-cochains satisfying the following conditions:

• α(ei) = d∗i for 1 ≤ i ≤ p,• α(xij(l)) is a 1-cocycle representing xij(l)∗ for 1 ≤ i, j ≤ p, 1 ≤ l ≤ dij ,• For all Z ∈ B(N − 1) and for all m ≥ 0, we have

∑X′,X′′∈B(N−1)

X′X′′=Z

α(X ′′)m+1 α(X ′)m = 0.

114 EIVIND ERIKSEN

We know that a family of 1-cochains with the above properties exists, since we can finda lifting MN of ξ2 to HN and this deformation must have some M-free resolution. So wechoose one such family α(X) : X ∈ B(N−1) and fix this choice. This means that we havefixed a deformation MN ∈ DefM(HN ) with an M-free resolution given by the correspondingdifferentials. So (HN ,MN ) is a pro-representing hull for DefM restricted to ap(N).

Lemma 7.1. Let π : R→ S be any small surjection in ap, let φ : H → S be any morphism,and denote by Mφ ∈ DefM(S) the deformation induced by φ. Then we can lift φ to amorphism φ : T1 → R making the diagram

T1φ

R

π

Hφ

S

commutative, and the obstruction o(π,Mφ) for lifting Mφ to R is given by

o(π,Mφ) =∑i,j,l

yij(l)∗ ⊗ φ(fij(l)).

Proof. By construction and functoriality, the obstruction o(π,Mφ) is given as the restrictionof the composition φ o to the k-linear subspace (Ext2A(Mj ,Mi)∗) ⊆ T2. Since yij(l) is ak-linear basis for this subspace, we get the desired expression for the obstruction.

Let us define bN ⊆ T1 to be the ideal bN = IaN +aNI = IN+1, and consider the naturalmap rN : RN → HN , where RN = T1/bN = T1

N+1. By construction, rN is a small surjectionin ap, and the natural surjection φN : T1 → RN makes the diagram

T2 oT1

φNRN

rN

HφN

HN

commutative. Let B′(N) be the set of all monomials in T1 of degree N . Since ker(rN ) =IN/IN+1, we see that X : X ∈ B′(N) is a monomial basis for ker(rN ). Moreover, letB′(N) = B′(N) ∪B(N − 1). Then clearly X : X ∈ B′(N) is a monomial basis for RN .

Since rN is a small surjection, there is an obstruction o(rN ,MN ) for lifting MN to RN ,and we see from lemma 7.1 that this obstruction can be expressed as

o(rN ,MN ) =∑i,j,l

yij(l)∗ ⊗ φN (fij(l))

=∑i,j,l

yij(l)∗ ⊗ f ij(l)

=∑i,j,l

∑X∈B′(N)

yij(l)∗ ⊗ (alij(X) ·X),

where f ij(l) and X denote the images of fij(l) and X in RN .


We say that the family D(N) = α(X) : X ∈ B(N − 1) of 1-cochains is a definingsystem for the matric Massey products of order N ,

〈x∗;X〉 for X ∈ B′(N).

Let X ∈ B′(N) be any monomial of type (i, j). We define the matric Massey product 〈x∗;X〉to be the coefficient of X in the obstruction o(rN ,MN ) above. Then we immediately seethat this matric Massey product has value

〈x∗;X〉 =rij∑l=1

alij(X) · yij(l)∗.

In other words, the coefficient of X in the power series fij(l) is given by the matric Masseyproduct 〈x∗;X〉 above as

alij(X) = yij(l)(〈x∗;X〉)

for 1 ≤ l ≤ rij .We notice that the matric Massey products of order N defined above are immediately

defined. In other words, they can be expressed in terms of the matric Massey productsof section 6. In fact, the defining system D(N) induces a defining system α(X ′) : X ′ |X, X ′ = X in the sense of section 6, and the value of the corresponding matric Masseyproduct 〈x∗;X〉 is exactly the coefficient of X in the obstruction o(rN ,MN ).

On the other hand, we can calculate the obstruction o(rN ,MN ) using the defining systemD(N), and therefore also the coefficient of X in this obstruction for each X ∈ B′(N).A straight-forward calculation show that this coefficient is given by the 2-cocycle y(X)defined by

y(X)m =∑

X′,X′′∈B(N−1)X′X′′=X

α(X ′′)m+1 α(X ′)m

for all m ≥ 0. This means that the matric Massey product 〈x∗;X〉 is represented by y(X),so we can easily calculate all matric Massey products of order N using the defining systemD(N). This determines the truncated power series fij(l)N , since we have

fij(l)N =∑

X∈B′(N)

alij(X) ·X =∑

X∈B′(N)

yij(l)(〈x∗;X〉) ·X

for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij .Let hN : HN+1 → HN be the natural map. Then ker(hN ) = IN/aN+1, so we can find

a subset B(N) ⊆ B′(N) of monomials in T1 of degree N such that X : X ∈ B(N) is amonomial basis for ker(hN ). Let B(N) = B(N) ∪B(N − 1), then clearly X : X ∈ B(N)is a monomial basis for HN+1. So for each monomial X ∈ T1 with |X| ≤ N , we have aunique relation in HN+1 of the form

X =∑

X′∈B(N)

β(X,X ′) X ′,

116 EIVIND ERIKSEN

with β(X,X ′) ∈ k for all X ′ ∈ B(N). Since we have o(hN ,MN ) = 0, we deduce that

∑|X|=N

〈x∗;X〉 β(X,X ′) = 0

for all X ′ ∈ B(N). Notice that β(X,X ′) = 0 if the monomials X and X ′ do not have thesame type. Therefore, it makes sense to consider the 1-cocycle

∑|X|=N

β(X,X ′) y(X),

and by the relation above, this is a 1-coboundary. It follows that we can find a 1-cochainα(X ′) such that

d α(X ′) = −∑|X|=N

β(X,X ′) y(X),

and we fix such a choice. Consider the family α(X) : X ∈ B(N). This defines an M-freecomplex over HN+1 if and only if we have

∑|X|=N

β(X,Z)∑

X′,X′′∈B(N)X′X′′=X

α(X ′′) α(X ′) = 0

for all Z ∈ B(N). By the definition of α(X ′) when X ′ ∈ B(N), this condition holds,and we denote by MN+1 ∈ DefM(HN+1) the deformation with the complex defined byα(X) : X ∈ B(N) as M-free resolution. It is clear from the construction that MN+1 is alifting of MN , so (HN+1,MN+1) is a pro-representing hull for DefM restricted to ap(N +1).

Let bN+1 ⊆ T1 be the ideal bN+1 = IaN+1+aN+1I = IN+2+I(fN )+(fN )I, and considerthe natural map rN+1 : RN+1 → HN+1, where RN+1 = T1/bN+1. By construction, rN+1

is a small surjection in ap, and it is clear that the natural morphism φN+1 : T1 → RN+1

makes the diagram

T2 oT1

φN+1RN+1

rN+1

HφN+1

φN

HN+1

hN

HN

commutative. We see that ker(rN+1) = aN+1/bN+1, which we can re-write in the follow-ing way:

ker(rN+1) = (IN+1 + (fN ))/(IN+2 + I(fN ) + (fN )I)

= (fN )/(I(fN ) + (fN )I) ⊕ IN+1/(IN+2 + I(fN ) + (fN )I)


Let us write c(N + 1) = IN+1/(IN+2 + I(fN ) + (fN )I). Then c(N + 1) ⊆ ker(rN+1) is anideal, and we can clearly find a set B′(N +1) of monomials in T1 of degree N +1 such thatX : X ∈ B′(N + 1) is a monomial basis for cN+1. Let us choose B′(N + 1) such thatfor every X ∈ B′(N + 1), there is a monomial X ′ ∈ B(N) such that X ′ | X, this is clearlypossible. We let B′(N + 1) = B′(N + 1) ∪B(N), then

X : X ∈ B′(N + 1) ∪ fij(l)N : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij

is a basis for RN+1. So for each monomial X ∈ T1 with |X| ≤ N + 1, we have a uniquerelation in RN+1 of the form

X =∑

X′∈B′(N+1)

β′(X,X ′)X ′ +∑i,j,l

β′(X, i, j, l)f ij(l)N ,

with β′(X,X ′), β′(X, i, j, l) ∈ k for all X ′ ∈ B′(N + 1), 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij .Since rN+1 is a small surjection, there is an obstruction o(rN+1,MN+1) for lifting MN+1

to RN+1, and we see from lemma 7.1 that this obstruction can be expressed as

o(rN+1,MN+1) =∑i,j,l

yij(l)∗ ⊗ φN+1(fij(l))

=∑i,j,l

yij(l)∗ ⊗ f ij(l)

=∑i,j,l

yij(l)∗ ⊗ (f ij(l)N +

∑X∈B′(N+1)

alij(X) ·X),

where f ij(l), f ij(l)N and X denote the images of fij(l), fij(l)N and X in RN+1.We say that the family D(N + 1) = α(X) : X ∈ B(N) is a defining system for the

matric Massey products of order N + 1,

〈x∗;X〉 for X ∈ B′(N + 1)

Let X ∈ B′(N + 1) be any monomial of type (i, j). We define the matric Massey prod-uct 〈x∗;X〉 to be the coefficient of X in the obstruction o(rN+1,MN+1) above. Then weimmediately see that this matric Massey product has value

〈x∗;X〉 =rij∑l=1

alij(X) · yij(l)∗.

In other words, the coefficient of X in the power series fij(l) is given by the matric Masseyproduct 〈x∗;X〉 above as

alij(X) = yij(l)(〈x∗;X〉)for 1 ≤ l ≤ rij .

On the other hand, we can calculate the obstruction o(rN+1,MN+1) using the defin-ing system D(N + 1), and therefore also the coefficient of X in this obstruction for each

118 EIVIND ERIKSEN

X ∈ B′(N + 1). A straight-forward calculation show that this coefficient is given by the2-cocycle y(X) defined by

y(X)m =∑

|Z|≤N+1

β′(Z,X)∑

X′,X′′∈B(N)X′X′′=Z

α(X ′′)m+1 α(X ′)m

for all m ≥ 0. This means that the matric Massey product 〈x∗;X〉 is represented by y(X),so we can easily calculate all matric Massey products of order N + 1 using the definingsystem D(N + 1).

By the construction in the proof of theorem 5.2, we have that HN+2 is the quotient ofRN+1 by the ideal generated by the obstruction o(rN+1,MN+1). On the other hand, weknow that HN+2 = T1/(IN+2 +(fN+1). This implies that for all monomials X ∈ B′(N+1)of degree N + 1, the coefficient alij(X) = 0 for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . In other words,the truncated power series fij(l)N+1 is determined by the matric Massey products of orderN + 1 above, since we have

fij(l)N+1 = fij(l)N +∑

X∈B′(N+1)

alij(X) ·X

= fij(l)N +∑

X∈B′(N+1)

yij(l)(〈x∗;X〉) ·X

for 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij .Let hN+1 : HN+2 → HN+1 be the natural map, and consider its kernel. By definition,

we have

ker(hN+1) = aN+1/aN+2 = ((fN ) + IN+1)/((fN+1) + IN+2),

so we can clearly find a subset B(N + 1) ⊆ B′(N + 1) of monomials of degree N + 1 suchthat X : X ∈ B(N + 1) ∪ f ij(l)N : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij is a basis for ker(hN+1).Let B(N + 1) = B(N + 1) ∪B(N), then clearly

X : X ∈ B(N + 1) ∪ f ij(l)N : 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij

is a monomial basis for HN+2. So for each monomial X ∈ T1 with |X| ≤ N + 1, we have aunique relation in HN+2 of the form

X =∑

X′∈B(N+1)

β(X,X ′) X ′ +∑i,j,l

β(X, i, j, l) fij(l)N ,

with β(X,X ′), β(X, i, j, l) ∈ k for all X ′ ∈ B(N + 1), 1 ≤ i, j ≤ p, 1 ≤ l ≤ rij . Since wehave o(hN+1,MN+1) = 0, we deduce that

∑|X|≤N+1

〈x∗;X〉 β(X,X ′) = 0


for all X ′ ∈ B(N + 1). Notice that β(X,X ′) = 0 if the monomials X and X ′ do not havethe same type. Therefore, it makes sense to consider the 1-cocycle

∑|X|≤N+1

β(X,X ′) y(X),

and by the relation above, this is a 1-coboundary. It follows that we can find a 1-cochainα(X ′) such that

dα(X ′) = −∑

|X|≤N+1

β(X,X ′) y(X),

and we fix such a choice. Consider the family α(X) : X ∈ B(N + 1). This defines anM-free complex over HN+2 if and only if we have

∑|X|≤N+1

β(X,Z)∑

X′,X′′∈B(N+1)X′X′′=X

α(X ′′) α(X ′) = 0

for all Z ∈ B(N + 1). By the definition of α(X ′) when X ′ ∈ B(N + 1), this conditionholds, and we denote by MN+2 ∈ DefM(HN+2) the deformation with the complex definedby α(X) : X ∈ B(N + 1) as M-free resolution. It is clear from the construction thatMN+2 is a lifting of MN+1, so (HN+2,MN+2) is a pro-representing hull for DefM restrictedto ap(N + 2).

It is clear that we can continue in this way. For every k ≥ 1, we can calculate thecoefficients in the truncated power series fij(l)N+k, and therefore find HN+k+1. At the sametime, we find the defining systems α(X) : X ∈ B(N+k) necessary to calculate the matricMassey products of order N + k + 1, and these defining systems completely determine thedeformation MN+k+1. We have described how to do this in the case k = 1, and the generalcase is similar.

We conclude that the method that we have described above can be used to calculate thepro-representing hull (Hn,Mn) for the deformation functor DefM restricted to ap(n) forany n ≥ N . We can therefore, in principle, find the hull

H = lim← Hn

of DefM, and also the corresponding versal family defined over H,

ξ = M = lim← Mn.

It follows that the pro-representing hull (H, ξ) of the deformation functor DefM can becalculated using matric Massey products.

8. An example

Let k be an algebraically closed field of characteristic 0, and let A = A2(k) be the secondWeyl algebra over k. We shall think of A as the ring of differential operators in the planedefined over k with coordinates x and y. Thus, we can write A = k[x, y]〈∂x, ∂y〉, where

120 EIVIND ERIKSEN

∂x = ∂/∂x and ∂y = ∂/∂y. In other words, A is the k-algebra generated by x, y, ∂x, ∂ywith relations [∂x, x] = [∂y, y] = 1.

Let us consider the family of left A-modules M = M1,M2,M3,M4, where Mi = A/Iifor 1 ≤ i ≤ 4 and Ii ⊆ A are left ideals given by

I1 = A(∂x, ∂y) I2 = A(∂x, y)

I3 = A(x, ∂y) I4 = A(x, y)

We immediately notice that the left A-modules in the family M have the following freeresolutions:

0←M1 ← A

∂x∂y

←−−−− A2

(∂y−∂x)←−−−−−−− A← 0

0←M2 ← A

∂xy

←−−−− A2

(y−∂x)←−−−−−− A← 0

0←M3 ← A

x∂y

←−−−− A2

(∂y−x)←−−−−−− A← 0

0←M4 ← A

xy

←−−− A2

(y−x)←−−−−− A← 0

We consider the elements of the free A-modules An as row vectors, and the maps in the freeresolutions above as right multiplication of these row vectors by the given matrices. Noticethat for 1 ≤ i ≤ 4, the free A-module Lm,i in the free resolution of Mi does not dependupon i. We shall therefore write Lm = Lm,i for all m ≥ 0, 1 ≤ i ≤ 4.

It is known that M is a family of simple holonomic left A-modules, so this family satisfythe finiteness condition (FC). Therefore, there exists a pro-representing hull (H, ξ) for thedeformation functor DefM : a4 → Sets by theorem 5.2. We shall use the methods fromsection 7 to construct this hull explicitly.

Let us start by calculating ExtnA(Mi,Mj) for n = 1, 2, 1 ≤ i, j ≤ 4. We need both thedimensions and k-linear bases for these vector spaces, where each basis vector is representedby a cocycle in the corresponding Yoneda complex. The calculations are straight-forward,so we only state the results here:

dimk Ext1A(Mi,Mj) =

1 if i = 1 or i = 4 and j = 2 or j = 3, orif i = 2 or i = 3 and j = 1 or j = 4,

0 otherwise

dimk Ext2A(Mi,Mj) =

1 if (i, j) = (1, 4), (2, 3), (3, 2), (4, 1),0 otherwise


We denote the basis vectors of Ext1A(Mj ,Mi) by x∗ij since there is at most one for eachpair of indices (i, j). From the dimensions listed above, we see that we have the followingbasis vectors:

x∗12, x∗13, x

∗21, x

∗24, x

∗31, x

∗34, x

∗42, x

∗43

We choose a Yoneda representative for each vector x∗ij in this list, and we denote thisrepresentative by α(xij). From the free resolutions above, we see that we can write each ofthese representatives in the form

α(X) = α(X)0, α(X)1,

where α(X)0 : L1 → L0 is right multiplication by a matrix ( ab ) with entries a, b ∈ A, andα(X)1 : L2 → L1 is right multiplication by a matrix ( c d ) with entries c, d ∈ A for eachmonomial X = xij . We find the following representatives:

α(x12) = α(x21) = α(x34) = α(x43) = ( 01 ) , ( 1 0 )

α(x13) = α(x31) = α(x24) = α(x42) = ( 10 ) , ( 0 −1 )

Similarly, we denote the basis vectors of Ext2A(Mj ,Mi) by y∗ij since there is at most onefor each pair of indices (i, j). From the dimensions listed above, we see that we have thefollowing basis vectors:

y∗14, y∗23, y

∗32, y

∗41

We choose a Yoneda representative for each vector y∗ij in this list, and we denote thisrepresentative α(yij). From the free resolutions above, we see that we can write each ofthese representatives in the form

α(Y ) = α(Y )0,

where α(Y )0 : L2 → L0 is given by right multiplication of an element a ∈ A for eachmonomial Y = yij . We find the following representatives:

α(y14) = α(y23) = α(x32) = α(x41) = ( 1 )

This completes the calculations of ExtnA(Mi,Mj) for n = 1, 2 and 1 ≤ i, j ≤ 4. We knowthat these calculations determine the hull at the tangent level, (H2, ξ2).

The next step is to find the the hull H and the versal family ξ, and we shall employthe notations and methods of section 7 to accomplish this. Let N = 2, we know that thischoice is always possible. As usual, we let T1 be the formal matrix algebra generated bythe monomials xij in the above list, and let I = I(T1) be its radical. Furthermore, denoteby fij = o(yij) ∈ I2

ij for (i, j) = (1, 4), (2, 3), (3, 2), (4, 1), and by fnij the correspondingtruncated power series for each n ≥ N .

First, we have to find a defining system α(X) : |X| < 2 for the matric Massey products〈x∗;X〉 when X is any monomial of degree 2 in T1. This is easily done: The 1-cocycle α(ei)is the free resolution of Mi for 1 ≤ i ≤ 4, and the 1-cocycle α(X) was chosen above for eachmonomial X = xij of degree 1.

122 EIVIND ERIKSEN

Let us calculate the matric Massey products of order 2: Using the defining system givenabove, we find that the cocycles y(X) representing the matric Massey products 〈x∗;X〉 aregiven by

y(X)0 =

−1 if X = x12x24, x21x13, x34x42, x43x31,1 if X = x13x34, x24x43, x31x12, x42x21,0 otherwise

for all monomials X of degree 2 in T1. This means that the corresponding matric Masseyproducts are given by

〈x12, x24〉 = −y14 〈x13, x34〉 = y14

〈x21, x13〉 = −y23 〈x24, x43〉 = y23

〈x31, x12〉 = y32 〈x34, x42〉 = −y32〈x42, x21〉 = y41 〈x43, x31〉 = −y41,

and all other matric Massey products of order 2 are zero. This translates to the followingtruncated power series f2

ij :

f214 = x13x34 − x12x24

f223 = x24x43 − x21x13

f232 = x31x12 − x34x42

f241 = x42x21 − x43x31

By the general theory, we therefore have H3 = T1/(f214, f

223, f

232, f

241) + I3. We know that

we can find a lifting ξ3 of ξ2 to H3, and that (H3, ξ3) is a pro-representing hull of DefMrestricted to a4(3).

In order to find ξ3, we let B(2) = X : |X| = 2 \ x13x34, x24x43, x31x12, x42x21. Wealso let B(2) = B(2) ∪ B(1), where B(1) = X : |X| ≤ 1. Then X : X ∈ B(2) isa monomial basis for H3. We observe that if we choose α(X) = 0 for all X ∈ B(2), thefamily α(X) : X ∈ B(2) defines an M-free complex over H3. In other words, this familycompletely defines the deformation ξ3 ∈ DefM(H3) lifting ξ2.

Clearly, we could continue in this way. But after the last computations, it is temptingto think that fij = f2

ij for (i, j) = (1, 4), (2, 3), (3, 2), (4, 1). Let us check if this is the case:We put T = T1/(f2

14, f223, f

232, f

241), and choose a monomial basis B of T containing B(2).

Furthermore, we let α(X) be as before when X ∈ B(2) and let α(X) = 0 for all monomialsX ∈ B of degree at least 3. This choice corresponds to maps dT0 , d

T1 of M-free modules over

T , and a computation shows that

dT0 dT1 = (1⊗ (f214 + f2

23 + f232 + f2

41)) = 0.

So the family α(X) : X ∈ B defines an M-free complex over T , and therefore a deforma-tion ξ ∈ DefM(T ) lifting ξ3. This proves that H = T , or in other words, that

H = T1/(x13x34 − x12x24, x24x43 − x21x13, x31x12 − x34x42, x42x21 − x43x31)

is a pro-representing hull of DefM. In particular, fij = f2ij for all i, j. Moreover, the family

α(X) : X ∈ B defines the versal family ξ ∈ DefM(H).


Appendix A. Yoneda and Hochschild representations

Let k be an algebraically closed (commutative) field, let A be an associative k-algebra,and let M,N be left A-modules. In this appendix, we recall several different descriptionsof the k-vector space ExtnA(M,N) for n ≥ 0. In particular, we show how to realize thiscohomology group using the Yoneda and Hochschild complexes.

A.1. The Yoneda representation. Fix free resolutions (L∗, d∗) of M and (L′∗, d′∗) of N .

We shall write di : Li+1 → Li and d′i : L′i+1 → L′i for the differentials, and denote theaugmentation morphisms by ρ : L0 →M and ρ′ : L′0 → N .

For all integers n ≥ 0, the cohomology group ExtnA(M,N) is defined to be the n’thcohomology group of the complex HomA(L∗, N),

ExtnA(M,N) = Hn(HomA(L∗, N)).

Notice that in general, this Abelian group does not have a left A-module structure, butonly a left C(A)-module structure, where C(A) is the centre of A. In particular, if A iscommutative, then ExtnA(M,N) has the structure of an A-module, and if A is a k-algebra,then ExtnA(M,N) has the structure of a k-vector space.

We denote by Hom∗(L∗, L′∗) the Yoneda complex given by the given free resolutions. Thiscomplex is defined in the following way: For each integer n ≥ 0, let Homn(L∗, L′∗) be theleft A-module

Homn(L∗, L′∗) = i HomA(Li+n, L′i).

Moreover, let the differential dn : Homn(L∗, L′∗) → Homn+1(L∗, L′∗) for n ≥ 0 be theA-linear map given by the formula

dn(φ)i = φidn+i + (−1)n+1d′iφi+1

for all i ≥ 0, where we write φ = (φi) with φi ∈ HomA(Li+n, L′i) for all i ≥ 0. It is easyto check that this map is a well-defined differential, so the Yoneda complex is a complex ofAbelian groups. We shall write Hn(Hom(L∗, L′∗)) for the cohomology groups of the Yonedacomplex. Since the differential d = dn is left C(A)-linear, these cohomology groups have anatural structure as left C(A)-modules.

Lemma A.1. For all integers n ≥ 0, there is a canonical isomorphism of left C(A)-modules

Hn(Hom(L∗, L′∗)) ∼= ExtnA(M,N).

Proof. There is a natural map fn : Homn(L∗, L′∗) → HomA(Ln, N), given by f(φ) = ρ′φ0,where φ = (φi) ∈ Homn(L∗, L′∗). It is easy to see that these maps are compatible with thedifferentials, and a small calculation show that fn induces an isomorphism on cohomologyHn(Hom(L∗, L′∗))→ ExtnA(M,N) for all integers n ≥ 0.

A.2. Definition of Hochschild cohomology. Let Q be an A-A bimodule. We definethe Hochschild complex of A with values in Q in the following way: Let HCn(A,Q) =Homk(⊗nkA,Q) for all n ≥ 0. So any ψ ∈ HCn(A,Q) corresponds to a k-multilinear mapfrom n copies of A into Q, and we shall therefore write ψ(a1, . . . , an) in place of ψ(a1⊗· · ·⊗

124 EIVIND ERIKSEN

an) for ψ ∈ HCn(A,Q), a1, . . . , an ∈ A. Moreover, let dn : HCn(A,Q) → HCn+1(A,Q) forn ≥ 0 be the k-linear map given by the formula

(7) dn(ψ)(a0, . . . , an) = a0 ψ(a1, . . . , an) +n∑i=1

(−1)iψ(a0, . . . , ai−1ai, . . . , an)

+(−1)n+1ψ(a0, . . . , an−1)an

for all ψ ∈ HCn(A,Q), a0, . . . , an ∈ A.

Lemma A.2. HC∗(A,Q) is a complex of k-vector spaces.

Proof. Let ψ ∈ HCn(A,Q). Then ψ′ = dn(ψ) is a sum of n+ 1 summands, and we denotethese by ψ′0, . . . , ψ

′n, in the order they appear in formula 7. We let ψ′′ = dn+1ψ′ = dn+1dnψ.

Each dn+1ψ′i for 0 ≤ i ≤ n is a sum of n + 2 summands, and we denote these by ψ′′ij for0 ≤ j ≤ n+ 1 in the order they appear in formula 7. A straight-forward calculation showsthat we have ψ′′i,j + ψ′′j,i+1 = 0 for all indices i, j with 0 ≤ j ≤ n + 2, j ≤ i ≤ n + 1. Sinceψ′′ =

∑ψ′′ij , it follows that ψ′′ = 0 in HCn+2(A,Q). Consequently, HC∗(A,Q) is a complex

of k-vector spaces.

We define the Hochschild cohomology of A with values in Q to be the cohomology of theHochschild complex HC∗(A,Q), so we have

HHn(A,Q) = Hn(HC∗(A,Q)) = ker(dn)/ Im(dn−1)

for all n ≥ 0. In particular, the cohomology groups HHn(A,Q) have a natural structure ask-vector spaces.

Let ψ ∈ HC1(A,Q), then ψ is a 1-cocycle if and only if ψ(ab) = aψ(b) + ψ(a)b for alla, b ∈ A. So we have ker(d1) = Derk(A,Q). We say that a derivation ψ ∈ Derk(A,Q) istrivial if there is an element q ∈ Q such that ψ is of the form ψ(a) = aq − qa for all a ∈ A.Clearly, the set of trivial derivations is the image Im(d0). So HH1(A,Q) ∼= Derk(A,Q)/Twhere T is the trivial derivations of A into Q.

A.3. The Hochschild representation. We remark that Q = Homk(M,N) is an A-Abimodule in a natural way: For any a ∈ A, let La : M → M denote left multiplication onM by a, and L′a : N → N left multiplication on N by a. The bimodule structure is givenby aφ = L′aφ, φa = φLa for a ∈ A, φ ∈ Homk(M,N). We shall consider the Hochschildcohomology of A with values in Q = Homk(M,N).

By definition, we have that HH0(A,Q) = HomA(M,N) when Q = Homk(M,N). So wehave a natural isomorphism of k-vector spaces Ext0A(M,N) ∼= HH0(A,Q). Notice that sincek ⊆ C(A), ExtnA(M,N) has a natural k-vector space structure for all n ≥ 0. It is possibleto extend the above isomorphism to the higher cohomology groups:

Proposition A.3. For all integers n ≥ 0, there is an isomorphism of k-vector spaces

σn : ExtnA(M,N)→ HHn(A,Homk(M,N)).

Proof. From Weibel [9], lemma 9.1.9, there is an isomorphism of k-vector spaces betweenHHn(A,Homk(M,N)) and ExtnA/k(M,N) for n ≥ 0. But since k is a commutative field,there is a canonical isomorphism between ExtnA/k(M,N) and ExtnA(M,N), see theorem8.7.10 in Weibel [9].


We shall give an explicit identification of k-vector spaces between Ext1A(M,N) andHH1(A,Homk(M,N)): Let (L∗, d∗) be a free resolution of M , with augmentation morphismρ : L0 → M , and let τ : M → L0 be a k-linear section of ρ. For any 1-cocycle φ ∈HomA(L1, N), let ψ = ψ(φ) ∈ Derk(A,Homk(M,N)) be the following derivation: For anya ∈ A, m ∈M , let x = x(a,m) ∈ L1 be such that d0(x) = aτ(m)−τ(am). Notice that suchan x exists, and is uniquely defined modulo the image Im d1. We define ψ by the equationψ(a)(m) = φ(x) with x = x(a,m). Since φ is a cocycle, ψ is a well-defined homomorphismin Homk(A,Homk(M,N)), and a straight-forward calculation shows that ψ is a derivation.

Lemma A.4. Assume that Ext1A(M,N) is a finite dimensional k-vector space. Thenthe assignment φ → ψ(φ) defined in the above paragraph induces an isomorphism σ1 :Ext1A(M,N)→ HH1(A,Homk(M,N)).

Proof. Assume that φ is a co-boundary, so φ = d0(φ′), where φ′ ∈ HomA(L0, N). Thenψ = d0(φ′), where ψ′ = φ′τ ∈ Homk(M,N), so φ is a trivial derivation. Consequently,the assignment induces a well-defined map of k-linear spaces. This map is furthermoreinjective: Assume that ψ is a trivial derivation, so ψ = d0(ψ′), where ψ′ ∈ Homk(M,N).Then, we can construct an A-linear map φ′ ∈ HomA(L0, N) in the following way: Choosea basis for L0, and for each basis vector y ∈ L0, choose y′ ∈ L1 such that d0(y′) =y−ψ′ρ(y). Then we define φ′(y) = ψ′ρ(y)+φ(y′) for each basis vector y ∈ L0. We obtain amorphism φ′ ∈ HomA(L0, N) by A-linear extension, and d0(φ′) = φ, so φ is a co-boundary.To show that σ1 is an isomorphism as well, it is enough to notice that dimk Ext1A(M,N) =dimk HH1(A,Homk(M,N)) by proposition A.3, since Ext1A(M,N) has finite k-dimension.

The identification σn : ExtnA(M,N) → HHn(A,Homk(M,N)) for n ≥ 2 can beconstructed in a similar way.

References

[1] N. Bourbaki, Algebre commutative, Elements de mathematique, Masson, 1985.[2] Olav Arnfinn Laudal, Formal moduli of algebraic structures, Lecture notes in mathematics,

no. 754, Springer-Verlag, 1979.[3] ——, A generalized burnside theorem, Preprint Series no. 42, University of Olso, 1995.[4] ——, Noncommutative deformations of modules, Preprint Series no. 2, University of

Oslo, 1995.[5] ——, Noncommutative algebraic geometry, Preprint Series no. 28, University of Olso, 1996.[6] ——, Noncommutative algebraic geometry II, Preprint Series no. 12, University of Olso, 1998.[7] ——, Noncommutative algebraic geometry, Preprint Series no. 115, Max Planck Institute of

Mathematics, 2000.[8] ——, Noncommutative deformations of modules, Homology, Homotopy and Applications 4

(2002), no. 2, 357–396.[9] Charles A. Weibel, An introduction to homological algebra, Cambridge studies in advanced

mathematics, no. 38, Cambridge University Press, 1994.

Institute of Mathematics, University of Warwick, Coventry CV4 7AL, UK


SYMMETRIC FUNCTIONS, NONCOMMUTATIVE SYMMETRICFUNCTIONS AND QUASISYMMETRIC FUNCTIONS II

by

MICHIEL HAZEWINKEL

CWI, POBox 94079, 1090GB Amsterdam, The Netherlands

Abstract. Like its precursor this paper is concerned with the Hopf algebra of noncom-mutative symmetric functions and its graded dual, the Hopf algebra of quasisymmetricfunctions. It complements and extends the previous paper but is also selfcontained.Here we concentrate on explicit descriptions (constructions) of a basis of the Lie alge-bra of primitives of NSymm and an explicit free polynomial basis of QSymm. As beforeeverything is done over the integers. As applications the matter of the existence ofsuitable analogues of Frobenius and Verschiebung morphisms is discussed.

MSCS: 16W30, 05E05, 05E10, 20C30, 14L05

Key words and key phrases: symmetric function, quasisymmetric function, noncom-mutative symmetric function, Hopf algebra, divided power sequence, endomorphismof Hopf algebras, automorphism of Hopf algebras, Frobenius operation, Verschiebungoperation, Adams operator, power sum, Newton primitive, Solomon descent algebra,cofree coalgebra, free algebra, dual Hopf algebra, lambda-ring, Leibniz Hopf algebra,Lie Hopf algebra, Lie polynomial, formal group, primitive of a Hopf algebra shufflealgebra, overlapping shuffle algebra.

1. Introduction

As said before, [24], the symmetric functions are an exceedingly fascinating object ofstudy; they are best studied from the Hopf algebraic point of view (in my opinion), althoughthey carry quite a good deal more important structures, indeed so much that whole booksdo not suffice, but see [26, 27, 31, 33, 34].The first of the two generalizations to be discussed is the Hopf algebra, NSymm, of noncom-mutative symmetric functions (over the integers). As an algebra, more precisely a ring, thisis simply the free associative ring over the integers, Z, in countably many indeterminates

NSymm = Z〈Z1, Z2, . . .〉 (1.1)

and the coalgebra structure is given by the comultiplication determined by

µ : Zn →∑i+j=n

Zi ⊗ Zj , where Z0 = 1 (1.2)

and i and j are in N ∪ 0 = 0, 1, 2, · · · . The augmentation is given by

ε(Zn) = 0, n = 1, 2, 3, . . . (1.3)

SYMM, NSYMM AND QSYMM FUNCTIONS 127

(and, of course ε(Z0) = ε(1) = 1). The Hopf algebra NSymm is a noncommutative coveringgeneralization of the Hopf algebra of symmetric functions,

Symm = Z[z1, z2, . . .] (1.4)

where the zn are seen as either the elementary symmetric functions en or the completesymetric functions hn. The interpretation of the zn as the hn seems to work out somewhatnicer, for instance in obtaining the standard inner product autoduality of Symm in termsof the natural duality between NSymm and QSymm, the Hopf algebra of quasisymmetricfunctions, see [24], section 6. QSymm will be described and discussed later in this paper.

The projection is given by

NSymm −→ Symm, Zn → Zn (1.5)

and is a morphism of Hopf algebras.The systematic investigation of NSymm as a noncommutative generalization of Symm

was started in [14] and continued in a whole slew of subsequent papers, e.g. [7, 8, 9, 20, 21,22, 23, 25, 28, 29, 30, 32, 46].

It is amazing how much of the theory of Symm has natural noncommutative analogues.This includes Newton primitives, Schur functions, representation theoretic interpretations,determinental formulas now involving the quasideterminants of Gel’fand - Retakh, [12, 13]),Capelli and Sylvester identities, and much more. And, not rarely, the noncommutativeversions are more elegant than their commutative counterparts.

Note, however, that in most of these papers the noncommutative symmetric functionsare studied over a fixed field K of characteristic zero and not over the integers (or a field ofpositive characteristic). This makes quite a difference, see section 3 below. The papers [19,20, 21, 22, 23] focuss on the case over the integers, as does the present paper.

It should be stressed that NSymm attracts a lot of attention not only as a naturalgeneralization of Symm. It turns up spontaneously. For instance in terms of representationsof the Hecke algebras at zero, [8, 24, 30, 46] and as the direct sum of the Solomon descentalgebras of the symmetric groups, [1, 10, 14, 35, 43, 44] and [39], Ch. 9. Moreover there aree.g. applications to noncommutative continued fractions, Pade approximants, and a varietyof interrelations with quantum groups and quantum enveloping algebras, [2, 14, 29, 37].Further, the duals, the quasisymmetric functions, first turned up (under that name) in thetheory of plane partitions and counting permutations with given descent sets, [15, 16, 45].Actually, QSymm, precisely as the graded dual of NSymm, goes back at least to 1972 in thetheory of noncommutative formal groups, [5]. See [20] for an outline of the role played byQSymm in that context. An application of NSymm to chromatic polynomials is in [11].Given a Hopf algebra H, with multiplication m and comultiplication µ, a primitive in H isan element P of H such that

µ(P ) = 1⊗ P + P ⊗ 1 (1.6)

The primitives of a Hopf algebra form a Lie algebra under the commutator product

[P1, P2] = P1P2 − P2P1 (1.7)

128 MICHIEL HAZEWINKEL

which is denoted Prim(H). For any Hopf algebra there is strong interest ina descriptionof its Lie algebra of primitives. For instance because of the Milnor - Moore theorem, [36],that says that a graded connected cocommutative Hopf algebra over a field of characteristiczero is isomorphic to to the universal enveloping algebra of its Lie algebra of primitives.Also, far from unrelated, let Q(H) = I(H)/I(H)2 be the module of indecomposables of agraded Hopf algebra H. Here I(H) is the augmentation ideal of H. Then there is an inducedduality between Q(H) and Prim(H∗), and there is the (classical) Leray theorem that saysthat for a connected commutative graded Hopf algebra H over a characteristic zero fieldany section of I(H) −→ Q(H) induces an isomorphism of the free commutative algebraover Q(A) to H. This last theorem now has been considerably generalized to the setting ofoperads, see [38], and the references quoted there.

The first main topic that is treated in some detail (but without proofs) in this survey isan explicit and algorithmic description of a basis over the integers of Prim(NSymm).A divided power sequence in a Hopf algebra H is a sequence of elements

d = (d(0) = 1, d(1), d(2), . . .) (1.8)

such that for all n

µH(d(n)) =∑i+j=n

d(i)⊗ d(j) i, j ∈ 1, 2, 3, . . . (1.9)

Note that d(1) is a primitive. Is is sometimes useful to write a DPS (divided power sequence)as a power series in a counting variable t:

d(t) = 1 + d(1)t+ d(2)t2 + d(3)t3 + · · · (1.10)

That makes it easier to talk about the inverse of a DPS (inverse power series), the productof two DPS’s (multiplication of power series) and shifted DPS’s: d(t) → d(tn), all operationsthat give new DPS’s from old ones. When written in the form (1.10) a DPS is often calleda curve.

It turns out that each primitive of Prim(NSymm) can be extended to a divided powersequence. This is important because it implies that as a coalgebra NSymm is the cocommu-tative cofree graded coalgebra over the module Prim(NSymm).Now let QSymm be the graded dual Hopf algebra (over the integers) of NSymm. For anexplicit description of QSymm, the Hopf algebra of quasisymmetric functions, see below insection 2.

A most important question concerning QSymm is whether it is free polynomial as acommutative algebra. This has been an important issue since 1972, since it is crucial for thedevelopment of certain parts of the theory of noncommutative formal groups, [5, 6, 17]. Thematter was finally settled in 1999, [21], in the positive sense that it is indeed free. A secondproof follows from the cofreeness of NSymm. However, both these proofs fail to produceexplicit generators. This has now also been taken care of, [23], and is the second main topicthat will be discussed in some detail below.

One most interesting and important aspect of the structure of Symm is the presenceof two families of Hopf algebra morphisms that are called Frobenius and Verschiebungmorphisms. They satisfy a large number of beautiful relations. The third main topic of thissurvey is to what extent these can be lifted to NSymm, respectively, extended to QSymm.


There are both positive and negative results. However, the matter has not yet been quitecompletely settled.

This paper is an expanded write-up of two talks that I gave on the subject: in Kras-noyarsk in August 2002 at the occasion of the International Conference “Algebra and itsapplications” in honour of the 70-th anniversary of V P Shunkov and the 65-th anniver-sary of V M Busarkin, and at the Z. Borewicz memorial conference in Skt Petersburg inSeptember 2002.

2. The Hopf algebra QSymm of quasisymmetric functions

Above, in the introduction, the graded Hopf algebra NSymm of noncommutative symmet-ric functions was defined. The grading is defined by

wt(Zn) = n (2.1)

and, more generally, if α = [a1, a2, . . . , am] is a nonempty word over the positive integersN = 1, 2, . . ., let Zα be the noncomutative monomial

Zα = Za1Za2 · · ·Zam(2.2)

then

wt(Zα) = wt(α) = a1 + · · ·+ am (2.3)

Let Z[ ] = 1, where [ ] is the empty word, then the Zα, α ∈ N∗, the monoid of words overN form a basis of NSymm (as a graded Abelian group). The empty word, and also Z[ ] = 1,has weight zero.

As a free Abelian graded group QSymm, the graded dual of NSymm can be taken to bethe free Abelian group with as basis N∗, the words over the set of natural numbers. Theduality is then

< Zα, β >= δβα (2.4)

The duality induced comultiplication is easy to describe. It is ‘cut’:

[a1, a2, . . . , am] →m∑i=0

[a1, . . . , ai]⊗ [ai+1, . . . , am] (2.5)

where of course [a1, . . . , ai] = [ ] = 1 if i = 0 and [ai+1, . . . am] = [ ] = 1 if i = m. Theduality induced multiplication is more difficult to describe. It is the socalled ‘overlappingshuffle multiplication’ which can be described as follows.

Let α = [a1, a2, . . . , am] and β = [b1, b2, . . . , bn] be two compositions or words. Take a‘sofar empty’ word with n + m − r slots where r is an integer between 0 and minm,n,0 ≤ r ≤ minm,n. Choose m of the available n +m − r slots and place in it the naturalnumbers from α in their original order; choose r of the now filled places; together with theremaining n+m−r−m = n−r places these form n slots; in these place the entries from β intheir orginal order; finally, for those slots which have two entries, add them. The product of


the two words α and β is the sum (with multiplicities) of all words that can be so obtained.So, for instance,

[a, b]×osh [c, d]= [a, b, c, d] + [a, c, b, d] + [a, c, d, b] + [c, a, b, d] + [c, a, d, b] + [c, d, a, b]+

+ [a+ c, b, d] + [a+ c, d, b] + [c, a+ d, b] + [a, b+ c, d] + [a, c, b+ d]+ (2.6)

+ [c, a, b+ d] + [a+ c, b+ d]

and [1]×osh [1]×osh [1] = 6[1, 1, 1] + 3[1, 2] + 3[2, 1] + [3].There is a concrete realization of QSymm much like the standard realization of Symm as

the ring of symmetric functions in infinitely many indeterminates x1, x2, . . . . See [34], Chap-ter 1 for some detail on how to work with infinitely many indeterminates in this context.

Let X be a finite or infinite set (of commuting variables) and consider the ring of poly-nomials, R[X], and the ring of power series, R[[X]], over a commutative ring R with unitelement in the commuting variables from X. A polynomial or power series f(X) ∈ R[[X]]is called symmetric if for any two finite sequences of indeterminates x1, x2, . . . , xn andy1, y2, . . . , yn from X and any sequence of exponents i1, i2, . . . , in ∈ N, the coefficients inf(X) of xi11 x

i22 . . . xinn and yi11 y

i22 . . . yinn are the same.

The quasi-symmetric formal power series are a generalization introduced by Gessel, [15],in connection with the combinatorics of plane partitions. This time one takes a totallyordered set of indeterminates, e.g. V = v1, v2, . . ., with the ordering that of the naturalnumbers, and the condition is that the coefficients of xi11 x

i22 . . . xinn and yi11 y

i22 . . . yinn are equal

for all totally ordered sets of indeterminates x1 < x2 < · · · < xn and y1 < y2 < · · · < yn.Thus, for example,

x1x22 + x2x

23 + x1x

23 (2.7)

is a quasi-symmetric polynomial in three variables that is not symmetric.Products and sums of quasi-symmetric polynomials and power series are again quasi-

symmetric (obviously), and thus one has, for example, the ring of quasi-symmetric powerseries QSymm∧ in countably many commuting variables over the integers and its subring

QSymm (2.8)

of quasi-symmetric polynomials in finite of countably many indeterminates, which are thequasi-symmetric power series of bounded degree. The notation is justified. The quasisym-metric functions in x1, x2, . . . in this sense are a concrete realization of the quasisymmetricfunctions as introduced above as the graded dual of NSymm.

In detail, given a word α = [a1, a2, . . . , am] over N, also called a composition in thiscontext, consider the quasi-monomial function

Mα =∑

i1<···<imxa1i1xa2i2. . . xam

im(2.9)

defined by α. It is now an easy exercise to verify that as power series in the x1, x2, . . . theMα satisfy

MαMβ = Mα×oshβ (2.10)

where ×osh is the overlapping shuffle product of words just defined above (2.6).


3. NSymm and QSymm over a field of characteristic zero

Let K be a field of characteristic zero, in particular the field of rational numbers Q. TheHopf algebras of noncommutative symmetric functions and quasisymmetric functions overK are denoted

NSymmK = NSymm⊗Z K, QSymmK = QSymm⊗Z K (3.1)

As remarked in the introduction, working over a field of characteristic zero tends to simplifythings considerably. Not that then everything becomes clear and easy and straightforward.Very far from it; witness the many papers quoted in the introduction. However, it is certainlytrue, that for the three groups of questions which form the main topic of this paper: prim-itives of noncommutative symmetric functions, freeness of the algebra of quasisymmetricfunctions, existence of Frobenius and Verschiebung morphisms, things either reduce toknown things, or become fairly straightforward. At least up to fairly explicitly given isomor-phisms. And in that connection it is well to reflect that knowing something well up to anisomorphism can be not all the same as really controlling things.

To start with, consider another Hopf algebra over the integers, the Lie Hopf algebra

U = Z〈U1, U2, . . .〉, µ(Un) = 1⊗ Un + Un ⊗ 1, ε(Un) = 0, n = 1, 2, . . . (3.2)

That is, as an algebra U is the free algebra over the integers in the noncommuting variablesUn and the coalgebra structure is given by requiring that the two right hand formulas of(3.2) are algebra homomorphisms.

The primitives of U are called Lie polynomials and they form the free Lie algebra over Zin the alphabet U1, U2, · · · . Also the universal enveloping algebra of Prim(U) is U.

Now, over the rationals (and hence over any field of characteristic zero or ringcontaining Q) NSymm1 and U are isomorphic. One particularly beautiful isomorphismNSymmQ

ϕ−→UQ is given by setting

1 + Z1t+ Z2t2 + Z3t

3 + · · · = exp(U1t+ U2t2 + U3t

3 + · · · )

=∞∑i=0

(U1t+ U2t2 + U3t

3 + · · · )i (3.3)

where t is a (counting) variable commuting with everything in sight. Equating equal powersof t on the left and right hand sides of (3.3) gives

Zn =∑

i1 + · · · ik = nij ∈ N

Ui1Ui2 · · ·Uikk!

(3.4)

For a proof that the algebra isomorphism defined by setting ϕ(Zn) equal to the right handside of (3.4) defines an isomorphism of Hopf algebras see [18].

Thus, up to the isomorphism ϕ, the matter of describing Prim(NSymm), comes down todescribing Prim(U), or, equivalently writing down some explicit bases for the free Lie algebra

1The Hopf algebra NSymm is sometimes called the Leibniz Hopf algebra.


generated by a countable set of indeterminates U1, U2, . . .. The matter of constructingbases for free Lie algebras has had a lot af attention. There are many more or less differentbases such as Hall bases, Shirshov bases, Lyndon bases, see [3, 39, 41, 42, 47].

Here, partly because most of the concepts will be needed below anyway, is a descriptionof the Lyndon basis (also called Chen - Fox - Lyndon basis).

Consider the free monoid N* of words on the alphabet N (or any other totally orderedalphabet for that matter). The lexicographic order (also called dictionary order, or alpha-betical order) on N* is defined as follows. If α = [a1, a2, . . . , am] and β = [b1, b2, . . . , bn] aretwo words of length m and n respectively, lg(α) = m, lg(β) = n,

α >lex β ⇔∃k ≤ minm,n such that a1 = b1, . . . , ak−1 = bk−1 and ak > bk

or lg(α) = m > lg(β) = n and a1 = b1, . . . , an = bn(3.5)

The empty word is smaller than any other word. This defines a total order. Of course,if one accepts the dictum that anything is larger than nothing, the second clause of (3.5)is superfluous.

The proper tails (suffixes) of the word α = [a1, a2, . . . , am] are the words [ai, ai+1, . . . am],i = 2, 3, . . . ,m. Words of length 1 or 0 have no proper tails. The prefix corresponding toa tail α′′ = [ai, ai+1, . . . am] is α′ = [a1, . . . , ai−1] so that α = α′ ∗ α′′ where * denotesconcatenation of words.

A word is Lyndon iff it is lexicographically smaller than each of its proper tails. Forinstance [4], [1, 3, 2], [1, 2, 1, 3] are Lyndon and [1, 2, 1] and [2, 1, 3] are not Lyndon.

For each Lyndon word α of length > 1 consider the lexicographically smallest proper tailα′′ of α. Let α′ be the corresponding prefix to α′′. Then α′ and α′′ are both Lyndon andα = α′ ∗ α′′ is called the canonical factorization of α.A basis of the free Lie algebra on U1, U2, . . ., i.e. a basis of Prim(U) ⊂ U, is now obtainedas follows. For each word α = [a1, a2, . . . , am] let Uα = Ua1Ua2 . . . Uam

be the correspondingmonomial. Now, by recursion in length, define for a word of length 1

Q[i] = Ui (3.6)

and for α Lyndon and of length lg(α) ≥ 2 let α = α′ ∗ α′′ be its canonical factorizationand set

Qα = [Qα′ , Qα′′ ] (3.7)

then the Qα : α Lyndon form a basis of Prim(U) ⊂ U. For a proof see e.g. [39], p. 105ff.The next topic to be taken up is the matter of the freeness of QSymm over the rationals.

The graded dual of U is the socalled shuffle algebra. As a free module over Z it has thewords over N as a basis and the product is the shuffle product which is like the overlappingshuffle product except that the overlap terms, i.e. those which involve additions of entriesare left out. Thus for example

[a, b]×sh [c, d] = [a, b, c, d] + [a, c, b, d] + [a, c, d, b] + [c, a, b, d] + [c, a, d, b] + [c, d, a, b]

(compare (2.6) above).It is well known that the shuffle algebra is free polynomial with as generators

(for example) the Lyndon words. See, for example, [39]. p. 111 for a proof. Thus via the


isomorphism ϕ, or rather its graded dual, it follows that QSymmQ is a free commutativealgebra. But the description of the generators is rather involved and they do not look verynice. Actually the situation is rather better and a modification of the proof of the freenessof the shuffle algebra (using a different ordering on words) gives that in fact QSymmQ

is commutative free polynomial on the Lyndon words. The ordering to be used is thewll-ordering. The acronym stands for weight first, than length, than lexicographic. See [20]for details.

The third main topic of this survey is the existence of Frobenius and Verschiebung typeHopf algebra endomorphisms of NSymm and QSymm which lift, respectively extend, thoseon Symm. Again, over the rationals, this is a relatively straightforward matter. Thoughthere are some unanswered questions.

Recall the situation for Symm, see [17, 24] for more details.On Symm there are two families of Hopf algebra endomorphims, called Frobenius andVerschiebung morphisms, denoted fn,vn, n ∈ N, which among others have the followingbeautiful properties:

(i) f1 = v1 = id(ii) fn is homogeneous of degree n, i.e. fn(Symmk) ⊂ Symmnk Here, for any graded Hopf

algebra, H,Hn is the homogeneous part of of weight n of H.(iii) vn is homogenous of degree n−1, i.e. vn(Symmk) ⊂ Symn−1k if n divides k, and

vn(Symmk) = 0 if n does not divide k.(iv) fnfm = fnm for all n,m ∈ N(v) vnvm = vnm for all n,m ∈ N(vi) fnvm = vmfn provided n and m are relatively prime, gcd(m,n) = 1(vii) vnfn = n, where n is the n-fold convolution of the identiy.

Now there is the natural projection

NSymm −→ Symm, Zn → hn (3.8)

and the natural (graded dual) inclusion

Symm ⊂ QSymm (3.9)

obtained by regarding a symmetric function as a special kind of quasisymmetric function.The question is whether there are lifts, respectively extensions, on NSymm, respectivelyQSymm, which also have the properties (i) - (vii).

Retaining property (vii) can be ruled out immediately for trivial reasons. The simplefact is that n on either Qsymm or NSymm simply is not a Hopf algebra endomorphism.So it is natural to concentrate on the other six properties. And then the answer over therationals is yes. But, as will be stated below, the answer over the integers is no. But thereare interesting substitutes.

Let

pn = xn1 + xn2 + xn3 . . . (3.10)

denote the power sums in Symm. They are related to the complete symmetric functions bythe recursion relation

nhn = pn + pn−1h1 + pn−2h2 + · · · p1hn−1 (3.11)


The Frobenius and Verschiebung morphisms on Symm are characterized by

fnpk = pnk, vnpk =

npk/n if n divides k,0 if n does not divide k.

(3.12)

On the polynomial generators hn this characterization of vn works out as

vnhk =

hk/n if n divides k,0 otherwise.

(3.13)

Define the (noncommutative) Newton primitives in NSymm by

Pn(Z) =∑

r1+···rk=n

(−1)k+1rkZr1Zr2 . . . Zrk, ri ∈ N = 1, 2, . . . (3.14)

or, equivalently, by the recursion relation

nZn = Pn(Z) + Z1Pn−1(Z) + Z2Pn−2(Z) + · · ·+ Zn−1P1(Z) (3.15)

Note that under the projection Zn → hn by (3.15) and (3.11) Pn(Z) goes to pn. It is easilyproved by induction, using (3.15), or directly from (3.14), that the Pn(Z) are primitivesof NSymm, and it is also easy to see from (3.15) that over the rationals NSymm is thefree associative algebra generated by the Pn(Z). Thus over the rationals the Lie algebra ofprimitives of NSymm is simply the free Lie algebra generated by the Pn(Z), giving a seconddescription of Prim(NSymmQ).

There are obvious candidate lifts of the vn on Symm to Hopf algebra endomorphisms onNSymm., viz

vn(Zk) =

Zk/n if k is divisible by n0 otherwise

(3.16)

By (3.14) or (3.15) this implies

vn(Pk) =

nPk/n if n divides k0 otherwise

(3.17)

Now on NSymmQ define the Frobenius morphisms as the algebra morphisms given by

fn(Pk(Z)) = Pnk(Z) (3.18)

It is now easily checked that the vn and fn as defined by (3.16) and (3.18) are Hopf algebraendomorphisms of NSymmQ, that they satisfy (the analogues on NSymmQ of) properties(i)-(vi) and that they descend to the usual Frobenius and Verschiebung morphisms on Symm.


A priori, the fn as defined by (3.18) are only defined over the rationals and indeednontrivial denominators show up almost immediately. For instance

f2(Z1) = 2Z2 − Z21

f2(Z2) = 2Z4 − 32Z1Z3 − 1

2Z3Z1 + Z22 + 1

2Z1Z2Z1 + 12Z

21Z2

(3.19)

On Symm a certain amount of coefficient magic sees to it that all coefficients become integral.But of course over Symm there are much better definitions of the Frobenius morphisms thatimmediately show that they are defined over the integers, see [24] or [17], §17.

As we shall see later, over the integers there are even no algebra endomorphisms fn ofNSymm that lift the fn on Symm such that together with the vn as defined by (3.16) theysatisfy (i)-(vi).

Note there is nothing unique about this solution (3.18) of the Frobenius-Verschiebunglifting problem over the rationals. For instance one could work instead with the seond setof Newton primitives defined by

P ′n(Z) =∑

r1+···rk=n

(−1)k+1r1Zr1Zr2 . . . Zrk, ri ∈ N = 1, 2, . . . (3.20)

and satisfying the recursion relation

nZn = P ′n(Z) + P ′n−1(Z)Z1 + P ′n−2(Z)Z2 + · · ·+ P ′1(Z)Zn−1 (3.20)

4. The primitives of NSymm

Above, some primitives of NSymm were already written down and they generate a freegraded Lie algebra contained in Prim(NSymm). Denote this Lie algebra by FrLie(P ) andits homogeneous part of weight n by FrLie(P )n. The Lie algebra Prim(NSymm) is alsograded of course. Let Prim(NSymm)n ⊂ NSymmn be the homogeneous part of degree nof Prim(NSymm). Both Prim(NSymm)n and FrLie(P )n are free Abelian groups of rankβn, the number of weight n Lyndon words2. The index of FrLie(P )n ⊂ Prim(NSymm)n asa function of n measures how large FrLie(P ) is in Prim(NSymm). As it turns out FrLie(P )is only a tiny part. Indeed, the value of the index alluded to is

Index of FrLie(P )n in Prim(NSymm)n =∏

α∈LYN, wt(α)=n

k(α)g(α)

(4.1)

where for a word α = [a1, a2, . . . , am] over the natural numbers g(α) is the gcd (greatestcommon divisor) of its entries a1, a2, . . . , am and k(α) is the product of its entries. Thus thevalues of (4.1) for the first six n are 1, 1, 2, 6, 576, 69120.

Thus taking iterated commutators of the known Newton primitives is not nearly goodenough. One can see this coming very quickly. Indeed [P1, P2] = 2(Z1Z2 − Z2Z1). It alsofollows that Prim(NSymm) is not a free Lie algebra over the integers. Rather it tries to besomething like a free divided power Lie algebra (though I do not know what such a thingmight be).

2The numbers βn are given by the identity (1 − t)−1(1 − 2t) =∏∞

n=1(1 − tn)βn which goes back to

Witt, [48].


Instead of taking commutators of primitives it turns out to be a good idea to work with wholeDPS’s (divided power sequences, see (1.8)). Accordingly, the next thing to be described aretechniques for producing new divided power sequences from known ones. There are twomore techniques for this (besides the ones mentioned in the introduction, which do notsuffice) coming from two socalled isobaric decomposition theorems.

For the first isobaric decomposition theorem consider the Hopf algebra

2NSymm = Z〈X1, Y1,X2, Y2, · · · 〉, µ(Xn) =∑i+j=n

Xi ⊗Xj , µ(Yn) =∑i+j=n

Yi ⊗ Yj (4.2)

and the two natural curves

X(s) = 1 +X1s+X2s2 + · · · , Y (t) = 1 + Y1t+ Y2t

2 + · · · (4.3)

and consider the commutator product

X(s)−1Y (t)−1X(s)Y (t) (4.4)

On the set of pairs of nonnegative integers consider the ordering

(u, v) <wl (u′, v′) ⇔ u+ v < u′ + v′ or (u+ v = u′ + v′ and u < u′) (4.5)

(Here the index wl on <wl is supposed to be a mnemonic for weight first, then lexico-graphic.)

4.6 Theorem. (first bi-isobaric decomposition theorem, Shay [40], Ditters). There are‘higher commutators’ (or perhaps better ‘corrected commutators’)

Lu,v(X,Y ) ∈ Z〈X,Y 〉, (u, v) ∈ N×N (4.7)

such that

X(s)−1Y (t)−1X(s)Y (t) =→∏

gcd(a,b)=1

(1 + La,b(X,Y )satb + L2a,2b(X,Y )s2at2b + · · · ) (4.8)

where the product is an ordered product for the ordering <wl just introduced, (4.5).Moreover

(i) Lu,v(X,Y ) = [Xu, Yv] + (terms of length ≥ 3) (4.9)(ii) Lu,v(X,Y ) is homogeneous of weight u in X and of weight v in Y . (4.10)(iii) For gcd(a, b) = 1, 1 + La,b(X,Y )satb + L2a,2b(X,Y )s2at2b + · · · is a 2-curve.


Here a 2-curve is a two dimensional version of a curve. A power series in two variableswith constant term 1

d(s, t) =∑i,j

d(i, j)sitj (4.11)

is a 2-curve iff

µ(d(m,n)) =∑

m1+m2=mn1+n2=n

d(m1, n1)⊗ d(m2, n2) (4.12)

This is not at all difficult to prove. The only thing needed is to observe that pure powers of sor t do not occur on the LHS of (4.8) and that each pair of nonnegative integers (u, v) occursjust once in one of the factors on the right of (4.8). That gives the decomposition. The factthat the factors are two curves then follows easily with induction from the observation thatthe LHS of (4.8) is a 2-curve.

Also (4.8) implies an explicit recursion formula for the Lu,v(X,Y ).4.13. Theorem. (Second bi-isobaric decomposition theorem, Hazewinkel [22]). There areunique homogeneous noncommutative polynomials Nu,v(Z) ∈ NSymm such that

Z(s)−1Z(t)−1Z(s+ t) =→∏

a,b∈Ngcd(a,b)=1

(1 +Na,b(Z)satb +N2a,2b(Z)s2at2b + · · · ). (4.14)

Moreover

(i) Nu,v(Z) =(u+ v

u

)Zu+v + (terms of length ≥ 2) (4.15)

(ii) Nu,v(Z) is homogeneous of weight u+ v (4.16)(iii) For each a, b ∈ N2, gcd(a, b) = 1,

1 +Na,b(Z)satb +N2a,2b(Z)s2at2b + · · · (4.17)is a 2-curve.

(iv) For each n ≥ 2, N1,n−1(Z) = Pn(Z) (4.18)

Again, the decomposition is not at all difficult to prove and the final observation resultsdirectly from the recursion formula implied by (4.14) compared to the recursion formula(3.15) for the Pn(Z).

There is now a sufficiency of tools to describe a basis of Prim(NSymm) and more. Theessential fact is that if d1, d2 are two divided power sequences in some Hopf algebra H, thanso is

Da,b(d1, d2) = (1, La,b(d1, d2), L2a,2b(d1, d2), . . .) (4.19)

where, as the notation suggests, Lu,v(d1, d2) is obtained from the Lu,v(X,Y ) of theorem 4.6by substituting d1(k) for Xk and d2(l) for Yl. This follows immediately from the fact thatfor gcd(a, b) = 1

1 + La,b(X,Y )satb + L2a,2b(X,Y )s2at2b + · · ·


is a 2-curve. Similarly, if d is a curve in any Hopf algebra H

Na,b(d) = 1 +Na,b(d)t+N2a,2b(d)t2 + · · · (4.20)

is another curve.3

Let LYN denote the set of Lyndon words over the natural numbers N = 1, 2, 3, . . ..Then to each α = [a1, . . . , am] ∈ LYN there are associated three things

(i) A number g(α) = gcda1, . . . , am(ii) A divided power sequence dα(iii) A primitive Pα

The items (ii) and (iii) are defined recursively as follows. If lg(α) = m is 1, d[n] =(1, Z1, Z2, . . .) and P[n] = Pn(d[n]) = Pn(Z). If lg(α) ≥ 2, let α = α′ ∗ α′′ be the canonicalfactorization of α (see just above (3.6)). Then

dα = (1.Lg(α′)/g(α),g(α′′)/g(α)(dα′ , dα′′), L2g(α′)/g(α),2g(α′′)/g(α)(dα′ , dα′′), . . .)

= Dg(α′)/g(α),g(α′′)/g(α)(dα′ , dα′′)(4.21)

and

Pα = Pg(α)(dα) (4.22)

Note that the divided power sequences associated to [a1, . . . , am] and [ra1, . . . , ram], r ∈ Nare the same.4.23. Theorem. The Pα, α ∈ LYN form a basis over the integers of Prim(NSymm). Eachof the Pα is the first term of a DPS.

The second property of theorem 4.23 guarantees that NSymm is the cofree cocommutativegraded coalgebra over the graded module Prim(NSymm), see the appendix of [22] for aproof of that. It follows that the graded dual QSymm is a free commutative algebra, andimplicitly specifies a set of generators for QSymm. However, this does not give a convenientdescription of such a set of generators.

The original proof of theorem 4.23, [22], is rather long and intricate. Fortunately thereis now a much shorter proof, which will be discussed in the next section.The basis Pα, α ∈ LYN of Prim(NSymm) has a number of nice properties, particularlywith respect to the Verschiebung endomorphisms vn of NSymm, see (3.16).

Consider the following ordering on words: α <wll β if and only if (wt(α) < wt(β) or(wt(α) = wt(β) and lg(α) < lgβ)) or (wt(α) = wt(β) and lg(α) = lgβ) and α <lex β)). Theacronym ‘wll’ stands for ‘weight first, than length, then lexicographic’.

3The operations on curves defined by (4.19) and (4.20) are functorial. Thus assigning to a Hopf algebra

H its group of curves gives a group valued functor with many operations on it, viz 2-ary operations La,b for

every pair a, b with gcd(a, b) = 1, and also unary operations Na,b for any such pair. In addition there are the

operations which come from the Verschiebung Hopf algebra endomorphisms on NSymm (see section 6) which

commute with the La,b and Na,b. And then there are also Frobenius type operations (see also section 6).

It is almost totally uninvestigated whether some subset of all this structure can be used for classification

purposes, say, of noncommutative formal groups.


4.24. Theorem. For each Lyndon word α = [a1, . . . , am]

(i) Pα = g(α)Zα + (wll larger terms) (4.25)

(ii) vn(Pα) =

nP[n−1a1,··· ,n−1am] if n divides g(α)0 otherwise.

(4.26)

The first property of theorem 4.24 results directly from the construction. The secondproperty comes from the fact that the curves

∑Lra,rb(X,Y )sratrb and

∑Nra,rb(Z)sratrb

of the two isobaric decomposition theorems 4.6 and 4.13 are particularly nice curves in that

vn(Lu,v(X,Y )) =

Ln−1u,n−1v(X,Y ) if n divides u, v,0 otherwise.

(4.27)

vn(Nu,v(Z)) =

Nn−1u,n−1v(Z) if n divides u, v0 otherwise.

(4.28)

where vn is defined on 2NSymm just like (3.16) for both the X’s and the Y ’s.

5. Free polynomial generators for QSymm over the integers

As in section 2 above, consider Symm and QSymm as symmetric functions in an infinityof ideterminates

Symm = Z[e1, e2, . . .] ⊂ QSymm ⊂ Z[x1, x2, . . .] (5.1)

where the ei are the elementary symmetric functions in the xj . There is a well known λ-ringstructure on Z[x1, x2, . . .] given by

λi(xj) =

xj if i = 10 if i ≥ 2

, j = 1, 2, . . . (5.2)

For information on λ-rings see [27]. The asscociated Adams operators, determined bythe formula

td

dtlog λt(a) =

∞∑n=1

(−1)nfn(a)tn (5.3)

where

λt(a) = 1 +∞∑n=1

λn(a)tn (5.4)

are the ring endomorphisms

fn : xj → xnj (5.5)


There are well-known determinantal relations between the λn and the fn as follows

n!λn(a) = det

f1(a) 1 0 · · · 0

f2(a) f1(a) 2. . .

......

.... . . . . . 0

fn−1(a) fn−2(a) · · · f1(a)n− 1fn(a) fn−1(a) · · · f2(a) f1(a)

(5.6)

fn(a) = det

λ1(a) 1 0 · · · 0

λ2(a) λ1(a) 1. . .

......

.... . . . . . 0

(n− 1)λn−1(a)λn−2(a) · · · λ1(a) 1nλn(a) λn−1(a) · · · λ2(a)λ1(a)

(5.7)

(which come from the Newton relations between the elementary symmetric functions andthe power sum symmetric functions).

It follows that the subrings Symm and QSymm are stable under the λn and fn becauseλn(QSymm⊗zQ) ⊂ QSymm⊗zQ by (1.6) and (QSymm⊗zQ)∩Z[x1, x2, . . .] = QSymm,and similarly for Symm. It follows that QSymm and Symm have induced λ-ring structures.On Symm this is of course the standard one.

It follows immediately from (5.5) that

fn([a1, a2, . . . , am]) = [na1, na2, . . . , nam] (5.8)

when α = [a1, a2, . . . , am] is seen as a monomial quasisymmetric function as in the realiza-tion of QSymm described in section 2.

It is more customary to denote the Adams operations (power operations) associated to aλ-ring structure by Ψ‘s. However, in the present case they coincide with the Frobenius endo-morphisms on Symm ([17], section E.2, p. 144ff), and their natural extensions to QSymm;so it seems natural to use f ’s instead in this case.

For any λ− ring R there is an associated mapping

Symm×R −→ R, (ϕ, a) → ϕ(λ1(a), λ2(a), . . . , λn(a), . . .) (5.9)

I.e. write ϕ ∈ Symm as a polynomial in the elementary symmetric functions e1, e2, . . . andthen substitute λi(a) for ei, i = 1, 2, . . .. For a fixed a ∈ R this is obviously a homomorphismof rings Symm −→ R. We shall often simply write ϕ(a) for ϕ(λ1(a), λ2(a), . . . , λn(a), . . .).

Another way to see (5.9) is to observe that for fixed a ∈ R(ϕ, a) → ϕ(λ1(a), λ2(a), . . .) =ϕ(a) is the unique homomorphism of λ-rings that takes e1 into a. (Symm is the free λ-ringon one generator, see also [27].) Note that

en(α) = λn(α), pn(α) = fn(α) = [na1, na2, . . . nam] (5.10)

The first formula of (5.10) is by definition and the second follows from (5.7) because therelations between the en and pn are precisely the same as between the λn(a) and the fn(a).


Now let P ∈ NSymm be a primitive. Then, by duality

〈P, αβ〉 = 〈µ(P ), α⊗ β〉 = 〈1⊗ P + P ⊗ 1, α⊗ β〉 = 0 (5.11)

for any words of length ≥ 1. Using the Newton relations

pn(α) = pn−1(α)e1(α)−pn−2(α)e2(α)+ · · · (−1)n−2p1(α)en−1(α)+(−1)n−1nen(α) (5.12)

it follows that for any primitive in NSymm

〈P, pn(α)〉 = ±n〈P, en(α)〉 (5.13)

Now for any α ∈ LYN , α = [a1, a2, . . . am] let αred = [g(α)−1a1, g(α)−1a2, . . . , g(α)−1am]and define

Eα = eg(α)(αred) (5.14)

5.15. Theorem. The Eα, α ∈ LYN form a free polynomial basis of QSymm over the integers

For a rather simple direct proof of this (based on Chen - Fox - Lyndon factorization, [4]),see [23].

Sometimes it is useful to relabel the Eα, α ∈ LYN a bit. Let eLYN be the set ofelementary Lyndon words, i.e those Lyndon words α for which g(α) = 1. Then the

en(α), α ∈ eLYN, n = 1, 2, 3, . . . (5.16)

are a sometimes convenient relabeling of the free polynomial basis Eα, α ∈ LYN . Notethat for a fixed α ∈ eLYN , the en(α) generate a subalgebra of QSymm that is isomorphicto Symm.

Now, for all weights n, consider the matrices of integers

(〈Pα, Eβ〉)α,β∈LYNn(5.17)

where LYNn is the set of Lyndon words of weight n and the columns and rows of (5.17) areordered by increasing wll-order.

Claim: this matrix is diagonal with entries ±1 on the main diagonal. The triangularityfollows from theorem 4.24 (ii). As to the diagonal part:

g(α)〈Pα, Eα〉= g(α)〈Pα, λg(α)(αred)〉 by definition=±〈Pα, pg(α)(αred)〉 by (5.13)=±〈Pα, α〉 by (5.10)=±g(α) by (4.25)

Using that there are precisely βn = #LY NnPα ‘s and Eα ‘s with α of weight n, theinvertibility (over the integers) of the matrix (5.17) immediately implies both that the Pαare a basis of Prim(NSymm) and that the Eα are a free polynomial basis for QSymm overthe integers


6. Frobenius and Verschiebung on NSymm and QSymm

To fix notations let

fSymmn and vSymmn

be the classical Frobenius and Verschiebung Hopf algebra endomorphisms over the integersof Symm, characterized by

fSymmn (pk) = pnk, vSymmn (pk) =

npk/n if n divides k0 otherwise

(6.1)

Now NSymm comes with a canonical projection NSymm −→ Symm,Zn → hn andthere is the inclusion Symm ⊂ QSymm (see section 2). The question arises whether therare lifts fNSymmn , vNSymmn on NSymm and extensions fQSymmn , vQSymmn on QSymm thatsatisfy respectively

(i) f?Symmn f?Symm

m = f?Symmnm

(ii) fn is homogeneous of degree n, i.e. fn(?Symmk) ⊂? Symmnk

(iii) f1 = v1 = id(iv) fmvn = vnfm if (n,m) = 1 (6.2)(v) v?Symm

n v?Symmm = v?Symm

nm

(vi) vn is homogeneous of degree n−1,

i.e. vn(? Symmk) ⊂

?Symmn−1k if n divides k0 otherwise.

Here ‘?’ can be ‘N’, or ‘Q’.Now there exist a natural lifts of the vSymmn to NSymm given by the Hopf algebra

endomorphisms

vNSymmn (Zk) =

Zk/n if n divides k0 otherwise

(6.3)

and there exist natural extension of the Frobenius morphisms on Symm to QSymm ⊃Symm given by the Hopf algebra endomorphisms

fQSymmn ([a1, . . . , am]) = [na1, . . . , nam] (6.4)

which, moreover, have the Frobenius-like property

fQSymmp (α) = αp mod p (6.5)

for each prime number p. These two families are so natural and beautiful that nothingbetter can be expected and in the following these are fixed as the Verschiebung morphismson NSymm and Frobenius morphisms on QSymm. They are also dual to each other.The question to be examined in this section is whether there are supplementary familiesof morphisms fn on NSymm, respectively vn on QSymm, such that (6.2) holds. The firstresult is negative


6.6. Theorem. There are no (Verschiebung-like) coalgebra endomorphisms vn of QSymmthat extend the vn on Symm, such that parts (iii)-(vi) of (6.2) hold. Dually there are no(Frobenius-like) algebra homomorphisms of NSymm that lift the fn on Symm such thatparts (i)-(iv) of (6.2) hold.

It is the coalgebra morphism property which makes it difficult for (6.2) (iii)-(vi) to hold.It is not particularly difficult to find algebra endomorphisms of QSymm that do the job. Forinstance define the vn on QSymm as the algebra endomorphisms given on the generators(5.16) by

vn(ek(α)) =

ek/n(α) if n divides k0 otherwise

(6.6)

It then follows from (5.12) that

vn(pk(α)) =

npk/n(α) if n divides k0 otherwise

(6.7)

and all the properties (6.2) follow.The last topic I would like to discuss is whether there are Hopf algebra endomorphisms

of QSymm (and dually, NSymm) such that some weaker versions of (6.2) hold.To this end we first discuss a filtration by Hopf subalgebras of QSymm. Define

Gi(QSymm) =∑

α, lg(α)≤iZα ⊂ QSymm (6.8)

the free subgroup spanned by all α of length ≤ i, and let

Fi(QSymm) = Z[en(α) : lg(α) ≤ i] (6.9)

the subalgebra spanned by those generators en(α), α ∈ eLYN , lg(α) ≤ i of length less orequal to i. Note that this does not mean that the elements of Fi(QSymm) are bounded inlength. For instance F1(QSymm) = Symm ⊂ QSymm contains the elements

[1, 1, . . . , 1]︸︷︷︸n

= en = en([1]) (6.10)

for any n.6.11. Theorem. Gi(QSymm) ⊂ Fi(QSymm)This is a consequence of the proof of the free generation theorem 5.15. Moreover,

Fi(QSymm)⊗z Q = Z[pn(α) : lg(α) ≤ i]Fi(QSymm) = Z[pn(α) : lg(α) ≤ i] ∩QSymm (6.11)

It follows from (6.11) and (6.12) that the Fi(QSymm) are not only subalgebras but subHopf algebras (because µ(pn(α)) =

∑α′∗α′′=α

pn(α′)⊗ pn(α′′)).4

4I believe the corresponding Hopf ideals in NSymm to be the iterated commutator ideals.


Now consider a coalgebra endomorphism of QSymm. Because of the commutative cofreenessof QSymm as a coalgebra over the module tZ[t] for the projection

QSymm −→ tZ[t], [ ] → 0, [n] → tn, α → 0 if lg(α) ≥ 2

or, equivalently, because of the freeness of NSymm over its submodule∑∞i=1 ZZi, a homo-

geneous coalgebra morphism of degree n−1 of QSymm is necessarily given by an expressionof the form

vϕ(α) =∑

α1∗···∗αr=α

ϕ(α1) · · ·ϕ(αr)[n−1wt(α1), . . . , n−1wt(αr)] (6.12)

for some morphism of Abelian groups ϕ : QSymm −→ Z. The endomorphism vϕ is a Hopfalgebra endomorphism iff ϕ is a morphism of algebras.

6.13. Proposition. vϕ(Fi(QSymm)) ⊂ Fi(QSymm)One particularly interesting family of ϕ’s is the family of ring morphisms given by

τn(en([1]) = τn(en) = (−1)n−1

τn(ek(α)) = 0 for k = n or lg(α) ≥ 2(α ∈ eLYN)(6.14)

Let vn be the Verschiebung type Hopf algebra endomorphism defined by τn according toformula (6.12). Then6.15. Theorem.

(i) vn([a1, . . . , am]) ≡nm[n−1a1, . . . n

−1am] mod(length m− 1) if n|ai ∀i0 mod(length m− 1) otherwise

(ii) vn extends vSymmn on Symm = F1(QSymm) ⊂ QSymm(iii) vpvq = vqvp on F2(QSymm)(iv) vnfn(α) = nlg(α)α mod(Flg(α)−1(QSymm))

And of course there is a corresponding dual theorem concerning Frobenius type endomor-phisms of NSymm.

This seems about the best one can do. One unsatisfactory aspect of theorem 6.15 is thatthere are also other families vn that work.To conclude I would like to conjecture a stronger version of theorem 6.6, viz that there is nofamily fn of algebra endomorphisms of NSymm over the integers that satisfies (6.2) (i)-(iii)and such that these fn descend to the fSymmn on Symm.

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QUOTIENT GROTHENDIECK REPRESENTATIONS

J.NDIRAHISHA

AND

F.VAN OYSTAEYEN

Dept. of Mathematics and Computer Science,

University of Antwerp (UIA)

B-2610 Wilrijk, Belgium

@-mail: [email protected]

Abstract. We generalize the concept of a Grothendieck representation to quotientGrothendieck representations; this allows the construction of a noncommutativeprojective geometry for geometrically graded rings that need not be positively graded.

Mathematics Subject Classifications (2000): 16 B 50; 14 A 15; 14 A 22.

Key words: (Quotient) Grothendieck representations; topological nerve; spectralrepresentation and scheme

Introduction

A Grothendieck representation of an arbitrary category B has been introduced in [9],with an aim to explain the appearance of canonical noncommutative topologies in noncom-mutative geometry. This point of view also provided a natural framework for a theory ofschemes for noncommutative algebras, including a general categorical version of Serre’sglobal sections theorem. Now the noncommutative theory of Proj may also be fit into thecontext of a Grothendieck representation but if one wants to relate the projective to thecorresponding “affine” theory, two Grothendieck representations of the same category haveto be compared. In this note we introduce the quotient of a Grothendieck representationwith respect to a so-called topological nerve with respect to B. We introduce some restrictingcondition, i.e. a spectral representation, allowing to retranslate certain properties holding inthe representing Grothendieck categories into properties related to a notion of spectrum in B.

The main result in Section 1, states that the quotient Grothendieck representation of aspectral representation is again a spectral representation.

In Section 2, we study an application to noncommutative algebra (or geometry). Asan example of the general construction in Section 1, we provide the construction andbasic noncommutative scheme theory for the scheme Proj of an arbitrary ZZ-graded ring(noncommutative and not necessarily positively graded). The main result following fromthese considerations is that Proj may indeed be viewed as a gluing of affine noncommuta-tive schemes.

1. Quotient Grothendieck Representations

Let B be a general category allowing products and coproducts. To R in B we asso-ciate a Grothendieck category Rep(R), to f : S → R in B we associate an exact func-tor F : Rep(R) → Rep(S) commuting with products and coproducts (we sometimes

148 J.NDIRAHISHA AND F.VAN OYSTAEYEN

write F = f0) such that: (IR)0 = IRep(R) for every R ∈ B, (f g)0 = g0 f0 forg: T → S, f : S → R in B.

A Grothendieck categorical representation (G. C.-representation) is a contra-variant functor Rep : B → R where R is the class consisting of the objects Rep(R), R ∈ Bwith HomR(Rep(R), Rep(S)) = HomB(S,R)0. The functor Rep associates to f : S → R inB the exact functor f0: Rep(R)→ Rep(S) given before.

We say that Rep: B → R measures B if for every R ∈ B there exists an object RRin Rep(R) such that to any f : S → R in B there is an associated morphism f∗: SS →Rep(f)(RR) in Rep(S), we put Rep(f)(RR) = SR, such that 1∗S is the identity of SS inRep(S) and for g: T → S, f : S → R in B, (f g)∗ = Rep(g)(f∗) g∗: TT →TS →TR, whereTR = Rep(g)Rep(f)(RR).

For detail on torsion theory (i.e localization theory, quotient categories, Serre localizingsubcategories, . . .) in a Grothendieck category, we refer to [2], [12].

For an arbitrary Grothendieck category G, the set of all torsion theories of G will bedenoted by Tors(G). It is known that Tors(G) is a complete distributive lattice. We usenotation as in [9], e.g we write κ, τ, γ, . . ., for torsion theories of G and Tκ, Tτ , Tγ , . . . resp., forthe classes of torsion objects in G. A partial order, ≤, is defined on Tors(G) by putting σ ≤ τif and only if Tσ ⊂ Tτ . AnM ∈ G is in Tσ∧τ if and only ifM is both in Tσ and Tτ ; on the otherhand Tσ∨τ is the torsion class generated by Tσ and Tτ . Recall that torsion classes in G areexactly those that are closed under taking: subobjects, images, finite products, extensions.For τ ∈ Tors(G), Tτ : G → G is the torsion functor corresponding to τ , i.e for M ∈ G, Tτ (M)is the largest object of Tτ contained in M . Any subfunctor T of the identity functor in G isof type Tτ (-) if and only if it is left exact and idempotent in the sense that T (M/T (M)) = 0.

As in [9] we abbreviate TorsRep to Top. To τ ∈ Tors(G) we associate the Serre quotientcategory (G, τ) together with the canonical functors:iτ : (G, τ)→ Gaτ : G → (G, τ), the reflector for τ , with aτ iτ the identity of (G, τ) and iτaτ = Qτ thelocalization functor G → G associated to τ .

To any exact functor F : Rep(R)→ Rep(S) that commutes with direct sums (coproducts)there corresponds a functor F 0: Top(S) → Top(R), defining F 0γ by letting its torsionclass consist of those objects X in Rep(R) such that F (X) ∈ Tγ . In case F derives from amorphism f : S → R in B, we write F 0 = f .

For any γ ∈ Top(A), A ∈ B, we put gen(γ) = τ, τ ≥ γ; for U ⊂ Top(A):

gen(∧U) ⊃ ∪τ∈U

gen(τ)

gen(∨U) = ∩τ∈U

gen(τ)

Hence, the subsets of the type gen(τ) for τ in Top(A) define a topology on Top(A). Foran exact functor F : Rep(R) → Rep(S) as above, F 0(γ) ≤ F 0(τ) if γ ≤ τ and F 0(∧U) =∧F 0(U) for any U ⊂ Top(S). Moreover, (F 0)−1(gen(τ)) = gen(ξτ ) for τ ∈ Top(R), whereξτ is such that Tξτ

is the torsion class generated by F (Tτ ). Note that F 0 is not necessarilya lattice morphism (F 0 does not necessarily respect ∨) but it is a continuous map in thegen-topologies.

In case γ ∈ Top(A) is perfect (Qγ is exact and commutes with arbitrary direct sums),then a0

γ defines a homeomorphism (in the gen topologies):Tors(Rep(A), γ) gen(γ) ( see also Lemma 1.4 in [9]).

QUOTIENT GROTHENDIECK REPRESENTATIONS 149

The foregoing expresses that open sets of type gen(τ) with perfect τ , behave as affineopen sets in scheme theory, i.e. viewed as topological spaces with the induced topologiesthey are homeomorphic to the topological space of a localized object.

In the sequel, we only consider G.C.-representations that are faithful in the sense thatfor any S −→

fR in B, Rep(f)(X) = 0 if and only if X = 0 (we write 0 for the zero object

in the category considered).

1.1. Definition. A faithful G.C.-representation that measures B is said to be spectral iffor every A ∈ B, γ ∈ Top(A) and τ ∈ gen(γ) we are given the following: A(γ) and fγ :A→ A(γ) in B such that the morphism in Rep(A), f∗γ :AA→ Rep(fγ)(A(γ)A(γ)), is exactlythe localization morphism AA→ Qγ(A); a morphism fτγ

: A(γ)→ A(τ) in B fitting into acommutative triangle:

Moreover, if ξ0(A) stands for the trivial element of Top(A) (i.e. the one having only the zero-object for its torsion class) then A(ξ0(A)) = A and the identity of A is the correspondingmorphism in B.

1.2. Proposition. Suppose Rep is spectral and γ ∈ Top(A) for A ∈ B, τ ∈ gen(γ). Writeτ for fγ(τ) ∈ Top(A(γ)) and fτ for the corresponding morphism in B, fτ : A(γ)→ A(γ)(τ),existing because of the spectral property of Rep (applied to A(γ) ∈ B). With notation as inDefinition 1.1. we have:

i) Rep(fτγ)(A(τ)A(τ)) = Qτ (A(γ)A(γ)).

ii) Rep(fγ)(Qτ (A(γ)A(γ))) = Qτ (AA).

Proof. Since Rep is spectral there exists an A(γ)(τ) in B together with a morphismA(γ) −→

fτ

A(γ)(τ) with corresponding morphism f∗τ in Rep(A(γ)), f∗τ : A(γ)A(γ) →Rep(fτ )(A(γ)(τ)A(γ)(τ)), which is nothing but the localization homomorphism A(γ)A(γ)→Qτ (A(γ)A(γ)), in Rep(A(γ)). On the other hand we may look at Rep(fγ)(f∗τ ): Qγ(AA) →Rep(fγ)(Qτ (A(γ)A(γ))), where the latter is in fact equal to: Rep(fγ)Rep(fτ )(A(γ)(τ)

A(γ)(τ)) = Rep(fτfγ)(A(γ)(τ)A(γ)(τ)).Looking at A −→

fγ

A(γ) −→fτ

A(γ)(τ), we obtain:


Now A(τ)A(τ) in Rep(A(τ)) is such that the localization morphismAA → Qτ (A) is exactly given by f∗τ , f∗τ : AA → Rep(fτ )(A(τ)A(τ)). Look at the objectMτ (γ) in Rep(A(γ)) defined by Mτ (γ) = Rep(fτγ

)(A(τ)A(τ)) and f∗τγ: A(γ)A(γ) →

Mτ (γ) the morphism in Rep(A(γ)) given by the fact that Rep is measuring. We haveτ(Mτ (γ)) ⊂ Mτ (γ) in Rep(A(γ)). By the definition of τ we have that Rep(fγ)(τMτ (γ))is τ -torsion in Rep(A), moreover the exactness of Rep(fγ) yields that Rep(fγ)(τMτ (γ)) ⊂Rep(fγ)(Mτ (γ)) = Rep(fτ )(A(τ)A(τ)), the latter being τ -torsionfree because it is equal toQτ (AA). ConsequentlyRep(fγ)(τMτ (γ)) = 0 and τMτ (γ) = 0 then follows from faithfulnessof Rep. Hence f∗τγ

:A(γ)A(γ)→Mτ (γ) factorizes over Bτ (γ)=A(γ)A(γ)/τ(A(γ)A(γ)). We havea sequence in Rep(A):AA −→

f∗γ

Qγ(AA) −−−−−−−−→Rep(fγ)(f∗

τγ)Qτ (AA). So it is clear that in Rep(A),

Rep(fγ)(Mτ (γ)/Bτ (γ)) is τ -torsion because the cokernel of Rep(fγ)(f∗τγ) is τ -torsion,

therefore Mτ (γ)/Bτ (γ) is τ -torsion in Rep(A(γ)). It follows that Mτ (γ) ⊂ Qτ (A(γ)A(γ))in Rep(A(γ)). By definition Qτ (A(γ)A(γ)) is τ -torsion over Bτ (γ) in Rep(A(γ)), henceRep(fγ)(Qτ (A(γ)A(γ))) is contained in Qτ (AA) as it is τ -torsion over AA/τ(AA). SinceMτ (γ) ⊂ Qτ (A(γ)A(γ)) the functor Rep(fγ) takes the value Qτ (A) for both objects andagain from exactness and faithfulness of Rep(fγ) it follows that: Mτ (γ) = Qτ (A(γ)A(γ)).This proves i) and() yields that the morphisms Rep(fγ)(f∗τ )f∗γ and Rep(fγ)(f∗τγ

)f∗γ arethe same. Now ii) is obtained from i) by applying Rep(fγ) to both members, so it followsagain from the exact faithfulness of Rep(fγ) that i) holds to.

1.3. Corollary. With conventions and notation as before, consider δ ≤ τ and γ ≤ τ withτ1 ∈ Top(A(δ)) and τ2 ∈ Top(A(γ)). We obtain the following diagram of morphisms in B:

with objects:

M1 =A(δ)(τ1) A(δ)(τ1) in Rep(A(δ)(τ1))M2 =A(δ)(τ2) A(γ)(τ2) in Rep(A(γ)(τ2))M =A(τ) A(τ) in Rep(A(τ))

such that:

i) Rep(fτ1)(M1) = Qτ (A(δ)A(δ)) = Rep(fτδ)(M)

ii) Rep(fτ2)(M2) = Qτ (A(γ)A(γ)) = Rep(fτγ)(M)

iii) Rep(fδ)Rep(fτ1)(M1) = Qτ (AA) = Rep(fγ)Rep(fτ2)(M2)= Rep(fτ )(A(τ)A(τ)).

This may be compared to the classical fact for localization in R-mod for a ring R,Qτ (R) = Qτ (Qγ(R)) for τ ≥ γ, where Qτ is a localization in Qγ(R)-mod. In our abstractsetting Rep(A(γ)) replaces Qγ(R)-mod and we have traced the basic properties of a


G.C.-representation (i.e. spectral) necessary to generalize the foregoing classical fact to themore general situation.

Note that A(δ)(τ1) and A(γ)(τ2) as in the corollary are a priori not related so not neces-sarily isomorphic in B. Up to introducing restrictive conditions on B and Rep, e.g. in termsof properties making AA a generator for Rep(A) etc. . ., one could study this further, howeverwe do not aim to push category theoretical generality to the limit, contenting ourselves withthe observation that the example(s) considered later will have an even better behaviour.

Let us write Top(A)0 for the opposite lattice of Top(A). It would be natural to considerthe functor AP: Top(A)0 → Rep(A), AP(τ) = Qτ (AA), as a structural presheaf (in fact:sheaf) of AA with values in Rep(A). The spectral property of Rep allows to “realize” thisstructure sheaf inside B, in some sense, by considering P: Top(A)0 → B, defined by P(τ) =A(τ) with structure morphism fτ : A→ A(τ). We have P(ξ0(A)) = A with IA: A→ A as thecorresponding morphism. Moreover for γ ≤ τ we take fτγ

: A(γ) → A(τ) as the restrictionmorphism from γ to τ (note that in Top(A)0 the partial order is reversed when viewing γand τ in Top(A) as open sets). For P to be a presheaf we need an extra property of Rep, asfollows. A spectral Rep is said to be schematic if in addition to the properties in Definition1.1. we have for every triple γ ≤ τ ≤ δ in Top(A) for any A ∈ B, a commutative diagram:

1.4. Corollary. If Rep is schematic then P: Top(A)0 → B is a presheaf of B-objects overthe lattice Top(A), for every A ∈ B.Proof. The composition property of “sections” follows from fδγ

= fδτfτδ

(in the proof of the foregoing proposition one may start from a triangle and

then use the same arguments for the triangles and to retract information

further to the Rep(A)-level).

If f : S → R is a morphism in B then we have defined

F 0 = f : Top(S)→ Top(R),

and we observed that f is a continuous map with respect to the gen-topologies on Top(S)andTop(R), (see remarks before Definition 1.1).


A topological nerve η associates to each A ∈ B a ηA ∈ Top(A) such that for any f :A→ B in B we have ηB ≤ f(ηA) in Top(B). The fact that the latter domination relation isassumed for every f : A→ B makes the topological nerve rather narrowly defined; in prac-tical situations the choice of ηA in each Top(A) follows from a radical-type of construction.A perfect example will be given by the torsion theory associated to the positive part A+

of a positively graded K-algebra or ring, where B is then the category of positively gradedK-algebras, resp. rings. We adopt the following notation:

iA: (Rep(A), ηA)→ Rep(A),aA: Rep(A)→ (Rep(A), ηA),QA: QηA

= iAaA.

To a G.C.-representation Rep and a topological nerve η we associate (Rep, η) such thatfor A ∈ B we have (Rep, η)(A) = (Rep(A), ηA) and for any morphism f : A → B in B weconsider the functor aAFiB : (Rep(B), ηB)→ (Rep(A), ηA) where F = f0. For g: B → C inB we obtain that the composition(aAFiB)(aBGiC): (Rep(C), ηC) → Rep(A, ηA) corresponds to g f , so we have to relatethis to aAFGiC . This will follow from the following proposition.

1.5. Proposition. With notation as before, for any f : A→ B in B we have aAFiBaB(M) =aAF (M) for every M ∈ Rep(B).

Proof. Clearly QB(M)/M is ηB-torsion in Rep(B), hence it is ηA-torsion because ηB ≤ ηAand ηA is defined by f as usual, ηA = f(ηA). By definition of ηA (and using exactnessof F ) we obtain that FiBaB(M)/F (M) is ηA-torsion in Rep(A). The exact sequence 0 →ηB(M)→M → iBaB(M) in Rep(B) yields an exact sequence in Rep(A):

0→ FηB(M)→ FM → FiBaB(M)→ T → 0, ()

where T is ηA-torsion in Rep(A) in view of the foregoing. Clearly ηB(M) is ηA-torsion,hence FηB(M) is ηA-torsion and it follows thatQA(F (M)) = QA(FiBaB(M)), or also aAF (M) = aAFiBaB(M).

We introduce a notion slightly more general than a G.C.-representation, say a general-ized G.C.-representation by weakening the definition of a G.C.-representation such thatfor f : S → R in B, F : Rep(R)→ Rep(S) is only assumed to commute with finite products.

1.6. Corollary. With notation as before: aAFiBaBGiC = aAFGiC corresponds to A −→f

B −→g

C. Hence, associating f = aAFiB to f : A → B, defines a generalized G.C.-

representation (Rep, η) as before (observe in particular that aAIM iA is IM by definition ofiA and aA, for all M ∈ (Rep(A), ηA).

Proof. The first statement is clear from the Proposition 1.5, so we only have to verify thataAFiB is exact and commutes with finite products (and coproducts i.e direct sums). This isobvious, except perhaps the exactness needs a word of explanation, since iB is not an exactfunctor in general. Exactness in the quotient categories may be interpreted as exactness upto torsion so the presence of aA in aAFiB (and the exactness of F ) makes

aAFiB : (Rep(B), ηB)→ (Rep(A), ηA)


exact. Explicitely, if 0→M ′ →M →M ′′ → 0 is exact in (Rep(B), ηB) then:

0→ iBM′ → iBM → iBM

′′ → TB → 0

is exact in Rep(B) where TB is ηB-torsion in Rep(B). Applying the exact functor F yieldsan exact sequence in Rep(A):

0→ FiB(M ′)→ FiB(M)→ FiB(M ′′)→ F (TB)→ 0.

Since TB is ηB-torsion it is ηA-torsion and thus F (TB) is ηA-torsion in Rep(A). Applyingthe exact functor

aA : Rep(A)→ (Rep(A), ηA)

we obtain an exact sequence in (Rep(A), ηA):

0→ aAFiB(M ′)→ aAFiB(M)→ aAFiB(M ′′)→ 0.

The representation (Rep, η) associated to a topological nerve η is called the quotientgeneralized G.C.-representation of Rep.

1.7. Observation. If η1 and η2 are two topological nerves then η1∧η2 defined by associatingη1A ∧ η2

A to A ∈ B is again a topological nerve but this does not hold for ∨, see the remarkspreceding Definition 1.1. (i.e. F 0(∧U) = ∧F 0(U) for any U ⊂ Top(S)).

1.8. Theorem. Let Rep and η be as before, then we have:

i) If Rep is measuring B then so is (Rep, η).ii) If Rep is weakly spectral, i.e same definition as spectral but without assuming the

faithfulness, then so is (Rep, η).

Proof. i) Since Rep is measuring, there are AA ∈ Rep(A) for A ∈ B and for f : S → R in B, acorresponding f∗: SS →S R = Rep(f)(RR) such that I∗S = I

SS and (fg)∗ = Rep(g)(f∗)g∗for g: T → S in B. Now take aA(AA) ∈ (Rep(A), ηA) for A ∈ B. To f : A → B in B weassociate f = aAf

0iB as in the Corollary 1.6. and then f : aA(AA) → f(aB(BB)). Inview of Proposition 1.5. we now have f(aB(BB)) = aAf

0(BB) = aA(AB) and for f wejust take aA(f∗), the localized map of f∗. Obviously (IA) = IaA(A) and the compositionrule follows as in Corollary 1.6.

ii) For every γ ∈ Top(A), A ∈ B, we have A(γ) ∈ B and fγ : A→ A(γ) in B such that f∗γ :AA → f0

γ (A(γ)A(γ)) is exactly the localization morphism AA → Qγ(AA) on Rep(A). Nowif we consider some γ ∈ Tors(Rep(A), ηA) then this γ corresponds to a γ ∈ TorsRep(A) =Top(A) such that γ ≥ ηA. (Since aA is exact and commutes with finite products, we may


take γ to be given by its torsion class consisting ofM ∈ Rep(A) such that aA(M) is γ-torsionin (Rep(A), ηA). We obtain aA(f∗γ ) = f γ as follows:

f γ : aA(AA) aAf0γ iA(γ)aA(γ)(A(γ)A(γ))

||aAf

0γ (A(γ)A(γ)) (Proposition 1.5)

||aA(Qγ(AA)) (choice ofA(γ))

Now since γ ≥ ηA we have QAQγ(AA) = Qγ(AA) = QγQA(AA) (this is also a specialcase of compatibility of localization functors cf. [13]) = QγQA(AA). So we arrive ataA(Qγ(AA)) = aγQA(AA)) = aγ(AA), and f γ is the localization map corresponding to γ.

1.9. Remarks. 1. The whole theory of G.C.-representations including the results of [9]extends to the case of generalized G.C.-representations. Since the case where the functorsF are “restrictions of scalars” functors over rings is very interesting we have made thedefinition of G.C.-representation as we did.

2. Given f : A → B then aAf0iB(M) = 0 if and only if iB(M) is ηA-torsion. Since

ηB ≤ ηA this means that M is ηA-torsion, ηA induced on (Rep(B), ηB) by ηA. Thus even ifRep is faithful, (Rep, η) need not be.However since f(M) = 0 exactly if M is ηA-torsion, we see for τ ≥ γ in Tors(Rep(A), ηA),corresponding to τ ≥ γ ≥ ηA in Top(A), that fγ(τ) ≥ fγ(γ) ≥ fγ(ηA) ≥ ηA(γ) in notation ofProposition 1.2; in particular we see that a γ = fγ(γ)-torsion object is certainly a τ = fγ(τ)-torsion object. This allows to obtain an equivalent of Proposition 1.2 for (Rep, η).

1.10. Proposition. Let Rep be a spectral G.C.-representation then(Rep, η) is a weakly spectral generalized G.C.-representation such that:

i) QηA(fτγ

(A(τ)A(τ))) = Qτ (A(γ)A(γ))ii) f

γ (aτ (A(γ)A(γ)) = Qτ (AA) = Qτ (QA(AA))

Proof. Follow the lines of proof of the Proposition 1.2 when definingMτ (γ) as fτγ

(A(τ)A(τ)),and replacing f0

γ = Rep(fγ) by fγ , f

∗τ by f τ , etc. . ., we obtain that f(Mτ (γ)) is ηA-

torsionfree by definition, hence Mτ (γ) is ηA-torsionfree. Thus fγ (τMτ (γ)) = 0 leads to

τMτ (γ) being ηA-torsion, contradicting the foregoing unless τMτ (γ) = 0. As in the proof ofProposition 1.2, one then arrives at f

γ (Qτ (A(γ)A(γ))/Mτ (γ)) = 0, hence Qτ (A(γ)A(γ)) isηA-torsion over Mτ (γ) and this yields i). We know already (the weakly spectral property)that f

γ fτγ

(A(τ)A(τ)) = Qτ (QA(AA)) hence fγ Mτ (γ) is τ -closed so f

γ (Qτ (A(γ)A(γ))) =fγ Mτ (γ) follows, or ii) follows too.

1.11. Example. Let B be the category of positively ZZ-graded rings with graded morphismsof degree zero for the morphisms. For A ∈ B we let Rep(A) be the category of graded (left)A-modules and graded A-linear morphisms. Full detail on the theory of graded rings andmodules may be found in [7], [8]. We shall write Gr(A) = A−gr for the category as definedabove. To a graded ring morphism f : A → B in B we associate the restriction of scalarsfunctor F = f0: B-gr → A-gr. It is easily seen that we have obtained a G.C.-representation


Gr with Gr(A) = A-gr for A ∈ B. In Tors(A-gr) we may consider the graded torsiontheory κA(+) by taking for its torsion class the graded A-modules such that every elementmay be annihilated by some power An+, n ∈ IN , of the ideal A+ = A1 ⊕ . . .⊕ An ⊕ . . .. Forany f : A → B in B we have that f(A+) ⊂ B+ it follows easily that κB(+) ≤ f(κA(+));consequently we have defined a topological nerve κ(+) for Gr. The quotient generalizedG.C.-representation (Gr, κ(+)) is denoted by Proj. It is not hard to see that Gr ismeasuring, just take AA = A viewed as a left graded A-module, and also spectral (for agraded (rigid) τ in Tors(Gr(A)) we have a graded ring of quotients Qgτ (A) and a gradedmorphism of rings jgτ : A→ Qgτ (A), then we look at Qg

τ (A)Qgτ (A) i.e Qgτ (A) as a left module

over itself). Detail on rigid graded localization may be found in [7]; a first attempt at aprojective scheme theory already appeared in [15], using results of [17].

The following section is devoted to a generalization of the foregoing example, to a moregeneral useful Proj.

2. Proj of a ZZ-graded ring

Let R = ⊕n∈ZZ

Rn be a ZZ-graded ring. Then δ =∑n=0

R−nRn is an ideal of R0 and

I = δ ⊕ ( ⊕n=0

Rn) is an ideal of R that is obviously graded and we have: I0 = δ. An ideal Iof R0 is said to be invariant if for all n ∈ ZZ, RnIR−n ⊂ I. Obviously, δ is an invariantideal. In fact, one easily verifies that an ideal I of R0 is invariant if and only if I ⊕ ( ⊕

n=0Rn)

is an ideal of R. Observe that every ideal I of R0 that contains δ is necessarily invariant!Indeed, for all n ∈ ZZ we have that:

RnIR−n ⊂ RnR−n ⊂ δ ⊂ I.

On R0-mod we may define a torsion theory defined by the Gabriel filter L(δ), L(δ) =L left ideal ofR0, L contains an ideal I ofR0 such that δ ⊂ rad(I).Such torsion theories were called (symmetric) radical torsion theories in [16]. An arbitrarytorsion theory σ on R0-mod, say σ is given by its filter L(σ) and is said to be invariant ifI ∈ L(σ) entails that

∑n=0

RnIR−n ∈ L(σ).

In this section we assume that R0 is a (left) Noetherian ring.

2.1. Observation. L(δ) is invariant.

Proof. Pick I ∈ L(δ) and consider I ′ =∑n=0

RnIR−n. Look at rad(I) and let m ∈ IN be

such that: rad(I)m ⊂ I.¿From (Rnrad(I)R−n)m ⊂ Rnrad(I)mR−n it follows that P ⊂ I ′ for some prime idealP of R0, yields P ⊃ ∑

n=0

Rnrad(I)R−n and consequently: P ⊃ ∑n=0

RnδR−n. In particular

P ⊃ RnR−nRnR−n, hence P ⊃ RnR−n, and this holds for all n = 0 in ZZ. Therefore wearrive at P ⊃ δ. Hence rad(I ′) ⊃ δ, or I ′ ∈ L(δ).

An ideal I of R0 is said to be strongly invariant whenever RnI = IRn for all n ∈ ZZ.For example, if R is a centralizing extension of R0 then every ideal I of R0 is stronglyinvariant. Clearly a strongly invariant ideal is invariant in general.

The properties of interest in this section will all follow in case R is a centralizing extensionof R0; in fact, an acceptable generalization of projective scheme theory for a positively


graded algebra A with A0 = k, a field in the centre of A would already be obtained bylooking at ZZ-graded rings with A0 ⊂ Z(A). Nevertheless it is somewhat remarkable that amuch less restrictive condition will suffice. We say that the ZZ-graded ring R is 0-normalif for all a ∈ ZZ we have: R−a(RaR−a) ⊂ (RaR−a)R−a.

2.2. Proposition. Let R be a 0-normal graded ring with R0 being (left) Noetherian. Ifδ = R0 then there exists a d ∈ ZZ such that RdR−d = R0 = R−dRd.

Proof. Fix an a = 0 in ZZ and look at (RaR−a)n for n ∈ IN . Assuming that n ≥ 1, weobtain:

(RaR−a)n = (RaR−a)(RaR−a)(RaR−a) . . . ⊂ Ra(RaR−a)R−a(RaR−a) . . . .

The latter equals R2aR

2−a(RaR−a) . . . and repeated application of the passing RaR−a over

some R−a, to the left, we arrive at

(RaR−a)n ⊂ RnaRn−a ⊂ RnaR−na.

Consequently RaR−a ⊂ rad(RnaR−na), for any n ∈ IN . Now observe that:

(R−aRa)(R−aRa) = R−a(RaR−a)Ra ⊂ RaR−aR−aRa

and the latter is contained in RaR−a since this is an ideal of R0. Consequently R−aRa ⊂rad(RaR−a). Now if δ = R0 the 1 ∈ δ and thus there exist finitely many a1, a2, . . . , am ∈ ZZsuch that

1 ∈ Ra1R−a1 +Ra2R−a2 . . .+RamR−am

()

Since we have observed that R−aRa ⊂ rad(RaR−a) for all a ∈ ZZ, it follows that we mayinterchange ai and −ai in the expression () and obtain

1 ∈ rad(Ra1R−a1 + · · ·+RamR−am

),

where we now have ai ∈ IN for i = 1, . . . ,m (up to renaming).Since RaR−a ⊂ rad(RnaR−na) for any n ∈ IN we obtain that:

rad(Ra1R−a1 + · · ·+RamR−am

) ⊂ rad(RdR−d)

where d ∈ IN is the lowest common multiple of a1, . . . , am ∈ IN . It follows then that

R0 = rad(RdR−d) orR0 = RdR−d.

Now look at R−dRd and calculate:

R−dRd = R−d(RdR−d)Rd = R−d(RdR−dRd) ⊂ R−d(R−dRdRd)


the latter following from 0-mormality applied to −d.¿From R−dRd ⊂ R−dR−dRdRd, we obtain:

Rd(R−dRd)R−d ⊂ Rd(R−dR−dRdRd)R−d,

but applying RdR−d = R0 then leads to: R0 ⊂ R−dRd, hence R0 = R−dRd, as well.

For d ∈ IN we let R(d) be the ZZ-graded ring defined by R(d)n = Rnd (the dth Veronesesubring of R). A ZZ-graded ring R is said to be d-strongly graded if R(d) is stronglygraded (i.e R(d)1R(d)−1 = R(d)−1R(d)1 = R(d)0).

2.3. Corollary. Let R be a 0-normal (left) Noetherian ZZ-graded ring. If δ = R0 then R isd-strongly graded for some d ∈ IN .

2.4. Proposition. Let R be a left Noetherian ZZ-graded 0-normal ring and consider aperfect rigid torsion theory τ on R-gr, say with graded filter Lg(τ). If δ = 0 and Rδ ∈ Lg(τ)then the graded localization Qgτ (R) is a d-strongly graded ring for some d ∈ IN .

Proof. Since R is left Noetherian, R0 is left Noetherian too.Hence rad(δ) is finitely generated as a left ideal and so we obtain that:

rad(δ) = rad(Ra1R−a1 + · · ·+RamR−am

),

with a1, . . . , am ∈ IN , using arguments formally similar to those used in the proof of Propo-sition 2.2. Then we also obtain that

rad(δ) = rad(RdR−d)

for some d ∈ IN , in fact we may take for d the lowest common multiple of a1, . . . , am ∈ IN .Since τ is perfect and Rδ ∈ Lg(τ), we obtain that Sδ = S, where we put S = Qgτ (R).Looking at the parts of degree zero, we obtain

S0δ = S0, thus S0rad(RdR−d) = S0.

Then for any m ∈ IN we also have that

S0rad(RdR−d)m = S0

and because for some m0 ∈ IN ,

(rad(RdR−d))m0 ⊂ RdR−d,

it also follows that

S0RdR−d = S0.

¿From the obvious inclusions Sd ⊃ Rd, S−d ⊃ R−d, it then follows that

SdS−d = S0.


Applying the foregoing argument to R−dRd, noting that

rad(RdR−d) = rad(R−dRd),

we arrive at

SdS−d = S−dSd = S0.

The equivalent of Proposition 2.4 in case δ = 0 and assuming I ∈ Lg(τ) does hold whenS is 0-normal graded; however at this moment we do not know whether the 0-normality ofS follows from the 0-normality of R.

2.5. Lemma. In case R is a domain, δ = 0 if and only if R is positively or negativelygraded, i.e resp. R = R≥0 or R = R≤0.

Proof. Rn = 0 for some n > 0, then for any −m < 0 we have Rn−mRmn = 0 because

R−nmRnm = 0 as δ = 0. Now Rmn = 0 and thus Rn−m and R−m are zero.

For homogeneous Ore sets T in R, putting T (d) = T ∩R(d), one easily verifies that T (d)is an Ore set in R(d); indeed, for t ∈ T (d) and r ∈ R(d) we find t′ ∈ T, r′ ∈ R, such thatt′r = r′t hence (t′)dr = (t′)d−1r′t, with (t′)d ∈ T (d) and (t′)d−1r′ ∈ R(d). However, notevery Ore set of R(d) is of the form T (d). Such problems may be circumvented by developinga “weighted” space theory generalizing the commutative case but we do not go into thathere. We content ourselves to pointing out an interesting case, allowing the noncommutativescheme theory and an interpretation in terms of quotient Grothendieck representations asin Section 1.

The ZZ-graded ring R is said to be geometrically graded if R is a Noetherian, R0 iscentral in R (hence certainly 0-normalizing) and R is generated over R0 by R1 ∪ R−1 asa ring.

2.6. Proposition. Consider a geometrically graded ring R and a perfect rigid torsion theoryτ on R-gr given by its graded filter Lg(τ).

1. If δ = 0 and Rδ ∈ Lg(τ) then S = Qgτ (R) is strongly graded.2. If δ = 0 and I ∈ Lg(τ) then S0 = S−1S1 = S−nSn for n ≥ 0.

In case τ corresponds to an Ore set T of R that is homogeneous and not contained inR0, then S is strongly graded.

Proof. An arbitrary r ∈ R can be written as a sum of monomials of type r0x1x2 . . . xnwhere r0 ∈ R0 and each xi is either in R1 or in R−1. In case xi ∈ R1 and xi+1 ∈ R−1, orconversely, then xixi+1 ∈ R0 and therefore it is central in R. Consequently, such a monomialis in Rd1R

e−1 or in Re−1R

d1 for suitable e and d in IN . In fact, if r ∈ Rn with n ≥ 0, then

we see in the same way that r ∈ Rn1 by putting factors in degree zero at the beginningof the monomials in the expression of r as above; for m ≤ 0 we find that Rm = R−m−1 .Summarizing, for n ≥ 0 we have Rn = Rn1 , R−n = Rn−1.

1. When δ = 0 then we establish for some d ∈ IN that

S0RdR−d = S0R−dRd = S0,

just as in the proof of Proposition 2.4. However:

RdR−d = Rd1Rd−1 = (R1R−1)d


follows from foregoing remarks. Then it follows fromS0(R1R−1)d = S0 thatS0(R1R−1) = S0.

Similarly we arrive at S0(R−1R1) = S0 from S0R−dRd = S0. Clearly we then obtain thatS1S−1 = S0 = S−1S1.

2. In this situation δ = 0 if and only if

R1R−1 = R−1R1 = 0orRnRm = 0

for n > 0, m < 0. Consequently δ = 0 if and only if R is either positively or negativelygraded depending whether R1 = 0, or resp. R−1 = 0.

Let us treat the positively graded case (the negatively graded case may be treated in asimilar way, note however that the statement in the proposition should then be given asS1S−1 = S0). So R = ⊕

n≥0Rn, I = ⊕

n>0Rn. The assumption I ∈ Lg(τ) and τ being a perfect

torsion theory, leads to S = SI, hence by looking at the parts of degree zero:

S0 =∑n>0

S−nIn =∑n>0

S−nRn.

Look at a typical element s−nrn with s−n ∈ S−n, rn ∈ Rn. For some I ∈ Lg(τ) we have:

Ips−nrn ⊂ Rp−nRn ⊂ Rp, ()

because Ips−n ⊂ Rp−n for all p. Thus, for any p > 0;

S−pIps−nrn ⊂ S−pRp = S−pRp−11 R1 ⊂ S−1R1 ⊂ S−1S1.

Now∑p>0

S−pIp = (SI)0 = S0 because I ∈ Lg(τ), this leads to S0s−nrn ⊂ S−1S1 where

s−n ∈ S−n, rn ∈ Rn as well as n were arbitrary. ¿From S0 =∑n>0

S−nRn it thus follows

that S0 = S−1S1. Note that in () we may indeed use that I0 = 0 because if I ∈ Lg(τ)then I ∩ I ∈ Lg(τ) has (I ∩ I)0 = 0 and we may replace I by the smaller I ∩ I in(). Note also that from the foregoing information it does not follow that S1S−1 = S0!However if τ is associated to a (left)Ore Set, T say, then the strongly graded conditiondoes follow. Indeed, for y ∈ S0 look at ytm for some tm ∈ T ∩ Rm, m > 0. Since tmis invertible in S and t−1

m ∈ S−m we may consider (ytm)t−1m = y ∈ SmS−m. ¿From

S−1S1 = S0 we may derive Sm = Sm1 , indeed Sm = SmS0 = SmS−1S1 yields Sm = Sm−1S1

and by repetition of the argument (Sm−1 = Sm−2S1 etc. . .) we obtain Sm = Sm1 . Nowy ∈ Sm1 S−m = S1(Sm−1

1 S−m) ⊂ S1S−1. Thus S0 = S1S−1.

To a ZZ-graded ring R we associate a torsion theory on R-gr, denoted by κR, definedby taking for its graded filter Lg(κR) the graded filter generated by Rδ and I. Note thatthe ideal I = δ ⊕ ( ⊕

n=0Rn) is automatically in Lg(κR) because it contains Rδ, in case

δ = 0. We can now define schematically graded rings by looking at the class of gradedring R such that there is a finite number of homogeneous Ore sets T1, . . . , Tm such thatκR = κT1 ∧ . . . ∧ κTm

(if so desired one may weaken the definition to κTithat are only

perfect rigid torsion theories).


We entend to use the κR in defining a topological nerve as mentioned after the proof ofCorollary 1.4 and used in Proposition 1.5 and Theorem 1.8. For that we need the followingeasy lemma.

2.7. Lemma. Let R and S be ZZ-graded rings with either δR = 0 and δS = 0 or elseδR = δS = 0. If f : R→ S is a morphism of graded rings then κS ≤ f(κR).

Proof. Since f(Rn) ⊂ Sn for every n ∈ ZZ, it is clear that f(δR) ⊂ δS and also thatf(IR) ⊂ IS . Now L ∈ L(f(κR)) means that S/L is a κR-torsion as an R-module, i.e Lcontains some I ∈ L(κR). The statement now follows easily.

Consider the category B, consisting of ZZ-graded rings R with δR = 0 and graded ringmorphisms. The association of R-gr to R defines a Grotendieck representation. The forego-ing lemma entails that κR defines a nerve and therefore also a quotient representation withrespect to the nerve κR. Applying the methods of [15] we obtain a satisfactory theory forProjR which is defined by the noncommutative topology on (R-gr, κR) (see Section 1) andthe corresponding sheaf theory.

2.8. Conclusion. If R is geometrically graded and schematic then ProjR defined on (R-gr, κR) satisfies all the properties valid in the positively graded case, in particular theschematic property in combination with Proposition 2.6 yields the existence of an affinecovering (in the sense of [15], [17]), moreover the proof of Serre’s global section theoremgiven for ProjR of a positively graded ring carries over this situation too. All this followsfrom a trivial modification of the proofs given in the positively graded case, taking intoaccount the results included in this section, so we omit this repetition here.

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ON THE STRONG RIGIDITY OF SOLVABLE LIE ALGEBRAS

TOUKAIDDINE. PETIT1

Departement Wiskunde en Informatica, Universiteit Antwerpen,

B-2020 (Belgium)

Abstract. We call a finite-dimensional complex Lie algebra g strongly rigid if itsuniversal enveloping algebra Ug is rigid as an associative algebra, i.e. every formalassociative deformation is equivalent to the trivial deformation. The aim of this paperis to study the strong rigidity properties of solvable Lie algebras. First, we show thata strongly rigid Lie algebra has to be rigid as Lie algebra, this restricts the researchto rigid Lie algebras. In addition the second scalar cohomology group has to vanish.Therefore the nilpotent Lie algebras of dimension greater or equal than two are notstrongly rigid and the torus’s dimension of strongly rigid solvable Lie algebra has tobe one. Moreover, the Kontsevitch’s theory of deformation quantization helps to seethat every polynomial deformation of the linear Poisson structure on g∗ which inducesa nonzero cohomology class of g leads to a nontrivial deformation of Ug. Since therigidity is intimately related to cohomology, the cohomology groups are characterized.At last, we classify the n-dimensional strongly rigid solvable Lie algebras where n ≤ 6and give some remarks on linearizability of their corresponding Poisson structure.

1. Introduction

The deformation of of rings and algebras was introduced by M. Gerstenhaber in 1964([12]). He gave a tool to deform algebraic structure based on formal power series. Theinterest on deformation has grown with the development of quantum groups related toquantum mechanics ([2]). Examples of quantum groups may be obtained as Hopf algebradeformation of enveloping algebra of Lie algebra.

A formal deformation of an associative (resp. Lie) algebra (A, µ) is an associative (resp.Lie) algebra A[[t]] with a multiplication µt defined by

µt(p, q) = µ(p, q) + tµ1(p, q) + t2µ2(p, q) + · · · (1.1)

where p, q ∈ AThe algebra is said rigid if every formal deformation is isomorphic to a trivial deformation.

The rigidity theorem of Gerstenhaber [12] (resp. of Nijenhuis-Richardson [20]) insure that ifthe 2nd Hochschild cohomological group H2

H(A,A) (resp. Chevalley-Eilenberg H2CE(g, g)) of

an associative algebra A (resp. a Lie algebra g) vanishes then the algebra (rep. Lie algebra)is rigid. Therefore the semisimple associative (resp. Lie) algebras are rigid because theirsecond cohomology groups are trivial ([14]).

The rigidity of n-dimensional complex rigid Lie algebras was studied by R.Carles,Y.Diakite, M.Goze and J.M. Ancochea-Bermudez. Carles and Diakite established theclassification for n ≤ 7 ([6],[4]), and Ancochea with Goze did the classification for solvable

1Author supported by the Scientific Programme NOG of the European Science Foundation,

e-mail:[email protected].

ON THE STRONG RIGIDITY OF SOLVABLE LIE ALGEBRAS 163

Lie algebras for n = 8 and some classes ([10], [1]). The classification of associative rigidalgebras are known up to n ≤ 6 (see [19]). In this paper we are interested in the deformationand rigidity of enveloping algebras associated to solvable Lie algebras. We have introducedin ([3]) the notion of strong rigidity of a Lie algebra. A Lie algebra is said strongly rigid ifits enveloping algebra is rigid as an associative algebra.

The paper is organized as follows. In Section 2 we summarize the definitons and recallsome important and useful results, namely the Cartan-Eilenberg theorem and Hochschild-Serre factorization theorem. In Section 3 we introduce the strong rigidity of a Lie algebraand give some properties. We show that a strongly rigid Lie algebra has to be rigid asa Lie algebra. In addition, the scalar second cohomology group has to vanish. Therefore,it permits to construct some classes of non strongly rigid Lie algebras. As an exampleof a strongly rigid Lie algebra, we consider the 2-dimensional non abelian Lie algebra.We show by a direct calculation that the second Hochschild cohomology group of itsenveloping algebra with values in the algebra is trivial. Thus this Lie algebra is stronglyrigid. Section 4 is devoted to the deformation of the enveloping algebra via the Poissonstructures. We recall the result of [3] that every nontrivial polynomial deformation ofthe linear Poisson structure associated the the Lie algebra induces a nontrivial defor-mation of the enveloping algebra. In Section 5 we characterize the cohomology groupsHnCE(g,Ug)). At last, we apply the previous results to classify the strongly rigid Lie

algebras in small dimensions and deduce some remarks on the linearization of Poissonstructures.

2. Preliminaries

1. Let g be a finite dimensional decomposable solvable Lie algebra, i.e g = t ⊕ n wheren is the nilradical and t is an exterior torus of derivations in Malcev’s sense; that is tis an abelian subalgebra of g such that adX is semisimple for all X ∈ t. This class ofsolvable Lie algebra contains the rigid Lie algebras ([4]).

2. Let K be a commutative ring and g be a Lie algebra over K. Recall that a (left)g-representation of g is a K-module M and a K-homomorphism

g⊗M →Mx⊗ a → xa

(2.1)

such that x(ya) − y(xa) = [x, y]a. To each Lie algebra g, we associate an associativeK-algebra Ug such that every (left) g-representation may be viewed as (left) Ug-representation and vice-versa. The algebra Ug is constructed as follows

Let Tg be a tensor algebra of K-module g , Tg = T 0 ⊕ T 1 ⊕ · · · ⊕ Tn ⊕ · · · whereTn = g⊗ g⊗ · · · ⊗ g (n times). In particular T 0 = K1 and T 1 = g. The multiplicationin Tg is the tensor product. Every K-linear map g⊗M→M has a unique extensionto a map Tg ⊗ M → M. The g-module is a g-representation if and only if theelements of Tg of the form x ⊗ y − y ⊗ x − [x, y] where x, y ∈ g annihilate M.Consequently, we are led to introduce the two-sided ideal I generated by the elementsx ⊗ y − y ⊗ x − [x, y] where x, y ∈ g. We define the enveloping algebra Ug of g asTg/I. Thus, g-representations and the Ug-modules may be identified. Recall thatevery bimodule M is a g-module by (x,m)→ xm−mx, denoted byMa.

Assume that g is a free Lie algebra. Let xi be a fixed basis of g and yi be the imageof xi by the K-homomorphism i : g → Ug. We set yI = yi1 · · · yip with I a finite

164 TOUKAIDDINE. PETIT

sequence of indices i1, . . . , ip and yI = 1 if I = ∅. The Poincare-Birkhoff-Witt theoreminsures that the enveloping algebra Ug is generated by the elements yI correspondingto the increasing sequences I.

We denote by SV the symmetric algebra over a K-module V . If Q ∈ K, then thereexists a canonical bijection between Sg and Ug which is a g-module isomorphismbetween Sg and Uga ( [9, pp.78–79] )

3. Unless otherwise stated, K denotes an algebraically closed field of characteristic 0.Let K[[t]] be the power series ring with coefficients in K. For a K-vector space Ewe denote by E[[t]] the K[[t]]-module of the power series with coefficients in E. Let(A, µ0) be an associative (resp Lie) K-algebra, then (A[[t]], µ0) is an associative (resp.Lie) K[[t]]-algebra.(a) A formal deformation of an associative (resp. Lie algebra) A is an associative

(resp. Lie) K[[t]]-algebra (A[[t]], µt) such that

µt = µ0 + tµ1 + t2µ2 + · · ·+ tnµn + · · · ,

where µn ∈ HomK(A⊗K A,A). (resp. µn ∈ HomK(A ∧K A,A)).(b) Two deformations (A[[t]], µt) and (A[[t]], µ′t) are said equivalent if there exists a

formal isomorphism

ϕt = ϕ0 + ϕ1t+ · · ·+ ϕntn + · · · ,

with ϕ0 = IdA (Identity map on A) and ϕn ∈ End(A) such that

µ′t(a, b) = ϕ−1t (µt(ϕt(a), ϕt(b)) ∀a, b ∈ A.

(c) A deformation of A is said trivial if it is equivalent to (A[[t]], µ0).(d) An associative (resp. Lie) algebraA is said rigid if every deformation ofA is trivial.

4. The deformation theory is related to Hochschild cohomology in the case of associativealgebra and Chevalley-Eilenberg cohomology in the case of Lie algebra. We denoteby Hn

H(A,M) the n-th Hochschild cohomology group of an associative algebra Awith values in the bimodule M and by Hn

CE(g,M) the n-th Chevalley-Eilenbergcohomology group of a Lie algebra A with values in a g-module M. The secondHochschild cohomology group of an associative algebra (resp. Chevalley-Eilenbergcohomology group of a Lie algebra ) with values in the algebra may be interpretedas the group of infinitesimal deformations. It follows that if this group is trivial thenthe algebra is rigid. The third cohomology group corresponds to the obstructions toextend a deformation of order n to a deformation of order n+ 1 ([12],[13] and [20]).

5. The following classical theorem due to H.Cartan et S.Eilenberg, ([7, pp.277]) gives alink between the Hochschild cohomology of an enveloping algebra with values in anUg-bimodule M (in particular M = Ug) and the Chevalley-Eilenberg cohomology ofthe Lie algebra with values in the same module.

Theorem 2.1. Let g be a finite dimensional Lie algebra over K. Then

HnH(Ug,M) Hn

CE(g,Ma) ∀n ∈ N


In particular, if Q ⊂ K

HnH(Ug,Ug) Hn

CE(g,Uga) HnCE(g,Sg) ∀n ∈ N

6. The Hochschild-Serre theorem [17] gives the following factorization of the Chevalley-Eilenberg cohomology groups in the case of a decomposable solvable Lie algebra.

Theorem 2.2. Let g = n⊕t be a finite dimensional solvable Lie algebra over K, wheren is the largest nilpotent ideal of g and t the supplementary subalgebra of n, reductivein g, such that the t-module induced on Uag is semisimple, then for all positive integersp, we have

HpCE(g,Ua(g))

∑i+j=p

HiCE(t,K)⊗Hj

CE(n,Uag)t.

where HjCE(n,Uag)t denotes the subspace of the t-invariant elements.

3. Strongly rigid Lie algebras and properties

We recall here the notion of strong rigid Lie algebra introduced in [3].

Definition 3.1. A Lie algebra g is said strongly rigid if its enveloping algebra Ug is rigidas an associative algebra.

The semisimple Lie algebras give examples of strongly rigid Lie algebras. In fact, theWhitehead lemmas induce that the first and second cohomology groups of a Lie algebra gwith values in every finite dimensional K-module vanish. Therefore these Lie algebras arerigid as Lie algebra. Using the filtration of Sg and the Cartan-Eilenberg theorem we obtainH2H(Ug,Ug) = 0. Therefore, the enveloping algebra of a semisimple Lie algebra is rigid.

3.1. The rigidity of the Lie algebra.

Theorem 3.1. If g is a finite dimensional strongly rigid Lie algebra over K, then g is rigidas a Lie algebra.

Proof. We suppose that the enveloping algebra Ug of g is rigid, but not the Lie algebra g.Then there exists a nontrivial formal deformation (g[[t]], µt) of g with µt =

∑∞n=0 µnt

n andthe cohomology class of µ1 is nontrivial in H2

CE(g, g). Since g is finite dimensional, thenthe K[[t]]-module g[[t]] is isomorphic to the free module g ⊗K K[[t]]. Let yI := yi1 · · · yikbe the generators of the PBW basis of Ug, let y′I := y′i1 • · · · • y′ik be the generators ofPBW basis of U(g[[t]]

)over K[[t]] and that • is the multiplication in U(g[[t]]

). The map

Φ : Ug ⊗K K[[t]] → U(g[[t]])

defined by Φ(yI) := y′I is a K[[t]]-module isomorphism. Letπt : Ug⊗KK[[t]] × Ug⊗KK[[t]] → Ug⊗KK[[t]] the multiplication on the module Ug⊗KK[[t]]induced by • and Φ, i.e. πt(a, b) := Φ−1

(Φ(a) • Φ(b)

). The restriction of πt to elements of

Ug× Ug defined a K-bilinear map Ug× Ug→ Ug⊗K K[[t]] ⊂ Ug[[t]] which we denote alsoby πt, i.e. πt(u, v) =

∑∞n=0 t

nπn(u, v) for all u, v ∈ Ug where πn ∈ HomK(Ug ⊗ Ug,Ug).The K-bilinear map πt defined naturally a K[[t]]-bilinear associative multiplication over the


K[[t]]-module Ug [[t]] (which contains Ug⊗K K[[t]] as a dense submodule with the t-adiquetopology) :

πt

( ∞∑s

tsus,

∞∑s′=0

ts′vs′

):=

∞∑r=0

tr∑

s,s′,s′′≥0s+s′+s′′=r

πs′′(us, vs′)

In particular, the map π0 defined an associative multiplication over the vector space Ug, and(Ug [[t]], πt) is a formal associative deformation of (Ug, π0). For a finite increasing sequenceI, J we have π0(yI , yJ ) = Φ−1(y′I • y′J)|t=0. By ordering the product y′I • y′J we obtain thatπ0 is the multiplication of Ug and (Ug [[t]], πt) is a formal deformation of Ug. It follows thatπ1 is a Hochschild 2-cocycle of Ug, and the restriction of π1 to X,Y ∈ g satifies

µ1(X,Y ) = π1(X,Y )− π1(Y,X) ∀X,Y ∈ g. (3.1)

because the Lie algebra (g[[t]], µt) is a Lie subalgebra of U(g[[t]]) which may consideredas an associative subalgebra of (Ug [[t]], πt). The rigidity of Ug implies that there exists aformal isomorphism ϕt =

∑∞r=0 ϕrt

r, where ϕ0 = IdUg and ϕn ∈ HomK(Ug,Ug) such that

ϕt(πt(u, v)) = πt(ϕt(u), ϕt(v)), ∀u, v ∈ Ug,

which is equivalent to

∞∑r=0

tr∑

a,b≥0a+b=r

ϕa(πb(u, v)) =∞∑r=0

tr∑

a,b,c≥0a+b+c=n

πa(ϕb(u), ϕc(v)) ∀u, v ∈ Ug. (3.2)

If r = 1, the relation becomes

π1(u, v) = (δHϕ1)(u, v) ∀u, v ∈ Ug (3.3)

where δH is a Hochschild cobord operator (see [16]) with respect the multiplication π0 ofthe enveloping algebra.

Then the formulae (3.1) and (3.3) imply

µ1(X,Y )= (δHϕ1)(X,Y )− (δHϕ1)(Y,X)=Xϕ1(Y )− ϕ(XY ) + ϕ(X)Y − Y ϕ1(X) + ϕ(Y X)− ϕ(Y )X=(δCEϕ1)(X,Y ) ∀X,Y ∈ g (3.4)

where δCE is the Chevalley-Eilenberg cobord operator. (see [7]).Therefore the class of µ1 in H2

CE(g, g) is trivial. contradiction.

This result show that the class of strongly rigid Lie algebras is contained in the class ofrigid Lie algebras.


3.2. Second scalar cohomology group. In this Section we give a necessary conditionon the scalar Chevalley-Eilenberg cohomology group for the strong rigidity of a Lie algebra.

Let ω ∈ Z2CE(g,K) be a scalar 2-cocycle of the Lie algebra g. Let gω = g ⊕ Kc be a

central extension of g with ω such that the new bracket [ , ]′ is defined as usually by

[X + ac, Y + bc]′ := [X,Y ] + ω(X,Y )c ∀X,Y ∈ g; a, b ∈ K. (3.5)

Theorem 3.2. Let g be a finite dimensional Lie algebra over K such that the second scalarcohomology group H2

CE(g,K) is different from 0, then g is not strongly rigid.

Proof. Let ω ∈ Z2CE(g,K) be a 2-cocycle with a nonzero class and let gtω[[t]] be the one-

dimensional central extension of the Lie algebra g[[t]] = g ⊗K K[[t]] over K = K[[t]] (see(3.5)). The multiplication of the enveloping algebra U(gtω[[t]]) of gtω[[t]] is denoted by •.Let consider the two-sided ideal I := (1 − c′) • U(gtω[[t]]) = U(gtω[[t]]) • (1 − c′) (where c′

denote the image of c in U(gtω[[t]])) and the quotient algebra Utωg := U(gtω[[t]])/I. Let

e1, . . . , en be the K-basis of g. Then c, e1, . . . , en is a K[[t]]-basis of gtω[[t]]. Let y1, . . . , yn bethe images of the basis vectors in Ug and c′, y′1, . . . , y

′n be the images of the basis vectors

in U(gtω[[t]]). Let yI := yi1 . . . yik in Ug over K be the generators of the PBW basis. The

elements c′•i0 • y′I (where i0 ∈ N and c

′•i0 := 1) form a basis of U(gtω[[t]])

over K[[t]].(the Lie algebra is a free module over a commutative ring, see [7], p.271). In the quotientalgebra Utωg, the element c

′•i0 is identified to 1. We denote the multiplication in Utωg by ·and by the canonical projection, the images of y′1, . . . , y

′n by y′′1 , . . . , y

′′n the elements y′I give

y′′I := y′′i1 · . . . · y′′in . It follows that the elements y′′I form a basis of the quotient algebra Utωg.As in the proof of the previous theorem 3.1, the map Φ : Ug ⊗K K[[t]] → U(gtω[[t]]

)given

by yI → y′′I defines an isomorphism of free K[[t]]-modules. In a similar way we show thatthe multiplication induced on Ug ⊗K K[[t]] by the multiplication · of Utωg and Φ define asequence of πt =

∑∞r=0 πrt

r, where πr ∈ HomK(Ug⊗Ug,Ug) with the following properties:1. πt defines a formal associative deformation of (Ug, π0), 2. π0 is the usual multiplicationof the enveloping algebra Ug of g. Therefore, π1 is a Hochschild 2-cocycle of Ug, and forall X,Y ∈ g ⊂ Ug we have the relation: ω(X,Y )1 = π1(X,Y ) − π1(Y,X) because the Liealgebra gtω[[t]] is injected in the quotient algebra Utωg, then in Ug⊗K K[[t]] ⊂ Ug [[t]].

Suppose that Ug is rigid, then the deformation πt is trivial. Therefore there exists aHochschild 1-cocycle ϕ1 ∈ C1

H(Ug,Ug) such that π1 = δH(ϕ1). It follows ∀X,Y ∈ g:

ω(X,Y )1 = π1(X,Y )− π1(Y,X) = δH(ϕ1)(X,Y )− δH(ϕ1)(Y,X) = δCE(ϕ1)(X,Y ).

Then ω is a Chevalley-Eilenberg cobord and its class is trivial in H2CE(g,K), contradiction.

3.3. Examples of non strongly rigid Lie algebras. The previous theorems permit toshow that some classes of solvable Lie algebras are not strongly rigid.

Corollary 3.1. The following Lie algebras are not strongly rigid :

1. Every n-dimensional nilpotent Lie algebra g with n greater or equal than 2.2. Every Lie algebra g = t ⊕ n where the dimension of the torus t is greater or equar

than 2.

Proof. The first assertion is a consequence of a classical result of Dixmier concerning thenilpotent Lie algebras ([8]): H2

CE(g,K) = 0 if dim(g ≥ 2).For the second, we have H2

CE(t,K) = 0 for an abelian Lie subalgebra.


3.4. Example of a strongly rigid Lie algebra. In this section we prove that the2-dimensional non abelian solvable Lie algebra is strongly rigid. We denote by r2 the solvableLie algebra generated by X,Y such that [X,Y ] = Y .

Lemma 3.1. 1. ∀n,m ∈ N : Y Xn = (X − 1)nY , [X,Y m] = mY m and ∀n ∈ N,∀m ∈N∗ (m− 1)XnY m = [X,XnY m]−XnY m.

2. There exists a polynomial Pn+1(X) in X of degree n+ 1 such that :(a) P1(X) = X and Pn+1(X) = Xn+1 +

∑n+1k=2(−1)kCkn+2Pn+2−k(X), if n ≥ 1.

(b) (n+ 1)XnY = [Pn+1(X), Y ].

Proof. The first assertion may easily be proved by induction.Let us prove the property (2) by induction on n.

It is true for n = 0, because [P1(X), Y ] = [X,Y ] = Y . Assume that it is true until n. Wehave (a):

[Xn+2, Y ] =Xn+2Y − Y Xn+2

=Xn+2Y − (X − 1)n+2Y following (1)

=Xn+2Y −n+2∑k=0

(−1)kCkn+2Xn+2−kY

=(n+ 2)Xn+1Y −n+2∑k=2

(−1)kCkn+2Xn+2−kY

Applying the induction hypothesis on n + 2 − k with k ≥ 2, we obtain Xn+2−kY =[Pn+3−k(X), Y ], (the degree of Pn+3−k(X) = n + 3 − k ≤ n + 1). Then (b) becomes :(n+ 2)Xn+1Y = [Xn+2 +

∑n+2k=2(−1)kCkn+2Pn+3−k(X), Y ] = [Pn+2(X), Y ]

In the following we show by a direct calculation, for the Lie algebra r2, that the secondHochschild cohomology group of its enveloping algebra with values in the algebra is trivial.Thus this Lie algebra is strongly rigid.

Proposition 3.1. Let r2 be the 2-dimensional non abelian Lie algebra. We have

H2H(Ur2,Ur2) H2

CE(r2,Ur2) = 0

Thus, the Lie algebra r2 is strongly rigid.

Proof. By Cartan-Eilenberg theorem we have

H2H(Ur2,Ur2) H2

CE(r2,Ur2).

We will show that

∀Φ ∈ Z2CE(r2,Ur2) ∃f ∈ C1

CH(r2,Ur2) s.t. δCE(f) = Φ (∗)

Let XnY m : n,m ∈ N be the Poincare-Birkhoff-Witt basis of Ur2. Let Φ be an elementof Z2

CE(r2,Ur2). It is defined by Φ(X,Y ) =: u =:∑n,m∈N un,mX

nY m where un,m ∈ K


are nonzero for a finite number of n,m. Let f be an element of C1CH(r2,Ur2). It is defined

by two elements f(X) =: v =:∑n,m∈N vn,mX

nY m and f(Y ) = w =∑n,m∈N wn,mX

nY m

where vn,m, wn,m ∈ K are nonzero for a finite number of n,m.Then

∀u =∑n,m∈N

un,mXnY m ∈ Ur2 ∃v, w ∈ Ur2 tels que u = [X,w]− w + [v, Y ] (∗∗)

We study two casesCase 1: m = 1.We set wn,m = un,m

m−1 , then vn,m = 0 if m = 1 and v, w satisfy (∗∗) by lemma (3.1,(2)).Case 2: m = 1.We set vn,m = 1

n+1un,1Pn+1(X) then wn,m = 0 if m = 1 and v, w satisfy (∗∗) by lemma(3.1,(1)).

We conclude that the relation (∗∗) is satisfied. Therefore H2CE(r2,Ur2) = 0, and the Lie

algebra r2 is strongly rigid.

4. Deformation of enveloping algebras by quantification

In this section, we recall a result of [3] which said that a nontrivial polynomial deformationof the linear Poisson structure associated to the Lie algebra induces a nontrivial deformationof the enveloping algebra. We recall first the Poisson structure. We refer to Vaisman’s book([21]) for a complete description.

1. A Poisson algebra is a commutative associative algebra A over K with a bilinear map, : A×A → A satisfying for f, g, h ∈ A

(1) f, g = −g, f(2) f, g, h+ g, h, f+ h, f, g = 0 (Jacobi identity )(3) h, fg = h, fg + fh, g ( Leibniz relation)

We denote by (A, ·, , ) such an algebra. A manifold M is called a Poisson manifoldif the algebra of functions, C∞ (M), has a Poisson structure.

A Poisson structure is determined by a skew-symmetric bilinear form on T ∗M . Inother word there exists a tensor field P ∈ Γ(M,Λ2TM) (with TM the fibre bundleof M) such that f, g = P (df, dg) =

∑i,j P

ij∂if∂jg, where ∂i denotes the partialderivative with respect to the local coordinate xi. The tensor field P is called thePoisson bivector of (M, , ). A Poisson structure on M is given by a bivectorP ∈ Γ(M,Λ2TM) satisfying

∑hP ih∂hP

jk + P jh∂hPki + P kh∂hP

ij = 0

2. Let T ipoly = Γ(M,ΛiTM) be the space of all skew symmetric tensor fields of ranki on a manifold M , T 0

poly = C∞ (M) and Tpoly = (⊕n≥0Tipoly,∧) the algebra of

multivectors on M .

A bivecteur P ∈ T2poly defined a Poisson structure if and only if the Schouten-

Nijenhuis bracket [P, P ]s = 0. The operator δP := [P,−]s determines the so-calledPoisson cohomology.


3. Let g be a finite dimensional Lie algebra over K and g∗ its algebraic dual. The symmet-ric algebra Sg is identified to the algebra of polynomial functions on g∗. The Lie algebrastructure of g induces a linear Poisson structure on g∗. f, g(x) = x([df(x), dg(x)])with f, g ∈ Sg and x ∈ g∗. Let (ei)i=1...n be a basis of g, (ei)i=1...n the dual basis andx =

∑ni=1 xie

i ∈ g∗, f, g = P0(df, dg) with P0 the bivector defined by

P0 =12

∑i,j

P ij0 ∂i ∧ ∂j ou P ij0 (x) =∑k

Ckijxk (4.1)

where Ckij are the structure constants of g. Therefore, the Poisson algebra structureon S(g).

4. In the theory of deformation by quantification ([2]), one associates to the Poissonstructure a formal deformation of the associative commutative algebra C∞(M), calledstar product, see e.g. [21] for the definition. The existence of a star product for everyPoisson structure was established by Kontsewitsh see [18]. Using this result we haveproved in [3] the following theorem

Theorem 4.1. Let g be a finite dimensional Lie algebra over K. Let P0 be the bivectordefining the linear Poisson structure on g∗.

Assume that it exists a sequence (Pn)n∈N of polynomial bivectors (Pn ∈ Sg⊗∧2g∗) suchthat

∑i+j=n[Pi, Pj ]s = 0 for n ∈ N and P1 is not cohomologeous to 0.

Then Pt =∑n≥0 t

nPn is a nontrivial deformation of the Poisson structure P0 and itinduces a nontrivial deformation of the enveloping algebra Ug. Therefore, the Lie algebra gis not strongly rigid.

5. Some cohomological properties

Let g = t ⊕ n be a finite-dimensional decomposable solvable Lie algebra, where n is thelargest nilpotent ideal and t is an exterior torus of derivations of g such that the center of tis t (this condition holds for rigid Lie algebras ([4])). The group HH(Ug,Ug) is isomorphicto HCE(g,Uga) using the Cartan-Eilenberg theorem 2.1 where the enveloping algebra Ugis considered as Ug-bimodule and Uag is considered as an adjoint g-module with X.u :=[X,u] := Xu− uX, where X ∈ g, u ∈ Ug [9]. Let Un be the two-sided ideal of Ug generatedby n and Z(Ug) be the center of the enveloping algebra. We denote by Ugt (resp. U t

n, U tt ),

the t-invariant elements of Ug (resp. Un, Ut).The group HCE(g,Uga) may be deduced from the t-invariant cohomology group

HCE(n,Un)t under some assumptions on the torus t (over g).The g-adjoint module Uga is an inductive limit of adjoint sub-g-modules (Ukg)k≥0 where

(Ukg)k≥0 is the canonical filtration Ug ([9]). In order to simplify the notation, we denotenext the adjoint module Uga by Ug.

By the Hochschild-Serre factorization theorem 2.2 we obtain :

Proposition 5.1. Let g = t⊕ n be a decomposable solvable Lie algebra. Then

H1CE(g,Ug) t∗ ⊗ Z(Ug) ⊕ H1

CE(n,Ug)t (5.1)H2CE(g,Ug) (∧2t∗)⊗ Z(Ug) ⊕ t∗ ⊗H1

CE(n,Ug)t ⊕ H2CE(n,Ug)t


The different t-modules deduced canonically from the t action on Ug are locallysemisimple. The exact sequence of t-modules :

0→ Un → Ug→ Ug/n→ 0 (5.2)

implies a cohomological exact sequence, which corresponds if we restrict to t-invariant groupsthe exact sequence :

0→H0CE(n,Un)t → H0

CE(n,Ug)t → H0CE(n,Ug/n)t → H1

CE(n,Un)t

→· · · → HpCE(n,Ug/n)t → Hp+1

CE (n,Un)t → Hp+1CE (n,Ug)t →

→Hp+1CE (n,Ug/n)t → · · · (5.3)

Assume that HpCE(n,Ug/n)t = 0 for p ≥ 1, and (Un)t = 0. This conditions holds if it exists

an element X of t such that the eigenvalues of adX|n are positive in an ordered subfield ofK. Recently, M. Goze and E. Remm showed that H2

CE(g,C) = 0 if and only if λ = 0 is aneigenvalue ([11]).

The sequence (5.3) implies the exact sequence

0→ Ugt → Utt → H1CE(n,Un)t → H1

CE(n,Ug)t → 0 (5.4)

and the isomorphisms :

Hp+1CE (n,Un)t Hp+1

CE (n,Ug)t for all p ≥ 1 (5.5)

Then, the Chevalley-Eilenberg cohomology groups of g with values in Ug become:

H1CE(g,Ug) t∗ ⊗ Z(Ug) ⊕H1

CE(n,Un)t/(Ut/Z(Ug)). (5.6)

H2CE(g,Ug) ∧2t∗ ⊗ Z(Ug) ⊕ t∗ ⊗H1

CE(n,Un)t/(Ut/Z(Ug))⊕ H2

CE(n,Un)t (5.7)HnCE(g,Ug) ∧nt∗ ⊗ Z(Ug) + ∧n−1t∗ ⊗H1

CE(n,Un)t/(Ut)/Z(Ug)

+∑

i+j=n; j≥2

∧it∗ ⊗HjCE(n,Un)t

∀n ≥ 2 (5.8)

Now, we characterize the center. Suppose that there exists X0 ∈ t such that theeigenvalues of adX|n are positive in an ordered subfield of K and let Y0, . . . , Yr be abasis of n and X0, . . . , Xs be a basis of t. Suppose that the action of X0 on an elementu =

∑aj0...jsi0...ir

Xj00 . . . Xjs

s Yi00 . . . Y irr in Ug vanishes. If X0Yk = λkYk then 0 = X0u =∑

(λ1i1 + · · ·+ λrir)aj0...jsi0...ir

Xj00 . . . Xjs

s Yi00 . . . Y irr . Since λk > 0 then the center of Ug is K.

One can see that

Theorem 5.1. Let g = t⊕ n be a decomposable solvable Lie algebra. We suppose that thereexists an element X of t such that the eigenvalues of adX|n are positive in an orderedsubfield of K.


Then, we have the following properties:

Z(Ug)= K (5.9)H1CE(g,Ug) t∗ ⊕ (Der(n,Un)t/U+

t ) (5.10)H2CE(g,Ug)∧2t∗ ⊕ (Der(n,Un)t/U+

t )⊗ t∗ ⊕ H2CE(n,Un)t (5.11)

Where Der(n,Un)t denote the t-invariant exterior derivations and U+t = Ut/Z(Ug)

Since the center is nontrivial, we find again, the following necessary condition :

Corollary 5.1. If the solvable Lie algebra g = t⊕n is strongly rigid with trivial H2H(Ug,Ug)

then dim(t) ≤ 1.

Since Ug and Sg are isomorphic as g-module. We can replace in the previous cohomo-logical characterization the algebra Ug by Sg and the two-sided ideal Un by Sn.

6. The classification of strongly rigid solvable Lie algebras in low

dimensions

Let K be the complex field. The classification of n-dimensional rigid Lie algebras is knownuntil n ≤ 8 [10].

For 2-dimensional Lie algebras, there is one isomorphism class, namely the Lie algebrar2 which is strongly rigid (see proposition 3.1)

In dimension 3, there is no solvable rigid Lie algebras.In dimension 4, there is only one rigid Lie algebras, r2 + r2. Since the torus is

2-dimensional, then according to corollary (3.1) this algebra is not strongly rigid.In dimension 5, there is only one rigid class with 2-dimensional torus. There is no strongly

rigid Lie algebra.In dimension 6, there is 3 isomorphism classes of 6-dimensional rigid solvable Lie algebras.

Only one has a one-dimensional torus. Let us consider this Lie algebra, it is denoted in [10]by t1 ⊕ n5,6. Setting the basis X0,X1,X2,X3,X4,X5 the Lie algebra is defined by

[X0,Xi] = iXi i = 1, . . . , 5 (6.1)[X1,Xi] =Xi+1 i = 2, 3, 4 (6.2)[X2,X3] =X5 (6.3)

The other bracket are equal to 0 or deduced by skew-symmetry from the previous one. Inthe following we give a nontrivial deformation of the linear Poisson structure associated tothe Lie algebra t1 ⊕ n5,6.

Proposition 6.1. Let P0 be the Poisson structure associated to the Lie algebra t1 ⊕ n5,6

and P1 ∈ Sg⊗ ∧2g∗ defined by (α, β, γ ∈ C3 \ (0, 0, 0)):

P1 = βX22

∂

∂X1∧ ∂

∂X3+ γ(−X2X3

∂

∂X1∧ ∂

∂X4+X2X5

∂

∂X3∧ ∂

∂X4) + αX1X5

∂

∂X2∧ ∂

∂X4

Then [P1, P1]s = 0 = [P0, P1]s and the cohomology class of P1 is not 0.Thus, g is not strongly rigid.


Proof. Straightforward computation. Theorem 4.1 implies that the Lie algebra is notstrongly rigid.

Theorem 6.1. There is only one n-dimensional solvable strongly rigid Lie algebra forn ≤ 6, namely the 2-dimensional Lie algebra r2.

Given a Poisson structure, if there exists a formal isomorphism such that this Pois-sons structure is isomorphic to its linear part then one says that this Poisson structure islinearizable. This problem was formulated first by A.Weinstein (based on considerations bySophus Lie) ([22]). Using the theorem 4.1, we may deduce :

Proposition 6.2. Every Poisson structure which is a deformation of linear Poisson struc-ture of n-dimensional strong rigid solvable Lie algebra is linearizable.

It follows that every Poisson structure which is a deformation of linear Poisson structureof n-dimensional solvable Lie algebra, with 3 ≤ n ≤ 6, is linearizable. The Poisson structureP0 + P1 (defined in proposition 6.1) is not linearizable.

References

[1] J. M. Ancochea Bermudez, M. Goze, algebras de Lie rigides dont le nilradical est filiforme.Notes aux C.R.A.Sc.Paris, 312 (1991), 21–24.

[2] F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz, D. Sternheimer Deformationtheory and quantization I/ II, Ann. Phys. 111 (1978), 61–110, 111–151.

[3] M. Bordemann, A. Makhlouf, T. Petit, Deformation par quantification et rigidite desalgebres enveloppantes, Journal of Algebra (to appear).

[4] R. Carles, Sur la structure des algebres de Lie rigides, Ann. Inst. Fourier 34 (1984), 65–82.[5] R. Carles, Weight systems for complex Lie algebras. Preprint Universite de Poitiers, 96

(1996).[6] R. Carles, Y. Diakite: Sur les varietes d’algebres de Lie de dimension 7. J. of Algebra. 91,

53–63 (1984).[7] H. Cartan, S. Eilenberg Homological algebra. Princeton University Press (1946).[8] J. Dixmier Cohomologie des algebres de Lie nilpotentes, Acta Sci. Math. 16, Nos.3–4 (1955),

246–250.[9] J. Dixmier, algebres enveloppantes, Gauthier-Villars, Paris, (1974). enveloping algebras, GSM

AMS, (1996).[10] M. Goze, J. M. Ancochea Bermudez, On the classification of Rigid Lie algebras. J. Algebra,

245 (2001), 68–91.[11] M. Goze, E. Remm, valued deformation of Lie algebra. Preprint (2002).[12] M. Gerstenhaber, On the deformation of rings and algebras II, Ann. of Math., 79 (1964),

pp.59–103.[13] M. Gerstenhaber, The cohomology structure of an associative ring. Ann.of Math. 78, 2,

267–288 (1963).[14] M. Gerstenhaber, S. D. Shack: Relative Hochschild cohomology, rigid algebras, and the

Bockstein, J. of Pure and Appl. Alg. 43, 53–74 (1986).[15] P. J. Hilton, U. Stammbach, A Course in Homological Algebra, Springer, New York/Berlin,

1996.[16] G. Hochschild On the cohomology groups of an associative algebra. Ann. Math. 46 (1945),

58–87.[17] G. Hochschild, J-P. Serre, Cohomology of Lie algebras, Ann. Math. 57 (1953), 72–144.[18] M. Kontsevitch Deformation quantization of Poisson manifolds, arXiv:q-alg/9709040, 1997.


[19] A. Makhlouf, M. Goze, Classification of rigid associative algebras in low dimensions, in:Lois d’algebras et varietes algebriques Hermann, Collection travaux en cours 50 (1996).

[20] A. Nijenhuis, R. W. Richardson, Cohomology and deformations in graded Lie Algebras,Bull. Amer. Math. Soc. 72, 1, (1966).

[21] I. Vaisman, Lectures on the geometry of Poisson manifolds, Birkhauser (1994).[22] A. Weinstein, The local structure of Poisson manifold, J. of diff geometry. 18, 3, (1983).

THE ROLE OF A THEOREM OF BERGMAN IN INVESTIGATINGIDENTITIES IN MATRIX ALGEBRAS WITH SYMPLECTIC

INVOLUTION∗

TSETSKA GRIGOROVA RASHKOVA

University of Rousse ”A.Kanchev”

E-mail: [email protected]

The talk is a survey on a series of results considering as applications of a theorem ofBergman [1], which is connected with investigating a class of identities for matrix algebras.These applications are based both on the essential use of an analogue of the stated theoremconcerning matrix algebras with symplectic involution and its elegant proof via graph theoryas a method for proving other results as well related to the same theorem.

We recall that in the matrix algebra over a field Kmof characteristics zero M2n(K, ∗) thesymplectic involution ∗ is defined by

(ABCD

)∗=(Dt −Bt−Ct At

),

where A,B,C,D are n× n matrices and t is the usual transpose.For an algebra R with involution ∗ we have (R, ∗) = R+ ⊕ R−, where R+ = r ∈ R |

r∗ = r and R+ = r ∈ R | r∗ = −r.Let K〈X〉 be the free associative algebra. We call f(x1, . . . , xn) ∈ K〈X〉 a ∗-polynomial

identity for (R, ∗) in symmetric variables if f(r+1 , . . . , r+n ) = 0 for all r+1 , . . . , r

+n ∈ R+.

Analogously f(x1, . . . , xs) ∈ K〈X〉 is a ∗-polynomial identity for (R, ∗) in skew-symmetricvariables if f(r−1 , . . . , r

−s ) = 0 for all r−1 , . . . , r

−s ∈ R−.

Some of the investigations concerning such identities are based on the classical P.I. theoryas every identity in symmetric (or skew-symmetric) variables for M2n(K, ∗) is an ordinaryidentity for Mn(K).

Constructive results however in the symplectic case need stronger tools. They take intoaccount the following considerations.

The algebra R+ is a Jordan algebra with respect to the multiplication r+1 r+2 = r+1 r+2 +

r+2 r+1 ; r+1 , r

+2 ∈ R+ and the identities in symmetric variables are weak polynomial identities

for the pair (R,R+).Similarly, the algebraR− is a Lie algebra with respect to the new multiplication [r−1 , r

−2 ] =

r−1 r−2 − r−2 r−1 ; r−1 , r

−2 ∈ R− and the identities in skew-symmetric variables for (R, ∗) are

weak polynomial identities for the pair (R,R−).For polynomials in symmetric variables the Cayley-Hamilton theorem gives an identity

in two variables for M2n(K, ∗) of degree n2+3n2 . For n = 3 this identity appears to be of

minimal degree [5]. A partial linearization of it gives rise to a Bergman type identity, namely

∗Partially supported by Grant MM1106/2001 of the Bulgarian Foundation for Scientific Research.

176 TSETSKA GRIGOROVA RASHKOVA

a homogeneous (of degree k) and multilinear in y1, . . . , yn polynomial f(x, y1, . . . , yn) fromthe free associative algebra K〈x, y1, . . . , yn〉 which can be written as

(1) f(x, y1, . . . , yn) =∑

i=(i1,...,in)∈Sym(n)

v(gi)(x, yi1 , . . . , yin),

where gi ∈ K[t1, . . . , tn+1] are homogeneous (of degree k − n) polynomials in commutingvariables

gi(t1, . . . , tn+1) =∑

αptp11 . . . t

pn+1n+1

and

(2) v(gi) = v(gi)(x, yi1 , . . . , yin) =∑

αpxp1yi1 . . . x

pnyinxpn+1 .

For polynomials of type (1) A. Giambruno and A. Valenti [2] gave a lower bound of theirdegree as identities in skew-symmetric variables for M2n(K, ∗). For any n they constructeda special multilinear polynomial of degree 4n−1 and found a polynomial of minimal degreefor M4(K, ∗). It leads to the existence of an identity of minimal degree 7 of type (1) for theconsidered algebra. In [3] on its base a full description of the Bergman type identities inskew-symmetric variables for M4(K, ∗) was made.

In the survey we define polynomials of type (1) of minimal degree for M6(K, ∗). Twodifferent classes of Bergman type identities in skew-symmetric variables are given for n = 3.One of the class is related to the existence of central polynomials in skew-symmetric variablesdescribed in [4].

A polynomial c(x1, . . . , xm) ∈ K〈X〉 is central in skew-symmetric variables for the algebra(R, ∗) if it is non-zero in R− and [c(r−1 , . . . , r

−m), r−m+1] = 0 for any r−1 , . . . , r

−m+1 ∈ R−.

The existence of Bergman type identities in the general case is discussed in the talkas well.

In the sequel we use the following notation:

g2n,0 =∏

1 ≤ p < q ≤ n + 1(p, q) = (1, n + 1)

(t2p − t2q)(t1 − tn+1).

Before stating the main results we formulate the theorem of Bergman and its analoguefor the symplectic case.

Proposition 1. [1, Section 6, (27)](i) The polynomial v(gi) from (2) is an identity for Mn(K) if and only if

∏1≤p<q≤n+1

(tp − tq)

divides gi(t1, . . . , tn+1) for all i = (i1, . . . , in).(ii) The polynomial f(x, y1, . . . , yn) from (1) is an identity for Mn(K) if and only if every

summand v(gi) is also an identity for Mn(K).

ROLE OF A THEOREM OF BERGMAN IN INVESTIGATING IDENTITIES 177

Proposition 2. [7, Theorem 3] Considered in M2n(K, ∗), the polynomial f from (1) satisfiesf(a, r1, . . . , rn) = 0 for any skew-symmetric matrix a and all matrices r1, . . . , rn if and onlyif (t1 + tn+1)g2n,0 divides the polynomials gi(t1, . . . , tn+1) for all i = (i1, . . . , in).

Some of the main results included in the talk are the following:

Proposition 3. The linearization in y of the standard polynomial S3([x3, y], [x2, y], [x, y])is an identity in symmetric variables for M6(K, ∗) of minimal degree.

This proposition follows straightforward from [5, Theorem 2.2, Corollary 2.1].

Proposition 4. [3, Theorem 3] A polynomial f is a Bergman type identity in skew-symmetric variables for M4(K, ∗) if and only if it has the form

f = α(v(g1)(x, y1, y2) + v(g2)(x, y2, y1))

+ βv(g3)(x, y1, y2) + γv(g4)(x, y2, y1),

where

1. g1 = g4,0∏i(ait1 + bit2 + cit3), g2 = g4,0

∏i(dit1 − bit2 + (ci + di − ai)t3) and t1 + t3

is not a factor of the polynomials g1 and g2;2. The polynomial (t1 + t3)g4,0 divides g3 and g4 and3. The identity v(g1)(x, y1, y2)+v(g2)(x, y2, y1)= 0 follows from the identity f0(x, y1, y2)=∑

σ∈Sym(2) v(g4,0)(x, yσ(1), yσ(2)) = 0.

Proposition 5. [6, Theorem 3] All Bergman type identities in skew-symmetric variables ofdegree 14 for M6(K, ∗) are consequences of the identity

P (x, y1, y2, y3)= v(g6,0)(x, yσ(1), yσ(2), yσ(3)) + v(g6,0)(x, yσ(3), yσ(2), yσ(1))=0, σ ∈ Sym(3).

Theorem 1. A Bergman type polynomial of degree 15 is a ∗-identity in skew-symmetricvariables for M6(K, ∗) if and only if it has the form

f = α∑i

v(gi)(x, yi1 , yi2 , yi3) + β∑k

v(gkk)(x, yk1 , yk2 , yk3),

where

1. gi = (ait1 + bit2 + cit3 + dit4)g6,0, gi+3 = −(dit1 + cit2 + bit3 + ait4)g6,0, i = 1, . . . , 3and t1 + t4 is not a factor of these polynomials;

2. The polynomial (t1 + t4)g6,0 divides gkk and3. The identity

∑v(gi)(x, yi1 , yi2 , yi3) = 0 follows from the identity P (x, y1, y2, y3) = 0.

For proving the theorem we need the following lemma:

Lemma 1. The identities in skew-symmetric variables

v(gi)(x, yi1 , yi2 , yi3) + v(gi+3)(x, yi3 , yi2 , yi1) = 0


for

gi=(ait1 + bit2 + cit3 + dit4)g6,0gi+3 =−(dit1 + cit2 + bit3 + ait4)g6,0, i = 1, 2, 3

follow from the identity P (x, y1, y2, y3) = 0.

Proof. For simplicity from now on we write f1 = v(g1)(x, y1, y2, y3), f2 = v(g2)(x, y1, y3, y2),f3 = v(g3)(x, y2, y1, y3), f4 = v(g4)(x, y3, y2, y1), f5 = v(g5)(x, y2, y3, y1) and f6 =v(g6)(x, y3, y1, y2), where the polynomials gi will be further specified when needed.

We start with the identity

(3) F = f1 + f4 = 0,

following from Proposition 5 for σ being the identity permutation. The identity

(4) α(F (y1 = [y1, x]) + β(F (y2 = [y2, x]) + γ(F (y3 = [y3, x]) = 0

could be written as

αf1(x, y1x, y2, y3)− αf1(x, xy1, y2, y3) + αf4(x, y3, y2, y1x)− αf4(x, y3, y2, xy1)+βf1(x, y1, y2x, y3)− βf1(x, y1, xy2, y3) + βf4(x, y3, y2x, y1)− βf4(x, y3, xy2, y1)+γf1(x, y1, y2, y3x)− γf1(x, y1, y2, xy3) + γf4(x, y3x, y2, y1)− γf4(x, xy3, y2, y1)= v(g1(x, y1, y2, y3)) + v(g4(x, y3, y2, y1)).

Thus

g1 =α(t2 − t1) + β(t3 − t2) + γ(t4 − t3)=−αt1 + (α− β)t2 + (β − γ)t3 + γt4,

g4 =α(t4 − t3) + β(t3 − t2) + γ(t2 − t1)=−γt1 + (γ − β)t2 + (β − α)t3 + αt4.

Summing (4) for α = 1, β = γ = −1 and α = β = −1, γ = 1 and taking into account thezero characteristics of the field we get that

v(g10(x, y1, y2, y3)) + v(g40(x, y3, y2, y1)) = 0

for g10 = (a1t1 + b1t2 + c1t3 + d1t4)g6,0 and g40 = (a1t1 − c1t2 − b1t3 + d1t4)g6,0.Writing g40 = −(d1t1 + c1t2 + b1t3 + a1t4)g6,0 + ((a1 + d1)t1 + (d1 + a1)t4)g6,0 we get the

validity of Lemma 1 for i = 1.Making in (3) the substitution y2 ↔ y3 we continue with the identity f2 + f5 = 0.

Analogous to the above operations prove the statement for i = 2. The substitution y1 ↔ y2in (3) finishes the proof of Lemma 1.


Remark 1. The proof of Lemma 1 does not use the specific form of g6,0. Thus we come toits generalization:

Lemma 2. Let f = v(g1)(x, yi1 , yi2 , yi3) + v(g2)(x, yi3 , yi2 , yi1) = 0 be a ∗-identity in skew-symmetric variables for M6(K, ∗). Then the identityv(g3)(x, yi1 , yi2 , yi3) + v(g4)(x, yi3 , yi2 , yi1) = 0 for

g3 = (at1 + bt2 + ct3 + dt4)g1g4 =− (dt1 + ct2 + bt3 + at4)g2

is a consequence of the identity f = 0.

Proof of Theorem 1: ⇒ We write a Bergman type identity of degree 14 in the form f =∑fi =

∑i v(gi)(x, yi1 , yi2 , yi3) according to the notations at the beginning of Lemma 1.

Then the commutative polynomials corresponding to a consequence of f of degree 15 willbe (

∑4i=1 aiti)g1, (

∑4i=1 biti)g2, . . . , (

∑4i=1 eiti)g5 and (

∑4i=1 liti)g6, respectively. Applying

Lemma 1 we get that the part Q of f , corresponding to the commutative polynomials

(4∑i=1

aiti)g1 − (4∑i=1

a5−iti)g4 + (4∑i=1

biti)g2 −

(4∑i=1

b5−iti)g5 + (4∑i=1

citi)g3 − (4∑i=1

c5−iti)g6,

is a ∗-identity in skew-symmetric variables. We consider the remaining part of f correspond-ing to

(a4 + d1)t1 + (a3 + d2)t2 + (a2 + d3)t3 + (a1 + d4)t4g4 +(b4 + e1)t1 + (b3 + e2)t2 + (b2 + e3)t3 + (b1 + e4)t4g5 +(c4 + l1)t1 + (c3 + l2)t2 + (c2 + l3)t3 + (c1 + l4)t4g6.

This part is related to the identity F = A+B + C = 0, where

A = (a4 + d1)xf4 + (a3 + d2)f4(y3 = y3x)+(a2 + d3)f4(y2 = y2x) + (a1 + d4)f4x,

B = (b4 + e1)xf5 + (b3 + e2)f5(y2 = y2x)+(b2 + e3)f5(y3 = y3x) + (b1 + e4)f5x,

C = (c4 + l1)xf6 + (c3 + l2)f6(y3 = y3x)+(c2 + l3)f6(y1 = y1x) + (c1 + l4)f6x.

We calculate F (e22−e55+2(e33−e66)−3(e11−e44), e21−e45−e36, e15+e24+e14, e53+e62)and F (e22− e55 +2(e33− e66)−3(e11− e44), e21− e45 + e31− e46, e15 + e24−2e25, e53 + e62).

The resulting system

3(b4 + e1) + (b3 + e2)− 2(b2 + e3) + 2(b1 + e4) = 0− 3(b4 + e1)− (b3 + e2) + 2(b2 + e3)− 3(b1 + e4) = 0

leads to b1 + e4 = 0.


Then we make three more calculations, namely F (e11 − e44 − 2(e22 − e55) + 5(e33 −e66), e12 − e54 + e32 − e56, e23 − e65, e16 + e34), F (e11 − e44 + 2(e22 − e55)− 5(e33 − e66) +3(e21− e45) + e32− e56, e23− e65, e16 + e34 + e63, e12− e54) and F (e11− e44− 2(e22− e55) +3(e33 − e66), e36, e43 + e61, e12 − e54).

The resulting system for the unknowns a4 + d1, a3 + d2, a2 + d3 and a1 + d4 is thefollowing:

2 (a4 + d1)− (a3 + d2)− 3(a2 + d3) + 2(a1 + d4) = 0,2 (a4 + d1)− (a3 + d2) + 3(a2 + d3)− 3(a1 + d4) = 0,(a4 + d1) + (a3 + d2) + 5(a2 + d3) + 5(a1 + d4) = 0,(a4 + d1)− 5(a3 + d2) + 2(a2 + d3)− (a1 + d4) = 0.

It has only the trivial solution.Now we calculate B(b1 + e4 = 0) + C for x = 3(e22 − e55) − 5(e33 − e66) − e11 + e44,

y1 = e12− e54− e23 + e65− e16− e34 + e42 + e51 + e43 + e61, y2 = e12− e54 + e23− e65− e16−e34 + e42 + e51− e43− e61, y3 = e12− e54− e23 + e65− 4(e16 + e34)− 2(e42 + e51)+ e43 + e61.

Thus we come to a homogeneous system for the unknowns (b4 + e1), (b3 + e2), (b2 + e3),(c4 + l1), (c3 + l2), (c2 + l3) and (c1 + l4), whose matrix of the coefficients is

−1 5 −3 −4 20 −12 −42 −60 −44 83 −465 199 −83

−12 20 −4 3 −5 1 35 −1 −3 1 −2 −6 1−1 −3 5 2 6 −10 2

6 2 −10 3 1 −5 35 −93 −89 5 −93 −89 −5

−10 6 2 −5 3 1 −5

.

The rank of this matrix is 7 meaning that the considered system has only a trivialsolution, i.e.

d1 + a4 = d2 + a3 = d3 + a2 = d4 + a1 = 0,e1 + b4 = e2 + b3 = e3 + b2 = e4 + b1 = 0,l1 + c4 = l2 + c3 = l3 + c2 = l4 + c1 = 0.

These relations and Proposition 5 show that the part Q is an identity in skew-symmetricvariables and the identity f has the form stated in the theorem.

All calculations were made using the computer algebra system Mathematica.⇐ Let f be a Bergman type polynomial of degree 15 of the stated form. We consider its

part A = f1(x, y1, y2, y3) + f4(x, y3, y2, y1) in which

g1 = (a1t1 + a2t2 + a3t3 + a4t4)g6,0, g4 = −(a4t1 + a3t2 + a2t3 + a1t4)g6,0.

Lemma 1 shows that the polynomial A is an ∗-identity in skew-symmetric variables.Applying the same lemma to the other two parts f2 + f5 and f3 + f6 of the polynomial

f we get that f is a ∗-identity, as a consequence of the identity P (x, y1, y2, y3) = 0.This ends the proof of Theorem 1.An easy corollary of it using Lemma 2 is the following


Proposition 6. Every Bergman type polynomial of degree k of the form

f = α∑i

fi(x, yi1 , yi2 , yi3) + β∑s

v(gss)(x, ys1 , ys2 , ys3),

where

1. gi = g6,0∏k−14j=1 (ajit1 + bjit2 + cjit3 + djit4), gi+3 = g6,0

∏k−14j=1 (−(ajit1 + cjit2 +

bjit3 + djit4)), i=1, . . . ,3 and t1 + t4 is not a factor of these polynomials;2. The polynomial (t1 + t4)g6,0 divides gss and3. The identity

∑fi(x, yi1 , yi2 , yi3) = 0 follows from the identity P (x, y1, y2, y3) = 0, is

a ∗-identity in skew-symmetric variables for M6(K, ∗).

Now we could define another class of Bergman type identities.

Theorem 2. The polynomials f =∑σ∈Sym(3)(−1)σv(g)(x, yσ(1), yσ(2), yσ(3)) for g = (t22−

t23)(t1t2 + t2t3 + t3t4)kg6,0 are Bergman type identities in skew-symmetric variables forM6(K, ∗) of degree 16+2k.

Proof. According to [4, Theorem 12] the polynomialsf =

∑σ∈Sym(3)(−1)σv(g1)(x, yσ(1), yσ(2), yσ(3)) for

g1 =∏

1 ≤ p < q ≤ 4(p, q) = (1, 4)

(t2p − t2q)(t22 − t23)(t1t2 + t2t3 + t3t4)k

are central polynomials in skew-symmetric variables for M6(K, ∗) of degree 15+2k. NowTheorem 2 is straightforward due to the definition of central polynomials.

Now we are able to formulate a result for M2n(K, ∗) generalizing the “only if” part ofProposition 4 for n = 2 and Proposition 6 for n = 3.

Theorem 3. For n ≡ 2,3 (mod 4) every Bergman type polynomial of degree k of the form

f = α∑i

v(gi)(x, yi1 , . . . , yin) + β∑j

v(gj)(x, yj1 , . . . , yjn),

where

1. gi=g2n,0∏k−n2−2n+1l=1

∑nm=1 a

(l)i,mtm, gi+ n!

2=g2n,0

∏k−n2−2n+1l=1 (−∑n

m=1 a(l)i,n+1−mtm),

i = 1, . . . , n!2 and t1 + tn+1 is not a factor of these polynomials;

2. The polynomial (t1 + tn+1)g2n,0 divides gj and3. The identity

∑v(gi)(x, yi1 , . . . , , yin) = 0 follows from the identity v(g2n,0)(x, yi1 ,

yi2 , . . . , yin) + v(g2n,0)(x, yin , yin−1 = 0, . . . , yi1), (i1, i2, . . . , in) ∈ Sym(n), is a ∗-identity in skew-symmetric variables for M2n(K, ∗).


Proof. The proof follows the ideas of presenting a polynomial and its consequences of thenext degree as exposed at the beginning of the proof of Theorem 1 using the next Lemma(instead of Lemma 2) and Remark 2 as well.

Lemma 3. For n ≡ 2,3 (mod 4) the polynomial

C(x, y1, y2, . . . , yn) = v(g2n,0)(x, yi1 , yi2 , . . . , yin) + v(g2n,0)(x, yin , yin−1 , . . . , yi1)

for any (i1, i2, . . . , in) ∈ Sym(n) is a ∗-identity in skew-symmetric variables for M2n(K, ∗).Proof. Considering the factors tk ± tl of g(t1, . . . , tn) (except t1 + tn+1) we follow the proofof [7, Proposition 2]. It shows that it is sufficient to consider only the case when x =∑ni=1 ρi(eii − en+i,n+i) and to find an analogue of the stated proof concerning the factor

t1 + tn+1 which is not the case here.

For this analogue we evaluate

C = C(n∑i=1

ρi(eii − en+i,n+i), e12 − en+2,n+1, . . . , en−1,n − e2n,2n−1, en,n+1 + e1,2n).

The result is

C = Ae1,n+1 = (g2n,0(ρ1, . . . , ρn,−ρ1) + (−1)ng2n,0(ρ1,−ρn,−ρn−1, . . . ,−ρ1))e1,n+1.

The properties of the polynomial g2n,0(t1, . . . , tn+1) allow us to write

A = g2n,0(ρ1, . . . , ρn,−ρ1) + (−1)n−1g2n,0(ρ1, ρn, ρn−1, . . . , ρ2,−ρ1).

Changing the places of two neighbouring variables causes (−1) as a factor of g2n,0. Thus

A= g2n,0(ρ1, . . . , ρn,−ρ1) + (−1)n−1+n−2+···+1g2n,0(ρ1, ρ2, . . . , ρn,−ρ1)

= g2n,0(ρ1, . . . , ρn,−ρ1) + (−1)n−1+(n−1)(n−2)

2 g2n,0(ρ1, ρ2, . . . , ρn,−ρ1)

= g2n,0(ρ1, . . . , ρn,−ρ1) + (−1)n(n−1)

2 g2n,0(ρ1, ρ2, . . . , ρn,−ρ1).

For n ≡ 2,3 (mod 4) A ≡ 0.

Remark 2. Let denote ρi =(

1 2 . . . ni1 i2 . . . in

)and ρi−1 =

(1 2 . . . nin in−1 . . . i1

). Then for the

symmetric group Sym(n) the following holds:Sym(n) = ∪iρi, ρi−1.Proof. For any ρi ∈ Sym(n) the map ϕ(ρi) = ρi−1 is an automorphism of second order andϕ, e divides the elements of Sym(n) into orbits. Obviously each orbit contains 2 elements.No orbits share common elements and any element of Sym(n) is in an orbit.

Remark 3. It could be easily seen that modulo the identity∑σ v(g4,0)(x, yσ(1), yσ(2)) the polynomial g2 in Proposition 4 could be written as g2 =

−g4,0∏i(citi + biti + aiti) meaning that Theorem 2 generalizes the partial cases for n

investigating in Propositions 4 and 6.


References

[1] G.M. Bergman, Wild automorphisms of free P.I. algebras and some new identities (1981),preprint.

[2] A. Giambruno, A. Valenti, On minimal ∗-identities of matrices, Linear Multilin. Algebra 39(1995), 309–323.

[3] T.G. Rashkova, Bergman type identities in matrix algebras with involution, Proceedings of theUnion of Scientists - Rousse, ser. 5 Mathematics, Informatics and Physics 1 (2001), 26–31.

[4] T. Rashkova, Matrix algebras with involution and central polynomials, J. of Algebra 248 (2002),132–145.

[5] T.G. Rashkova, ∗-identities of minimal degree in matrix algebras of low order, Periodica Math-ematica Hungarica 34 (3) (1998), 229–233.

[6] T.G. Rashkova, One conjecture for the identities in matrix algebras with involution, Ann. ofSofia Univ. 94 (2002), to appear.

[7] Ts. Rashkova, V. Drensky, Identities of representations of Lie algebras and ∗-polynomial iden-tities, Rendiconti del Circolo Matematico di Palermo 48 (1999), 153–162.

THE TRIANGULAR STRUCTURE OF LADDER FUNCTORS

WOLFGANG RUMP

Institut fur Algebra und Zahlentheorie, Universitat Stuttgart

Pfaffenwaldring 57, D-70550 Stuttgart, Germany

e-mail: [email protected]

Ladder functors were introduced in [14] and [16] as a tool for the structural analysis ofcategories A with Auslander-Reiten sequences. The original idea of a ladder goes back toIgusa and Todorov [6] who considered chains of maps

· · · −→ a0λ0−→ a1

λ1−→ a2 −→ · · ·between two-termed complexes ai such that the mapping cone of each λi is an almost splitsequence. Being rare objects, such ladders were difficult to handle, but Igusa and Todorovsuccessfully applied them to obtain their characterization of the Auslander-Reiten quiversof representation-finite artinian algebras. The corresponding problem in dimension one, i. e.for orders over a complete discrete valuation domain, was solved 15 years later by Iyama[8], who showed that (modified) ladders of arbitrary length are obtained when the startingmorphism a0 ∈ A is special, i. e. if its isomorphism class is invariant modulo Rad2A. Forthis improvement, it has to be payed in return that at each step an of the ladder, trivialdirect summands 0→ A have to be discarded before passing to an+1.

In [16] we showed that the homotopy category M(A) of two-termed complexes is ratheruseful for the study of ladders. We constructed a functor L: M(A) → M(A) together withan augmentation λ: L→ 1 such that any morphism a ∈ A, regarded as an object in M(A),gives rise to a ladder

· · · −→ L2aλLa−→ La

λa−→ a

which generalizes the above mentioned constructions. There is no restriction on a, andthere are no trivial summands to be discarded. Moreover, the components of λ, regarded ascommutative squares in A, are exact, i. e. they are simultaneous pullbacks and pushouts. Thedual construction leads to an endofunctor L− with a natural transformation λ−: 1 → L−,which produces ladders in the reverse direction. Both functors L,L− determine each other,forming an adjoint pair L L−. The use of these ladder functors [14, 16] led to a betterunderstanding and improvement of previous work on categories with almost split sequences.In particular, Iyama’s criterion [8] for finite Auslander-Reiten quivers in dimension one wasreplaced by a characterization in terms of additive functions [16].

As the homotopy category M(A) can be viewed as a certain part of a triangulated cate-gory, there remains some fragment of triangulated structure in M(A). Our first aim in thisarticle will be to show how the ladder functors are related to triangles (Theorem 1). Weprove that each object a in M(A) determines a “triangle”

(0) TSaσa−→ La

λa−→ aπa−→ Sa,

2000 Mathematics Subject Classification. Primary: 16G70, 16G30, 16D90, 18E30. Secondary: 16G60.

Key words and phrases. Ladder functor, triangulated category, Auslander-Reiten sequence.

THE TRIANGULAR STRUCTURE OF LADDER FUNCTORS 185

where πa belongs to the localization M(A)PrL of M(A) by the full subcategory PrL =a ∈ ObM(A) | λa invertible, and σa belongs to the corresponding localization M(A)InLwith respect to L−. The objects Sa and TSa are semisimple in M(A)PrL and M(A)InL,respectively. (We call an object S left semisimple if every monomorphism S′ → S splits.Left and right semisimple objects are just called semisimple.) The functor T provides anequivalence between semisimple objects of M(A)PrL and M(A)InL, respectively. Since T isnot an endofunctor, it cannot be iterated. However, with that grain of salt, the sequences (0)behave just like ordinary triangles. For example, if ϕ: b → a is a morphism in M(A) withπaϕ = 0 in M(A)PrL, then ϕ factors uniquely through λa (as a morphism of M(A)!).

In addition to the triangle property, the morphism πa ∈ M(A)PrL is universal in thesense that any morphism ψ: a→ s in M(A)PrL with s semisimple factors uniquely throughπa. Therefore, the ladder functor L is uniquely determined by the full subcategory PrL.

Let us remark that for a category A of maximal Cohen-Macaulay modules over an isolatedsingularity Λ in the sense Auslander [2], if Λ is of dimension > 2, then projective objectsP in A are no longer characterized by the property τP = 0. Fortunately, the constructionof our ladder functor L makes no use of right almost split sequences 0 → ϑP → P . Onthe other hand, this means that the ladder functors do not tell the whole story about A.However, the missing structure is suggested by the triangles (0), as they define a connectionbetween semisimple objects in M(A)PrL and M(A)InL. While left and right semisimpleobjects coincide in these localisations, they form two different full subcategories SlM(A)and SrM(A) in M(A). We will show (Proposition 8) that an object a: A→ P in M(A) is leftsemisimple if and only if a is a right almost split morphism in A with τP = 0. Consequently,there are full embeddings of A into the categories of one-sided semisimple objects in M(A):

SlM(A)← A → SrM(A).

As a byproduct, it follows that A can be recovered from M(A). In case indA is finite,we show (Theorem 4) that the essential condition for A to be representable as a categoryA-mod over an artinian ring A, or as a category Λ-lat of lattices over an order Λ, consistsin a specific correspondence between SlM(A) and SrM(A).

1. Preliminaries

Let A be an additive category. For a full subcategory C, the ideal generated by theidentity morphisms 1C with C ∈ ObC will be denoted by [C]. By Mor(A) we denote thecategory with morphisms in A as objects and commutative squares as morphisms. There isa natural full embedding A → Mor(A) which maps A ∈ Ob A to 1A ∈ Ob Mor(A). If weregard the objects of Mor(A) as two-termed complexes 0 → A1 → A0 → 0, then the ideal[A] of Mor(A) consists of the morphisms which are homotopic to zero. We define mod(A)(resp. com(A)) as the factor category of Mor(A) modulo the ideal of morphisms

(1)

186 WOLFGANG RUMP

such that f0 factors through b (resp. f1 factors through a). Then mod(A) is equivalent tothe category of coherent functors Aop → Ab into the category of abelian groups, and

(2) com(Aop) = mod(A)op.

If A = R-proj, the category of finitely generated projective left modules over a ring R, thenmod(A) is equivalent to the category R-mod of finitely presented left R-modules. Notethat mod(A) need not be abelian.

Now let A be a Krull-Schmidt category, that is, an additive category such that everyobject is a finite direct sum of objects with local endomorphism rings. A morphism f : A→ Bin A is said to belong to the radical Rad A if 1 − gf is invertible for all g: B → A. (Ofcourse, this concept is left-right symmetric.) Every morphism f ∈ A admits a decompositionf = e⊕r into an isomorphism e and a morphism r ∈ Rad A. Therefore, the full subcategoryM(A) of objects a: A1 → A0 in Mor(A)/[A] with a ∈ Rad A is equivalent to Mor(A)/[A].If A+ (resp. A−) denotes the full subcategory of objects A+: 0 → A (resp. A−: A → 0) inM(A), then

(3) mod(A) ≈ M(A)/[A−]; com(A) ≈ M(A)/[A+].

This follows since a morphism in M(A) belongs to [A−] if and only if it vanishes in mod(A).In order to work with mod(A), an intrinsic characterization will be useful. Recall that

an object P of an additive category A is said to be projective if for every cokernel A B inA, the natural map HomA(P,A) → HomA(P,B) is surjective. Arrows (resp. ) standfor cokernels (kernels). We say that A has enough projectives if for each object A there isan epimorphism P → A with P projective. Injective objects and the property of havingenough injectives are defined dually. The full subcategory of projective (injective) objectsin A will be denoted by Proj(A) (resp. Inj(A)). For a full subcategory C of A, we defineadd C as the full subcategory of objects C ∈ ObA with 1C ∈ [C]. If idempotents split in A,this coincides with the usual definition (see [3]). There is a natural embedding

(4) A →mod(A)

which maps A ∈ Ob A to 0→ A.

Proposition 1. Let A be an additive category. Then

(5) addA = Proj(mod(A)).

An additive category M is equivalent to mod(A) for some additive category A if and onlyif the following are satisfied.

(a) Every morphism in mod(A) has a cokernel.(b) Every epimorphism in M is a cokernel.(c) M has enough projectives.(d) If A a→ B

c C is a sequence of morphisms in M with c = cok a, and a morphism p:P → B with P projective satisfies cp = 0, then p factors through a.

The proof makes use of the following lemma. A sequence of morphisms P1a→ P0

cM withP0, P1 projective and c = cok a is said to be a projective presentation of M .


Lemma 1. Let M be an additive category such that every object has a projective presenta-tion. Then condition (c) of Proposition 1 is satisfied whenever it holds with A,B projective.

Proof. For a given sequence Aa→ B

c C as in (c), consider a projective presentation

Q1b→ Q0

d B and a cokernel q: Q A with Q projective. Then there exists a morphismf : Q→ Q0 with aq = df , and we obtain a commutative diagram

where cd is a cokernel of (f b). Now the assertion follows immediately.

Proof of Proposition 1. For a morphism (1) in mod(A), it is easily checked that a cokernelis given by

(6)

This yields (a) and implies that the objects of A are projective in mod(A). Moreover, weinfer that any object a: A1 → A0 in mod(A) has a projective presentation

(7)

If a ∈ Proj(mod(A)), this gives a split epimorphism A0 a. Hence (5) holds. Therefore,(7) satisfies the property stated in (d). By Lemma 1, this implies that (d) holds in mod(A).Furthermore, (7) yields (c) for mod(A). To verify (b), let (1) be an epimorphism in mod(A).By (6) this means that (b f0) has a section

(hs

): B0 → B1 ⊕ A0, i. e. bh+ f0s = 1. In fact,

this implies that (1) is a cokernel:

(The composition is zero by virtue of the map (hf0 1− hb): A0 ⊕B1 → B1.)

188 WOLFGANG RUMP

Conversely, let M be an additive category which satisfies (a), (b), (c), and (d). Thenevery object M of M has a projective presentation P1

a→ P0 M , and it is readily seenthat M → a yields an equivalence M −→∼ mod(Proj(M)).

Recall that a monic and epic morphism is said to be regular.

Corollary. Let A be an additive category. Then every regular morphism in mod(A) isinvertible.

Proof. Let r: M → N be regular in mod(A). Since r is epic, we have r = cok f for somef ∈mod(A). Since r is monic, this gives f = 0. Hence r is invertible.

2. Semisimple objects

In this section we study existence properties for semisimple objects in mod(A), as far asneeded in connection with ladder functors. Let A be an additive category. We call an objectS left (right) semisimple if every monomorphism A→ S (epimorphism S → A) in A splits.If both conditions hold, we call S semisimple. The full subcategory of semisimple objectswill be denoted by S(A). We call A (co-)semilocal if S(A) is a (co-)reflective subcategoryof A, i. e. if the inclusion S(A) → A has a left (right) adjoint. If this adjoint functor mapsnon-zero objects to non-zero objects, we call A strongly (co-)semilocal.

Example. For a ring R, let R-Mod be the category of left R-modules. Then S(R-Mod)consists of the direct sums of simple modules ([1], Theorem 9.6), and there is no differencebetween left and right semisimple objects. Furthermore, R-Mod is semilocal if and only ifR is a semilocal ring. However, R-Mod is always co-semilocal. By [1], Lemma 28.3, R-Modis strongly semilocal if and only if R is left perfect. By [17], VIII, Proposition 2.5, R-Modis strongly co-semilocal if and only if R is left semi-artinian.

Recall that an additive functor S: A→ A together with a natural transformation π: 1→S is said to be a pointed functor [10]. For a reflective full subcategory C of A with reflector S:A→ C and inclusion I: C → A, the unit η: 1→ IS defines a pointed functor IS. This willbe called the reflection of C. In contrast to the above example, the components of η need notbe epic, in general. Dually, an additive endofunctor S with a natural transformation S → 1will be called an augmented functor, and if S comes from a coreflective full subcategory C,we call S the coreflection of C.

Proposition 2. Let A be an additive category. Then every right semisimple object inmod(A) is semisimple.

Proof. Let S be right semisimple in mod(A), and let f : A→ S be a monomorphism. Thenf has a cokernel c: S C. Since S is right semisimple, c has a section s: C → S, thatis, cs = 1. Now it is easy to verify that (f s): A ⊕ C → S is regular. By the Corollary ofProposition 1, it follows that (f s) is invertible, whence f is split monic.

Remark. It can be shown that for a strict τ -category (see §4), left and right semisimplicityis equivalent in mod(A).

Let A be a Krull-Schmidt category. A morphism u: A→ B in A is said to be right almostsplit if u ∈ Rad A, and every f : C → B in Rad A factors through u. When such a morphismu exists for each B ∈ ObA, we simply say that A has right almost split morphisms. Theleft-hand notions are defined in a dual fashion. The following proposition has a well-knownversion for (semi-)simple functors.


Proposition 3. Let A be a Krull-Schmidt category, and let P1a→ P0

p S be a projective

presentation in mod(A) with a ∈ RadA. Then S is semisimple if and only if a is rightalmost split in A.

Proof. Suppose that S is semisimple. Let r: P → P0 be in RadA. Then the cokernelc: S C of pr has a section s: C → S. Now scp: P0 → S lifts along p, say, scp = pe.Hence cp(1 − e) = cp − cscp = 0, and thus p(1 − e) = prg for some g: P0 → P . Therefore,1 − e − rg factors through a. So we get 1 − e ∈ RadA, whence e is invertible. Since(1 − sc)pe = scp − scpe = scp(1 − e) = 0, we get 1 − sc = 0. Thus pr = scpr = 0, whichproves that r factors through a.

Conversely, let a be right almost split, and let e: S → B be an epimorphism in mod(A).Consider a projective presentation Q1

b→ Q0

q B with b ∈ Rad A. Since e is a cokernel,

we get a morphism f : Q0 → P0 with ep · f = q. Since a is right almost split, we obtain acommutative diagram

This yields a morphism s: B → S with sq = pf . Hence (1 − es)q = q − epf = 0, and thuses = 1.

Lemma 2. Let A be a Krull-Schmidt category, and let r: P → Q be a morphism in Rad A.If S is a semisimple object in mod(A), then every morphism s: Q→ S in mod(A) satisfiessr = 0.

Proof. Choose a projective presentation P1a→ P0

p S with a ∈ Rad A. By Proposition 3,

a is right almost split in A. Therefore, we get a commutative diagram

which yields sr = 0.

Lemma 3. If an object M of an additive category admits a projective presentation, thenevery cokernel q: Q0 M with Q0 projective can be completed to a projective presentation.

Proof. Let P1a→ P0

p M be a projective presentation. There are morphisms f, g between

P0 and Q0 such that p = qf and q = pg. Then q is a cokernel of (1−fg fa): Q0⊕P1 → Q0.

Proposition 4. Let A be a Krull-Schmidt category. Then mod(A) is strongly semilocal ifand only if A has right almost split morphisms. If mod(A) is semilocal with reflection η:1→ S, then the components of η are epic.

190 WOLFGANG RUMP

Proof. Let mod(A) be semilocal with reflection η: 1 → S. For an object M of mod(A),the cokernel c: SM C of ηM has a section s: C → SM . Hence (1 + sc)ηM = ηM , andthus 1 + sc = 1 by the uniqueness property of η. So we get c = 0, which shows that ηM isepic. By Lemma 3, every A ∈ ObA gives rise to a projective presentation A′ a→ A

ηA SAin mod(A). Assume that mod(A) is strongly semilocal. Then A has no non-zero directsummand B with ηB = 0. Hence a ∈ RadA. By Proposition 3, this yields a right almostsplit morphism a. Conversely, suppose that A has right almost split morphisms. For a givenobject M in mod(A), consider a projective presentation P1

a→ P0

p M with a ∈ Rad A,

and a right almost split morphism u: P → P0 in A. Then a factors through u, and so weget a commutative diagram

with c = coku. By Proposition 3, SM is semisimple. Let f : M → S be any morphism inmod(A) with S semisimple. Then fpu = 0 by Lemma 2. Hence fp = gc for some g: SM →S. Thus (f − gηM )p = 0, which gives f = gηM . This proves that η: 1 → S is a reflection.If SM = 0, then u is epic, and thus a cokernel in mod(A). Hence u is split epic. So we getP0 = 0, and therefore, M = 0. This proves that mod(A) is strongly semilocal.

3. Quotient categories

For our purpose we need a generalization of Grothendieck’s quotient category ([5], 1.11).Let A be an additive category. We shall say that A has a quotient category if the regularmorphisms in A admit a calculus of left and right fractions [4]. The corresponding quotientcategory will be denoted by Q(A). Up to equivalence, the faithful embedding Q: A → Q(A)is characterized by the property that Q makes regular morphisms invertible, and that everymorphism in Q(A) can be written in the form r−1a = bs−1 with a, b, r, s ∈ A and r, s regular.

More generally, let S be a full subcategory of A. We define Σ(S) as the class of morphismsin A which are regular in A/[S]. We call an additive functor

(8) Q : A→ B

with S := KerQ := A ∈ ObA |Q(A) = 0 a quotient functor if the following are satisfied.

(a) For any r ∈ Σ(S), Q(r) is invertible in B.(b) Q(f) = 0 if and only if f ∈ [S].(c) Every morphism in B is of the form Q(r)−1Q(f) = Q(g)Q(s)−1 with f, g ∈ A and

r, s ∈ Σ(S).

It is easy to see that for a quotient functor (8), up to equivalence, the category B merelydepends on S, namely, there is an equivalence E: B −→∼ Q(A/[S]) such that EQ coincideswith the natural composition A→ A/[S]→ Q(A/[S]). Hence, for a full subcategory S of A,a quotient functor (8) with KerQ = add S exists if and only if A/[S] has a quotient category.Therefore, if Q(A/[S]) exists, we write AS := B. In particular, A0 ≈ Q(A) when A has aquotient category.


Remark. Obviously, a quotient functor Q: A→ AS solves the universal problem of makingthe morphisms of Σ(S) invertible. We call a full subcategory S of A thick if Σ(S) admits acalculus of left and right fractions. When A is abelian, it can be shown that this concept ofthickness coincides with the usual one (then S is also called a Serre subcategory), and thatAS is equivalent to Grothendieck’s quotient category A/S.

Proposition 5. Let A be an additive category and P a full subcategory. If GenP denotesthe full subcategory of objects M in mod(A) admitting a cokernel P M with P ∈ P, then

(9) mod(A/[P]) = mod(A)Gen P.

Proof. We have to show that the natural functor

Q : mod(A)/[Gen P] −→mod(A/[P])

is a faithful quotient functor. Thus let f : M → N be a morphism in mod(A) which is zeroin mod(A/[P]). Consider a commutative diagram

in mod(A) with cokernels a, b and A,B ∈ A. Then we have morphisms Ap→ P

q→ B withP ∈ P such that b(g − qp) = 0. The pushout

in mod(A) yields an object C ∈ GenP and bq · p = f · a implies that f factors through c.This proves that Q is faithful.

Now let (1) be a morphism f ∈mod(A/[P]). Then there are morphisms A1p→ P

q→ B0

with P ∈ P such that f0a− bf1 = qp. Consider the morphisms r:(ap

)→ a and s: b→ (b q)in mod(A)/[Gen P] given by

192 WOLFGANG RUMP

respectively. Then Q(r) and Q(s) are invertible, whence r, s are regular. Moreover, thereare morphisms g, h ∈ mod(A) with f · Q(r) = Q(g) and Q(s) · f = Q(h). Hence f =Q(g)Q(r)−1 = Q(s)−1Q(h).

It remains to show that Q makes regular morphisms invertible. More generally, lete ∈ mod(A)/[Gen P] be monic. Assume that f = Q(g)Q(r)−1 ∈ mod(A/[P]) satisfiesQ(e) · f = 0. Then Q(e)Q(g) = 0, which gives eg = 0, and thus g = 0. Hence Q(e) ismonic. Dually, we infer that Q preserves epimorphisms. Now the Corollary of Proposition 1completes the proof.

4. Ladder functors and triangles

Ladder functors were introduced [14, 16] in order to an analyse the Auslander-Reitenstructure of a strict τ -category A. We will show now that ladder functors are closely relatedto a kind of triangular structure of the homotopy category M(A).

Recall that a sequence of morphisms

(10) τAv ϑA

u→ A

in a Krull-Schmidt category A is said to be right almost split if u is right almost split, v isleft almost split, and v = keru. Up to isomorphism, such a sequence is uniquely determinedby the object A. In a dual fashion, left almost split sequences

(11) A→ ϑ−A τ−A

are defined. If left and right almost split sequences (10) and (11) exist for each object A ofA, then A is said to be a strict τ -category [7].

Let A be a strict τ -category. Every object a: A1 → A0 of M(A) admits a left standardform [16], that is, a representation as a matrix

(12) a =(b fts p

): B ⊕ U → C ⊕ P

with τP = 0 and t ∈ Rad A, such that there exists a left and right almost split sequence

(13)

Then a morphism λa: La→ a in M(A) is given by the commutative diagram

(14)


By [16], Proposition 4, this yields an additive functor L: M(A) → M(A) together with anatural transformation λ: L → 1. The augmented functor L will be called the left ladderfunctor. In a dual way, we get the right ladder functor λ−: 1→ L−. In [16] we showed thatthe components λa and λ−a are regular. Notice that the vertical morphisms in (14) are ina sense dual to each other. By this property, it follows ([16], Proposition 5) that L is leftadjoint to L−, i. e. there is a natural isomorphism

Φ : HomM(A)(La, b) −→∼ HomM(A)(a, L−b).

Let us write ϕ− := Φ(ϕ) and ψ+ := Φ−1(ψ). Then the correspondence ϕ → ϕ− is given bythe commutative diagram

(15)

Let us denote the full subcategory of objects A in A with τA = 0 (resp. τ−A = 0)by Projτ (A) (resp. Injτ (A)). In [16] we defined the full subcategories FixL and FixL−

of objects a in M(A) for which λa (resp. λ−a ) is invertible. An object x ∈ ObM(A) willbe called L-projective (L-injective) if (λa)∗: HomM(A)(x,La) → HomM(A)(x, a) (resp. λ∗a:HomM(A)(a, x)→ HomM(A)(La, x)) is surjective for all a ∈ ObM(A). The full subcategoryof L-projective (L-injective) objects will be denoted by PrL (resp. InL). The relationshipbetween these categories is given by

Proposition 6. Let A be a strict τ -category. Then PrL = FixL and InL = FixL−.

Proof. The inclusion FixL ⊂ PrL is trivial. Conversely, assume that a ∈ PrL. Then λahas a section ϕ: a → La. Since λa is regular, this implies that a ∈ FixL. By (15), we getFixL− ⊂ InL and a commutative diagram

(16)

for any b ∈ Ob M(A). Suppose that b ∈ InL. Then there is a morphism ϕ: L−b →b with εb = ϕλL−b. Since λL−b is componentwise epic, this gives λ−b ϕ = 1, and thusb ∈ FixL−.

Next we will prove that the localizations M(A)S with S = PrL or S = InL exist for astrict τ -category A. For a morphism ϕ ∈ M(A) we define the local (co-)kernel kerSϕ (resp.cokSϕ) as the (co-)kernel of ϕ in M(A)S. As a certain counterpart, we call ϕ ∈ M(A) a

194 WOLFGANG RUMP

global kernel of ψ ∈ M(A)S if ψϕ = 0 holds in M(A)S, and for any ϕ′ ∈ M(A) with ψϕ′ = 0in M(A)S there exists a unique ω ∈ M(A) with ϕω = ϕ′.

(17)

By definition, the global kernel, as well as its dual, the global cokernel, is unique up toisomorphism. We write ϕ = kerSψ (resp. cokSψ) for the global (co-)kernel of ψ.

Proposition 7. Let A be a strict τ -category with a full subcategory P, and let SP(mod(A))denote the full subcategory of objects S in mod(A) with HomA(P, S) = 0. Then there is anatural equivalence

(18) SP(mod(A)) −→∼ S(mod(A/[P])).

Proof. According to Proposition 3, let ϑAu→ A S be a projective presentation of

an object S in SP(mod(A)). Since HomA(P, S) = 0, it follows that A has no non-zero direct summand in P. As u is right almost split in A/[P], it determines an objectS in S(mod(A/[P])), and so S → S defines an additive functor F : SP(mod(A)) →S(mod(A/[P])). If F (ϕ) = 0 for a morphism ϕ ∈ SP(mod(A)), then ϕ ∈ [Gen P] by

Proposition 5. Hence ϕ = 0, and thus F is faithful. Let ϑA′ u′→ A′ S′ be a projective

presentation of another object S′ in SP(mod(A)). A morphism F (S) → F (S′) is given bya commutative diagram

(19)

in A/[P]. Then f can be modified modulo [P] such that (19) becomes commutative in A.Thus F is full. By Proposition 3, any semisimple object in mod(A/[P]) is given by a rightalmost split morphism a: A1 → A0 in A/[P], where we may assume that a ∈ Rad A, andthat A0 has no non-zero direct summand in P. Since the right almost split morphism u:ϑA0 → A0 in A belongs to Rad(A/[P]), there is a morphism f : ϑA0 → A1 with u−af ∈ [P].So if we replace a by some a′: A1 ⊕ P → A0 with P ∈ P, we may assume that u factorsthrough a′. Since a′ ∈ RadA, it follows that a′ is right almost split in A. This proves thatF is dense, and thus an equivalence.

Theorem 1. Let A be a strict τ -category. Then M(A)PrL ≈ mod(A/[Projτ (A)]) andM(A)InL ≈ com(A/[Injτ (A)]). For any a ∈ Ob M(A), there is a sequence of morphisms

(20) TSaσa−→ La

λa−→ aπa−→ Sa


with πa ∈ M(A)PrL and σa ∈ M(A)InL such

(a) πa = cokPrLλa and σa = kerInLλa.(b) λa = kerPrLπa = cokInLσa.(c) M(A)PrL is semilocal with reflection π: 1→ S.(d) T : S(M(A)PrL)→ S(M(A)InL) is an equivalence.

Remarks. 1. The theorem shows that (20) shares some properties with the distinguishedtriangles of a triangulated category. The main difference lies in the fact that (20)belongs to three different categories.

2. The natural transformation λ is close to an isomorphism since the end terms of (20)are semisimple in M(A)PrL resp. M(A)InL.

3. The ladder functors are determined by the full subcategory PrL, or equivalently, byInL.

Proof. Consider a cokernel (1) in mod(A) with a ∈ P := Projτ (A) ⊂ A ⊂ mod(A), i. e.A1 = 0 and A0 ∈ P. We may assume that b ∈ RadA. Then (6) shows that f0 is splitepic. Hence B0 ∈ P. Conversely, every object a: A → P in mod(A) with P ∈ P belongsto GenP. By (3) and Proposition 5, this implies that M(A)PrL ≈mod(A/[P]). The secondequivalence follows by duality.

Let us write out (20) explicitly.

(21)

Since (1 0)(f 00 1

)= (b f)

(0 01 0

), we have πaλa = 0 in mod(A), hence also in mod(A/[P]).

Moreover, πa is epic in mod(A/[P]) by (6). Now let

be a morphism ϕ: a → x in mod(A/[P]) with ϕλa = 0 in mod(A/[P]). Then x(v w) −(y z)

(b fts p

)∈ [P], and there is a morphism (g h): B′⊕P → X1 with (y z)

(f 00 1

)−x(g h) ∈ [P].

Hence we get a morphism

196 WOLFGANG RUMP

in mod(A/[P]). As (y z) − y(1 0) − x(0h) ∈ [P], this shows that ϕ factors through πa,whence πa = cokPrLλa. Note that the preceding proof remains valid if P is replaced by anyfull subcategory of A. Therefore, the dual argument yields (a).

Next let

be a morphism ϕ: x→ a in M(A) with πaϕ = 0 in mod(A/[P]). Then there is a morphismh: X0 → B ⊕ B′ = ϑC with y − (b f)h ∈ [P]. Thus y ∈ Rad A. By [16], Proposition 3,this implies that ϕ factors uniquely through λa. Hence λa = kerPrLπa. Again, the dualargument gives (b). Since Sa is right almost split in A/[P], we get Sa ∈ S(mod(A/[P]))by Proposition 3. To prove (c), let ρ−1ϕ: a → s ← s′ be a morphism in M(A)PrL withϕ, ρ ∈ M(A) such that ρ is regular modulo [PrL] and s′ semisimple. Then s is semisimple aswell. Regarding s: S1 → S0 as an object of mod(A/[P]), we may assume that S0

∼= τ−τS0

and S1 = ϑS0 by Proposition 7. Hence ϕλa = 0 in mod(A/[P]), and thus ρ−1ϕ factors(uniquely) through πa. This proves (c).

By Proposition 7, the objects of S(mod(A/[P])) are of the form u: ϑX X. This givesa left and right almost split sequence τX

v ϑXu X in A. If u′: ϑX ′ X ′ is another

such object, then Proposition 7 implies that there is a one-to-one correspondence betweenmorphisms u→ u′ in mod(A/[P]) and morphisms of complexes

(22)

modulo homotopy. Therefore, by symmetry, u → v defines an equivalence T : S(M(A)PrL)→S(M(A)InL). The compatibility with (21) is achieved by a slight modification of (21), replac-ing

(b′

f ′)

by(f ′

−b′)

and(

1 00 s

)by

(0−1s 0

).

Corollary 1. For a triangle (20) there is a short exact sequence Laλa a

πa Sa in M(A)PrL

and a short exact sequence TSaσa La

λa a in M(A)InL.

Proof. It suffices to prove that λa is monic in M(A)PrL and epic in M(A)InL. Thus letϕρ−1: x ← x′ → La be a morphism in M(A)PrL with λa(ϕρ−1) = 0 in M(A)PrL. Thenλaϕ ∈ [PrL], say, λaϕ = αβ with α: p → a, β: x → p, and p ∈ PrL. Hence α factorsthrough λa. Since λa is monic, we infer that ϕ ∈ [PrL]. Thus λa is monic in M(A)PrL. Thedual argument shows that λa is epic in M(A)InL.


Corollary 2. Let A be a strict τ -category. For an object a of M(A), the following equiva-lences hold.

(a) a is semisimple in M(A)PrL ⇔ πa is invertible.(b) a ∈ PrL⇔ λa is invertible.(c) a ∈ InL⇔ σa is invertible.

Proof. (a) follows by Theorem 1(c), and (b) is trivial, whereas (c) follows by Corollary 1.

By Theorem 1, every morphism ϕ: a→ b in M(A) gives rise to a diagram

(23)

with commutative squares in M(A)InL, M(A), and M(A)PrL, respectively. We will show that(23) is in fact a triangle morphism, that is, ψ = TSϕ.

Theorem 2. Let A be a strict τ -category. Every morphism ϕ: a → b in M(A) induces atriangle morphism (23).

Proof. We modify (21) as indicated in the proof of Theorem 1, thus replacing(b′

f ′)

by(f ′

−b′).

Up to isomorphism, any semisimple object in M(A)PrL comes from an object u: ϑA Ain M(A). Then (21) looks as follows.

(24)

If u′: ϑA′ A′ is another such object, the theorem obviously holds for morphisms ϕ:u→ u′. Thus it remains to show that it also holds in the special case ϕ = πa. Here we getthe following commutative diagram (23).

198 WOLFGANG RUMP

The homotopy(

0 01 0

): B′ ⊕ P → B ⊕ B′ yields the commutativity of the middle square

(in M(A)):(

1 00 t

)(f ′ 00 1

)− ( f ′

−b′)(1 0) =

(0 01 0

)(b′ tsf ′ p

)and (1 0)

(f 00 1

)= (b f)

(0 01 0

). Thus ψ =

TSϕ = 1 in this case.

As an immediate consequence of Theorem 1 and Theorem 2 we have

Corollary 3. If the two triangles in (23) together with any pair of adjacent verticalmorphisms α, β are given, such that α and β make up a commutative diagram in M(A)InL orM(A) or M(A)PrL, respectively, then (23) can be completed with ψ = TSϕ in a unique way.

5. Semisimplicity in M(A)

The preceding results make no use of one-sided almost split sequences in A. Moreover, forconcrete strict τ -categories A (e. g., module categories over an artinian algebra, categoriesof socle-projective modules, or of lattices over an order), there is, in addition, a specificrelationship between Projτ (A) and Injτ (A) (cf. Theorem 4.2 of [8], Theorem 4.4 of [9],Theorem 1 of [14], Theorem 4 of [16]). We will show that this connection is intimatelyrelated to the one-sided semisimple objects of the homotopy category M(A).

Proposition 8. Let A be a strict τ -category. An object a: A1 → A0 of M(A) is left (right)semisimple if and only if the morphism a ∈ A is right (left) almost split with τA0 = 0 (resp.τ−A1 = 0).

Proof. Let a: A → P be a left semisimple object in M(A). Then the monomorphism λa:La → a splits. Hence a ∈ FixL, i. e. τP = 0. As a factors through the right almost splitmap u: ϑP → P , say, a = ue, we get a morphism

(25)

in Mor(A), and it is easy to verify that (25) is monic in Mor(A)/[A]. Since a is left semisimplein M(A), we infer that (25) is split monic. Hence e is split epic, and thus u factors througha. This implies that a is right almost split in A.

Conversely, let a: A → P be an object in M(A) with τP = 0 and a ∈ A right almostsplit. Consider a monomorphism

(26)

in M(A). Every split monomorphism d: D X0 in A with gd ∈ Rad A induces a morphismδ: D+ → x in M(A) such that the composition of (26) with δ is zero. Hence δ = 0,which implies that g is split monic. So there is a morphism p: P → X0 with pg = 1.


The composition pu of p with u: ϑP → P induces a morphism ϕ: (ϑP )+ → x which isannihilated by the monomorphism (26). Hence ϕ = 0, i. e. pu = xq for some q: ϑP → X1.As above, a = ue for some e: A → ϑP . Since a is right almost split, there is a morphisms: ϑP → A with u = as. Hence u(es − 1) = 0, and thus es = 1 by virtue of τP = 0. Nowx(1−qef) = x− puef = x− paf = x− pgx = 0. So we have a commutative diagram

(27)

in Mor(A). Moreover, the composition (27) is homotopic to zero. In fact, uef(1−qef) =af(1−qef) = 0 yields ef(1−qef) = 0. Therefore, the morphism 1−se: A → A satisfies(1− se) · f(1− qef) = f · (1−qef) and a · (1−se) = ue(1−se) = 0 = g · 0. As (26) is monic,the left-hand morphism in (27) belongs to [A], i. e. there is a morphism h: A → X1 withh · f(1−qef) = 1− qef and x · h = 0. This implies that

is a retraction of (26). Namely, x · (qe + h(1−fqe)) = xqe = pue = p · a and (qe +h(1−fqe)) · f = qef + hf(1−qef) = qef + (1−qef) = 1. Hence a is left semisimple.

Let A be a strict τ -category. By the preceding result, every left semisimple object inM(A) is of the form u ⊕ V − with a right almost split sequence 0 → ϑP

u→ P . Let usdenote the full subcategory of M(A), consisting of these objects u (resp. of the dual objectsI → ϑ−I with τ−I = 0) by Ap (resp. Ai). The objects of A− (resp. Ap) will be called leftsemisimple of type 0 (resp. of type 1). Dually, the objects of A+ (resp. Ai) are called rightsemisimple of type 0 (resp. 1). Proposition 8 also shows that the full subcategory SlM(A)(resp. SrM(A)) of left (right) semisimple objects in M(A) is a Krull-Schmidt category,containing A ≈ A− ≈ A+ as a full subcategory. Therefore, the endomorphism ring of anindecomposable one-sided semisimple object in M(A) need not be a skew-field.

We define a source object (sink object) [13] of an additive category as an object A = 0such that every non-zero morphism X → A (resp. A → X) is split epic (split monic). Fora strict τ -category A, this means that A is indecomposable with ϑA = 0 (resp. ϑ−A = 0).

The one-sided semisimple objects of type 0 or 1 admit the following intrinsic characteri-zation in M(A).

Proposition 9. Let A be a strict τ -category without simultaneous source and sink objects.A left semisimple object s in M(A) is of type 0 if and only if HomM(A)(s,Ap) = 0. Anindecomposable left semisimple object is of type 1 if and only if it is a source object inSlM(A).

200 WOLFGANG RUMP

Proof. The first assertion follows since HomM(A)(A−,Ap) = 0. Let

be a morphism ϕ: u′ → u in Ap with u indecomposable. If g factors through u, thenϕ = 0 since u is monic. Otherwise, there is a section i: P → P ′ of g. As iu ∈ Rad A, wefind a morphism j: ϑP → ϑP ′ with iu = u′j, hence a morphism ψ: u → u′. Moreover,u(fj−1) = gu′j − u = giu − u = 0 gives fj = 1, whence ϕψ = 1. Conversely, let s be asource object in SlM(A). If s = A− for some A ∈ ObA, then A is a source object in A.Hence A is not a sink object. So there is an irreducible morphism A → P in A with Pindecomposable. Since A is a source object, this implies that τP = 0. Furthermore, A is adirect summand of ϑP . Therefore, we get a non-zero morphism

which cannot be split epic since HomM(A)(A−, u) = 0. This contradiction shows that s ∈ Ap.

Definition. Let A be a strict τ -category. We say that a left semisimple object s is connectedto a right semisimple object t in M(A) if there exists a morphism γ: s→ t such that everys′ → t with s′ left semisimple factors uniquely through γ, and every s → t′ with t′ rightsemisimple factors uniquely through γ.

The definition implies that up to isomorphism, the objects s and t determine each other.If s and t are indecomposable, then either s or t has to be of type 1. This follows sinceHomM(A)(A−,A+) = 0. Hence there are three types of connections between indecompos-ables.

(28)

Recall that a simultaneous pullback and pushout is said to be an exact square. We will saythat M(A) is ladder-finite if for each object a of M(A), there exists an n ∈ N with Lna ∈FixL and (L−)n ∈ FixL− (cf. [16], §4). The following theorem shows how connections inM(A) are related to invertible ladders.

Theorem 3. Let A be a strict τ -category. For a connection between indecomposableone-sided semisimple objects in M(A), the corresponding commutative square (28) is


exact. If M(A) is ladder-finite, then a morphism s → t in M(A) with indecomposables ∈ Sl(mod(A)) and t ∈ Sr(mod(A)) is a connection if and only if it is of the form

(29) s = s0λ1−→ s1

λ2−→ · · · λn−→ sn = t

with λi = λsi= λ−si−1

for i ∈ 1, . . . , n.

Proof. Let γ: s→ t be a connection (28). By definition, every morphism ϕ: A− → t factorsuniquely through γ. For the third square in (28), this is tantamount to the pullback property.For the first two squares in (28), the fact that ϑP → P is monic implies that ϕ = 0. Hencewe have a pullback by [16], Proposition 7. By duality, the commutative square (28) is exact.Now let M(A) be ladder-finite. If γ corresponds to the middle square in (28), the assertionfollows by [16], Theorem 1. For the third square in (28), assume that Lnt = S− with nminimal. If Lit: Bi ⊕ Ui → Ci ⊕ Pi is the left standard form of Lit, then Pn = 0 impliesthat Pi = 0 for all i ∈ 1, . . . , n. By [7], 3.6.1, it follows that U−i is a direct summand ofLit for these i. Since t is indecomposable, this gives Un = S and Ui = 0 for i < n. Hence γis of the form (29). Conversely, let γ: s → t be a morphism in M(A) with indecomposables ∈ Sl(mod(A)), t ∈ Sr(mod(A)), such that γ is of the form (29). By [16], Proposition 2,γ is regular, and every A− → t factors through γ. Since Ap ⊂ PrL, Proposition 8 impliesthat every morphism s′ → t with s′ ∈ Sl(mod(A)) factors uniquely through γ. Hence, byduality, γ is a connection.

There are two important special types of connections in M(A).

Definition. Let A be a strict τ -category. We say that M(A) has a connection of type 1if there is a one-to-one connection between the indecomposable left and right semisimpleobjects of type 1 in M(A). If every indecomposable left (right) semisimple object of type1 is connected to an indecomposable right (left) semisimple object of type 0, we say thatM(A) has a connection of type 0.

Let A be a strict τ -category. By ind A we denote any fixed representative system of theisomorphism classes of indecomposable objects in A. We conclude this article with a resultwhich shows that existence of a connection of type 0 or 1 is the essential condition for A tobe equivalent to a category of representations of an artinian ring or an order in case indA

is finite.

Definition. (cf. [15, 16]) Let R be a ring. An R-module E in R-mod is said to be anR-lattice if E has no simple submodules. The full subcategory of R-lattices in R-mod willbe denoted by R-lat. If R is a subring of a ring R′, then R′ is said to be a left overorder of Rif SocRR′ = 0, and the left R-module R′/a ∈ R |R′a ⊂ R is of finite length. Similarly, wedefine a right overorder of R. A left and right overorder will be called an overorder. We callR a one-dimensional order if R has an overorder

∏si=1 Mni

(Ωi) with (non-commutative)discrete valuation domains Ωi.

Theorem 4. Let A be a strict τ -category with |indA| <∞.

(a) There exists an artinian ring R with R-mod ≈ A if and only if the radical of A isnilpotent, and M(A) has a connection of type 0.

(b) There exists a one-dimensional order R with R-lat ≈ A if and only if A is ladder-finitewith

⋂∞i=1 RadiA = 0, and M(A) has a connection of type 1.

202 WOLFGANG RUMP

Proof. (a) follows by [14], Corollary of Proposition 6, using [7], Theorem 6.4. Part (b) is aconsequence of Theorem 3, together with [16], Theorem 4 and its corollary.

Remark. It can be shown (see [16]) that a lattice-finite one-dimensional order is a noether-ian order in a semisimple ring.

References

[1] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules, Springer New York -Heidelberg - Berlin 1974.

[2] M. Auslander: Isolated singularities and existence of almost split sequences, Springer LectureNotes in Mathematics, Vol. 1178 (1986), 194–242.

[3] M. Auslander, I. Reiten, S. O. Smalø: Representation Theory of Artin Algebras, CambridgeUniversity Press 1995.

[4] P. Gabriel, M. Zisman: Calculus of Fractions and Homotopy Theory, Berlin - Heidelberg -New York 1967.

[5] A. Grothendieck: Sur quelques points d’algebre homologique, Tohoku math. J. 9 (1957),119–221.

[6] K. Igusa, G. Todorov: Radical Layers of Representable Functors, J. Algebra 89 (1984),105–147.

[7] O. Iyama: τ -categories I: Radical Layers Theorem, Algebras and Representation Theory, toappear.

[8] O. Iyama: τ -categories III: Auslander Orders and Auslander-Reiten quivers, Algebras andRepresentation Theory, to appear.

[9] O. Iyama: The relationship between homological properties and representation theoretic real-ization of artin algebras, Preprint.

[10] G. M. Kelly: A unified treatment of transfinite constructions for free algebras, free monoids,colimits, associated sheaves, and so on, Bull. Austral. Math. Soc. 22 (1980), 1–83.

[11] B. Keller: Derived Categories and Their Uses, in: Handbook of Algebra (ed. M. Hazewinkel),vol. 1, Elsevier Science B. V., 1996, 671–701.

[12] S. Mac Lane: Categories for the Working Mathematician, Springer, New York - Heidelberg -Berlin, 1971.

[13] W. Rump: Derived orders and Auslander-Reiten quivers, An. St. Univ. Ovidius Constanta 8(2000), 125–142.

[14] W. Rump: Ladder functors with an application to representation-finite artinian rings, An.St. Univ. Ovidius Constanta 9 (2001), 107–124.

[15] W. Rump: Lattice-finite rings and their Auslander orders, Proc. 34th Symposium on RingTheory and Representation Theory (2001), 89–99.

[16] W. Rump: The category of lattices over a lattice-finite ring, Algebras and RepresentationTheory, to appear.

[17] B. Stenstrom: Rings of Quotients, Springer-Verlag, New York - Heidelberg - Berlin, 1975.

NON-COMMUTATIVE ALGEBRAIC GEOMETRY ANDCOMMUTATIVE DESINGULARIZATIONS

LIEVEN LE BRUYN

University of Antwerp (Belgium)

1. Introduction

Ever since the dawn of non-commutative algebraic geometry in the mid seventies, see forexample the work of P. Cohn [11], J. Golan [17], C. Procesi [38], F. Van Oystaeyen and A.Verschoren [47],[48], it has been ringtheorists’ hope that this theory may one day be relevantto commutative geometry, in particular to the study of singularities and their resolutions.

Over the last decade, non-commutative algebras have been used to construct canonical(partial) resolutions of quotient singularities. That is, take a finite group G acting on Cd

freely away from the origin then the orbit-space Cd/G is an isolated singularity. ResolutionsY Cd/G have been constructed using the skew group algebra

C[x1, . . . , xd]#G

which is an order with center C[Cd/G] = C[xi, . . . , xd]G or deformations of it. In dimensiond = 2 (the case of Kleinian singulariies) this gives us minimal resolutions via the connectionwith the preprojective algebra, see for example [14]. In dimension d = 3, the skew groupalgebra appears via the superpotential and commuting matrices setting (in the physicsliterature) or via the McKay quiver, see for example [13]. If G is Abelian one obtains fromthis study crepant resolutions but for general G one obtains at best partial resolutions withconifold singularities remaining. In dimension d > 3 the situation is unclear at this moment.Usually, skew group algebras and their deformations are studied via homological methodsas they are Regular orders, see for example [46]. Here, we will follow a different approach.

My motivation was to find a non-commutative explanation for the omnipresence of coni-fold singularities in partial resolutions of three-dimensional quotient singularities. Of courseyou may argue that they have to appear because they are somehow the nicest singularities.But then, what is the corresponding list of ‘nice’ singularities in dimension four? or five,six. . . ?? If my conjectural explanation has any merit the nicest partial resolutions of C4/Gshould contain only singularities which are either polynomials over the conifold or one ofthe following three types

C[[a, b, c, d, e, f ]](ae− bd, af − cd, bf − ce)

C[[a, b, c, d, e]](abc− de)

C[[a, b, c, d, e, f, g, h]]I

where I is the ideal of all 2× 2 minors of the matrix

[a b c de f g h

]

In dimension d = 5 the conjecture is that another list of ten new specific singularities willappear, in dimension d = 6 another 63 new ones appear and so on.

204 LIEVEN LE BRUYN

How do we come to these outlandish conjectures and specific lists? The hope is that anyquotient singularity X = Cd/G has associated to it a ‘nice’ order A with center R = C[X]such that there is a stability structure θ with the scheme of all θ-semistable representationsof A being a smooth variety (all these terms will be explained in the main text). If this isthe case, the associated moduli space will be a partial resolution

moduliθαAX = Cd/G

and has a sheaf of Smooth orders A over it, allowing us to control its singularities in acombinatorial way as depicted in the frontispiece.

If A is a Smooth order over R = C[X] then its non-commutative variety max A ofmaximal twosided ideals is birational to X away from the ramification locus. If P is a pointof the ramification locus ram A then there is a finite cluster of infinitesimally close non-commutative points lying over it. The local structure of the non-commutative variety maxA near this cluster can be summarized by a (marked) quiver setting (Q,α) which in turnallows us to compute the etale local structure of A and R in P . The central singularitieswhich appear in this way have been classified in [6] up to smooth equivalence giving us thesmall lists of conjectured singularities.

In these talks I have tried to include background information which may or may notbe useful to you. I suggest to browse through the notes by reading the ‘notes’ first. If theremark seems obvious to you, carry on. If it puzzles you this may be a good point to enterthe main text. More information can be found in the never-ending bookproject [28].

AcknowledgementThese notes are (hopefully) a streamlined version of three talks given at the workshop

“Schemas de Hilbert, algebre noncommutative et correspondance de McKay” held at CIRMin Luminy (France), October 27–31, 2003.

I like to thank the organizers, Jacques Alev, Bernhard Keller and Thierry Levasseur forthe invitation (and the possibility to have a nice vacation with part of my family) and theparticipants for their patience.

2. Non-commutative algebra

The organizers of this conference on “Hilbert schemes, non-commutative algebra and theMcKay correspondence” are perfectly aware of my ignorance on Hilbert schemes and McKaycorrespondence. I therefore have to assume that I was hired in to tell you something aboutnon-commutative algebra and that is precisely what I intend to do in these three talks.

2.1. Why non-commutative algebra? Let me begin by trying to motivate why youmight get interested in non-commutative algebra if you want to understand quotient singu-larities and their resolutions.

So let us take a setting which will be popular this week: we have a finite group Gacting on d-dimensional affine space Cd and this action is free away from the origin.Then the orbit-space, the so called quotient singularity Cd/G, is an isolated singularity

NON-COMMUTATIVE ALGEBRAIC GEOMETRY 205

and we want to construct ‘minimal’ or ‘canonical’ resolutions of this singularity. Thebuzzword seems to be ‘crepant’ in these circles. In his Bourbaki talk [40] Miles Reid assertsthat McKay correspondence follows from a much more general principleMiles Reid’s Principle: Let M be an algebraic manifold, G a group of automorphismsof M , and Y X a resolution of singularities of X = M/G. Then the answer to any wellposed question about the geometry of Y is the G-equivariant geometry of M .

Applied to the case of quotient singularities, the content of his slogan is that theG-equivariant geometry of Cd already knows about the crepant resolution Y Cd/G.

Men having principles are an easy target for abuse. So let us change this principle slightly:assume we have an affine variety M on which a reductive group (and for definiteness takePGLn) acts with algebraic quotient variety M//PGLn Cd/G

then, in favorable situations, we can argue that the PGLn-equivariant geometry of M knowsabout good resolutions Y . This brings us to our first entry in our

note: One of the key lessons to be learned from this talk is that PGLn-equivariant geometryof M is roughly equivalent to the study of a certain non-commutative algebra over Cd/G.In fact, an order in a central simple algebra of dimension n2 over the function field of thequotient singularity.

Hence, if we know of good orders over Cd/G, we might get our hands at ‘good’ resolutionsY by non-commutative methods.

2.2. What non-commutative algebras? For the duration of these talks, we will workin the following, quite general, setting :

• X will be a normal affine variety, possibly having singularities.• R will be the coordinate ring C[X] of X.• K will be the function field C(X) of X.

If you are only interested in quotient singularities, you should replace X by Cd/G,R bythe invariant ring C[x1, . . . , Xd]G and K by the invariant field C(x1, . . . , Xd)G in all state-ments below.

If you are an algebraist, you have my sympathy and our goal will be to construct lots ofR-orders A in a central simple K-algebra Σ.

206 LIEVEN LE BRUYN

If you do not know what a central simple algebra is, take any non-commutative K-algebraΣ with center Z(Σ) = K such that over the algebraic closure K of K we obtain fulln× n matrices

Σ⊗K K Mn(K)

There are plenty such central simple K-algebras:

Example 2.1. For any non-zero functions f, g ∈ K∗, the cyclic algebra

Σ = (f, g)n defined by (f, g)n =K〈x, y〉

(xn − f, yn − g, yx− qxy)

with q is a primitive n-th root of unity, is a central simple K-algebra of dimension n2. Often,(f, g)n will even be a division algebra, that is a non-commutative algebra such that everynon-zero element has an inverse.

For example, this is always the case when E = K[x] is a (commutative) field extensionof dimension n and if g has order n in the quotient K∗/NE/K(E∗) where NE/K is the normmap of E/K. See for example [37, Chp. 15] for more details, but if your German is a pointI strongly suggest you to read Ina Kersten’s book [21] instead.

Now, fix such a central simple K-algebra Σ. An R-order A in Σ is a subalgebras A ⊂ Σwith center Z(A) = R such that A is finitely generated as an R-module and contains aK-basis of Σ, that is

A⊗R K Σ

The classic reference for orders is Irving Reiner’s book [41] but it is hopelessly outdated andfocusses too much on the one-dimensional case. Here is a gap in the market for someoneto fill. . .

Example 2.2. In the case of quotient singularities X = Cd/G a natural choice of R-order might be the skew group ring: C[x1, . . . , Xd]#G which consists of all formal sums∑g∈G rg#g witn multiplication defined by

(r#g)(r′#g′) = rφg(r′)#gg′

where φg is the action of g on C[x1, . . . , Xd]. The center of the skew group algebra is easilyverified to be the ring of G-invariants

R = C[Cd/G] = C[x1, . . . , xd]G

Further, one can show that C[x1, . . . , Xd]#G is an R-order in Mn(K) with n the order ofG. If we ever get to the third lecture, we will give another description of the skew groupalgebra in terms of the McKay-quiver setting and the variety of commuting matrices.

However, there are plenty of other R-orders in Mn(K) which may or may not be relevantin the study of the quotient singularity Cd/G.


Example 2.3. If f, g ∈ R−0, then the free R-submodule of rank n2 of the cyclic K-algebraΣ = (f, g)n of example 2.1

A =n−1∑i,j=0

Rxiyj

is an R-order. But there is really no need to go for this ‘canonical’ example. Someone moretwisted may take I and J any two non-zero ideals of R, and consider

AIJ =n−1∑i,j=0

IiJjxiyj

which is an R-order too in Σ and which is far from being a projective R-module unless Iand J are invertible R-ideals.

For example, in Mn(K) we can take the ‘obvious’ R-order Mn(R) but one might alsotake the subring [

R IJ R

]

which is an R-order if I and J are non-zero ideals of R.If you are a geometer, our goal is to construct lots of affine PGLn-varieties M such that

the algebraic quotient M//PGLn is isomorphic to X and, moreover, such that there is aZariski open subset U ⊂ X

for which the quotient map is a principal PGLn-fibration, that is, all fibers π−1(u) PGLnfor u ∈ U .

The connection between such varieties M and orders A in central simple algebras maynot be clear at first sight. To give you at least an idea that there is a link, think of M as theaffine variety of n-dimensional representations repn A and of U as the Zariski open subsetof all simple n-dimensional representations.

Naturally, one can only expect the R-order A (or the corresponding PGLn-variety M)to be useful in the study of resolutions of X if A is smooth in some appropriate non-commutative sense.

Now, there are many characterizations of commutative regular domains R:

• R is regular, that is, has finite global dimension• R is smooth, that is, X is a smooth variety

and generalizing either of them to the non-commutative world leads to quite differ-ent concepts.

208 LIEVEN LE BRUYN

We will call an R-order A is a central simple K-algebra Σ:

• Regular if A has finite global dimension together with some extra features such asAuslander regularity or Cohen-Macaulay property, see for example [33].• Smooth if the corresponding PGLn-affine variety M is a smooth variety as we will

clarify later in this talk.

For applications of Regular orders to desingularizations we refer to the talks by MichelVan den Bergh at this conference or to his paper [46] on this topic. I will concentrate onthe properties of Smooth orders instead. Still, it is worth pointing out the strengths andweaknesses of both definitions right now

note: Regular orders are excellent if you want to control homological properties, for exampleif you want to study the derived categories of their modules. At this moment there is nolocal characterization of Regular orders if dimX ≥ 2.

Smooth orders are excellent if you want to have smooth moduli spaces of semi-stablerepresentations. As we will see later, in each dimension there are only a finite number oflocal types of Smooth orders and these are classified. The downside of this is that Smoothorders are less versatile as Regular orders.

In applications to canonical desingularizations, one often needs the good properties ofboth so there is a case for investigating SmoothRegular orders better than has been donein the past.

In general though, both theories are quite different.

Example 2.4. The skew group algebra C[x1, . . . , xd]#G is always a Regular order but wewill see in the next lecture, it is virtually never a Smooth order.

Example 2.5. Let X be the variety of matrix-invariants, that is

X = Mn(C)⊕Mn(C)//PGLn

where PGLn acts on pairs of n×n matrices by simultaneous conjugation. The trace ring oftwo generic n× n matrices A is the subalgebra of Mn(C[Mn(C)⊕Mn(C)]) generated overC[X] by the two generic matrices

X =

x11 · · · x1n

......

xn1 · · · xnn

and Y =

y11 · · · y1n...

...yn1 · · · ynn

Then, A is an R-order in a division algebra of dimension n2 over K, called the genericdivision algebra. Moreover, A is a Smooth order but is Regular only when n = 2, see [30].

2.3. Constructing orders by descent.

note: French mathematicians have developed in the sixties an elegant theory, called descenttheory, which allows one to construct elaborate examples out of trivial ones by bringing intopology. This theory allows to classify objects which are only locally (but not necessarilyglobally) trivial.

For applications to orders there are two topologies to consider: the well-known Zariskitopology and the perhaps lesser-known etale topology. Let us try to give a formal definitionof Zariski and etale covers aimed at ringtheorists.


A Zariski cover of X is a finite product of localizations at elements of R

Sz =k∏i=1

Rfisuch that (f1, . . . , fk) = R

and is therefore a faithfully flat extension of R. Geometrically, the ringmorphism R −→ Szdefines a cover ofX = specR by k disjoint sheets spec Sz = ispecRfi

, each correspondingto a Zariski open subset of X, the complement of V(fi) and the condition is that these closedsubsets V(fi) do not have a point in common. That is, we have the picture of figure 1.1:

Zariski covers form a Grothendieck topology, that is, two Zariski covers S1z =

∏ki=1Rfi

and S2z =

∏lj=1Rgj

nave a common refinement

Sz = S1Z ⊗R S2

z =k∏i=1

l∏j=1

Rfigj

For a given Zariski cover Sz =∏ki=1Rfi

a corresponding etale cover is a product

Se =k∏i=1

Rfi[x(i)1, . . . , x(i)ki]

(g(i)1, . . . , g(i)ki)with

∂g(i)1∂x(i)1

· · · ∂g(i)1∂x(i)ki

......

∂g(i)ki

∂x(i)1· · · ∂g(i)ki

∂x(i)ki

a unit in the i-th component of Se. In fact, for applications to orders it is usually enoughto consider special etale extensions

Se =k∏i=1

RFi[x]

(xki − ai) where aiis a unit in Rfi

Figure 1: A Zariski cover of X = spec R

210 LIEVEN LE BRUYN

Geometrically, an etale cover determines for every Zariski sheet spec Rfia locally iso-

morphic (for the analytic topology) multi-covering and the number of sheets may vary withi (depending on the degrees of the polynomials g(i)j ∈ Rfi

[x(i)1, . . . , x(i)ki]. That is, the

mental picture corresponding to an etale cover is given in figure 1.2 below.Again, etale covers form a Zariski topology as the common refinement S1

e ⊗R S2e of two

etale covers is again etale because its components are of the form

Rfigj[x(i)1, . . . , x(i)ki

, y(j)1, . . . , y(j)lj ](g(i)1, . . . , g(i)ki

, h(j)1, . . . , h(j)lj )

and the Jacobian-matrix condition for each of these components is again satisfied. Because ofthe local isomorphism property many ringtheoretical local properties (such as smoothness,normality etc.) are preserved under etale covers.

Now, fix an R-order B in some central simple K-algebra Σ, then a Zariski twisted formA of B is an R-algebra such that

A⊗R Sz B ⊗R Sz

for some Zariski cover Sz of R.If P ∈ X is a point with corresponding maximal ideal m, then P ∈ specRfi

for some ofthe components of Sz and as Afi

Bfiwe have for the local rings at P

Am Bm

Figure 2: An etale cover of X = spec R


that is, the Zariski local information of any Zariski-twisted form of B is that of Bitself.

Likewise, an etale twisted form A of B is an R-algebra such that

A⊗R Se B ⊗R Se

for some etale cover Se of R.This time the Zariski local information of A and B may be different at a point P ∈ X

but we do have that the m-adic completions of A and B

Am Bm

are isomorphic as Rm-algebras.

note: The Zariski local structure of A determines the localization Am, the etale local struc-ture determines the completion Am.

Descent theory allows to classify Zariski- or etale twisted forms of an R-order B by meansof the corresponding cohomology groups of the automorphism schemes. For more details onthis please read the book [23] by M. Knus and M. Ojanguren if you are a ringth-eorist andthat of S. Milne [35] if you are more of a geometer.

If one applies descent to the most trivial of all R-orders, the full matrix algebra Mn(R),one arrives at

2.4. Azumaya algebras. A Zariski twisted form of Mn(R) is an R-algebra A such that

A⊗R Sz Mn(Sz) =k∏i=1

Mn(Rfi)

Conversely, you can construct such twisted forms by gluing together the matrix ringsMn(Rfi

). The easiest way to do this is to glue Mn(Rfi) with Mn(Rfj

) over Rfifivia the

natural embedding

Rfi→ Rfifj

← Rfj

Not surprisingly, we obtain in this way Mn(R) back.But there are more clever ways to perform the gluing by bringing in the non-

commutativity of matrix-rings. We can glue

Mn(Rfi) →Mn(Rfifj

)gij ·g−1

ij−−−−−→ Mn(Rfifj)← Mn(Rfj

)

over their intersection via conjugation with an invertible matrix gij ∈ GLn(Rfifi). If the

elements gij for 1 ≤ i, j ≤ k satisfy the cocycle condition (meaning that the differentpossible gluings are compatible over their common localization Rfifjfl

), we obtain a sheafof non-commutative algebras A over X = spec R such that its global sections are notnecessarily Mn(R).

212 LIEVEN LE BRUYN

Proposition 2.6. Any Zariski twisted form of Mn(R) is isomorphic to

EndR(P )

where P is a projective R-module of rank n. Two such twisted forms are isomorphic asR-algebras

EndR(P ) EndR(Q) iff P Q⊗ I

for some invertible R-ideal I.

Proof. [sketch] We have an exact sequence of groupschemes

1 −→ Gm −→ GLn −→ PGLn −→ 1

(here, Gm is the sheaf of units) and taking Zariski cohomology groups over X we havea sequence

1 −→ H1Zar(X,Gm) −→ H1

Zar(X, GLn) −→ H1Zar(X, PGLn)

where the first term is isomorphic to the Picard group Pic(R) and the second term classifiesprojective R-modules of rank n upto isomorphism. The final term classifies the Zariskitwisted forms of Mn(R) as the automorphism group of Mn(R) is PGLn.

Example 2.7. Let I and J be two invertible ideals of R, then

EndR(I ⊕ J) [

R I−1JIJ−1 R

]⊂M2(K)

and if IJ−l = (r) then I ⊕ J (Rr ⊕R)⊗ J and indeed we have an isomorphism

[1 00 r−1

] [R I−1J

IJ−1 R

] [1 00 r

]=[R RR R

]

Things get a lot more interesting in the etale topology.

Definition 2.8. An n-Azumaya algebra over R is an etale twisted form A of Mn(R). If Ais also a Zariski twisted form we call A a trivial Azumaya algebra.

From the definition and faithfully flat descent, the following facts follow:

Lemma 2.9. If A is an n-Azumaya algebra over R, then:

1. The center Z(A) = R and A is a projective R-module of rank n2.2. All simple A-representations have dimension n and for every maximal ideal m of R

we have

A/mA Mn(C)


Proof. For (2) takeM∩R = m whereM is the kernel of a simple representation AMk(C),then as Am Mn(Rm) it follows that

A/mA Mn(C)

and hence that k = n and M = Am.

It is clear from the definition that when A is an n-Azumaya algebra and A′ is an m-Azumaya algebra over R,A⊗R A′ is an mn-Azumaya and also that

A⊗R Aop EndR(A)

where Aop is the opposite algebra (that is, equipped with the reverse multiplication rule).These facts allow us to define the Brauer group BrR to be the set of equivalence classes

[A] of Azumaya algebras over R where

[A] = [A′] iff A⊗R A′ EndR(P )

for some projective R-module P and where multiplication is induced from the rule

[A] · [A′] = [A⊗R A′]

One can extend the definition of the Brauer group from affine varieties to arbitrary schemesand A. Grothendieck has shown that the Brauer group of a projective smooth variety is abirational invariant, see [19]. Moreover, he conjectured a cohomological description of theBrauer group BrR which was subsequently proved by O. Gabber in [16].

Theorem 2.10. The Brauer group is an etale cohomology group

BrR H2et(XGm)torsion

where Gm is the unit sheaf and where the subscript denotes that we take only torsionelements. If R is regular, then H2

et(X,Gm) is torsion so we can forget the subscript.This result should be viewed as the ringtheory analogon of the crossed product theorem

for central simple algebras over fields, see for example [37].Observe that in Gabber’s result there is no sign of singularities in the description of the

Brauer group. In fact, with respect to the desingularization problem, Azumaya algebras areonly as good as their centers.

Proposition 2.11. If A is an n-Azumaya algebra over R, then

1. A is Regular iff R is commutative regular.2. A is Smooth iff R is commutative regular.

Proof. (1) follows from faithfully flat descent and (2) from lemma 2.9 which asserts that thePGLn-affine variety corresponding to A is a principal PGLn-fibration in the etale topology,which shows that both n-Azumaya algebras and principal PGLn-fibrations are classified bythe etale cohomology group H1

et(X, PGLn).

note: In the correspondence between R-orders and PGLn-varieties, Azumaya algebrascorrespond to principal PGLn-fibrations over X. With respect to desingularizations,Azumaya algebras are therefore only as good as their centers.

214 LIEVEN LE BRUYN

2.5. Reflexive Azumaya algebras. So let us bring in ramification in order to constructorders which may be more useful in our desingularization project.

Example 2.12. Consider the R-order in M2(K)

A =[R RI R

]

where I is some ideal of R. If P ∈ X is a point with corresponding maximal ideal m wehave that:

For I not contained in m we have Am M2(Rm) whence A is an Azumaya algebra in P .For I ⊂ m we have

Am [Rm Rm

Im Rm

]= M2(Rm)

whence A is not Azumaya in P .

Definition 2.13. The ramification locus of an R-order A is the Zariski closed subset of Xconsisting of those points P such that for the corresponding maximal ideal m

A/mA Mn(C)

That is, ram A is the locus of X where A is not an Azumaya algebra. Its complement azuA is called the Azumaya locus of A which is always a Zariski open subset of X.

Definition 2.14. An R-order A is said to be a reflexive n-Azumaya algebra iff

1. ram A has codimension at least two in X, and2. A is a reflexive R-module

that is, A HomR(HomR(A,R), R) = A∗∗.The origin of the terminology is that when A is a reflexive n-Azumaya algebra we have

that Ap is n-Azumaya for every height one prime ideal p of R and that A = ∩pAp wherethe intersection is taken over all height one primes.

For example, in example 2.12 if I is a divisorial ideal of R, then A is not reflexive Azumayaas Ap is not Azumaya for p a height one prime containing I and if I has at least height two,then A is often not a reflexive Azumaya algebra because A is not reflexive as an R-module.For example take

A =[

C[x, y] C[x, y](x, y) C[x, y]

]

then the reflexive closure of A is A∗∗ = M2(C[x, y]).Sometimes though, we get reflexivity of A for free, for example when A is a Cohen-

MacaulayR-module. An other important fact to remember is that forA a reflexive Azumaya,A is Azumaya if and only if A is projective as an R-module. If you want to know moreabout reflexive Azumaya algebras you may want to read [36] or my Ph.D. thesis [24].

Example 2.15. Let A = C[x1, . . . , xd]#G then A is a reflexive Azumaya algebra wheneverG acts freely away from the origin and d ≥ 2. Moreover, A is never an Azumaya algebra asits ramification locus is the isolated singularity.


In analogy with the Brauer group one can define the reflexive Brauer group β(R) whoseelements are the equivalence classes [A] for A a reflexive Azumaya algebra over R withequivalence relation

[A] = [A′] iff A⊗R A′ EndR(M)

where M is a reflexive R-module and with multiplication induced by the rule

[A] · [A′] = [(A⊗R A′)∗∗]

In [26] it was shown that the reflexive Brauer group does have a cohomological description

Proposition 2.16. The reflexive Brauer group is an etale cohomology group

β(R) H2et(Xsm,Gm)

where Xsm is the smooth locus of X.This time we see that the singularities of X do appear in the description so perhaps

reflexive Azumaya algebras are a class of orders more suitable for our project. This is evenmore evident if we impose non-commutative smoothness conditions on A.

Proposition 2.17. Let A be a reflexive Azumaya algebra over R, then:

1. if A is Regular, then ram A = Xsing, and2. if A is Smooth, then Xsing is contained in ram A.

Proof. (1) was proved in [27] the essential point being that if A is Regular then A is aCohen-Macaulay R-module whence it must be projective over a smooth point of X butthen it is not just an reflexive Azumaya but actually an Azumaya algebra in that point.The second statement can be further refined as we will see in the next lecture.

Many classes of well-studied algebras are reflexive Azumaya algebras,

• Trace rings Tm,n of m generic n× n matrices (unless (m,n) = (2, 2)), see [25].• Quantum enveloping algebras Uq(g) of semi-simple Lie algebras at roots of unity, see

for example [8].• Quantum function algebras Oq(G) for semi-simple Lie groups at roots of unity, see for

example [9].• Symplectic reflection algebras At,c, see [10].

note: Many interesting classes of Regular orders are reflexive Azumaya algebras. As aconsequence their ramification locus coincides with the singularity locus of the center.

2.6. Cayley-Hamilton algebras. It is about time to clarify the connection with PGLn-equivariant geometry. We will introduce a class of non-commutative algebras, the so calledCayley-Hamilton algebras which are the level n generalization of the category of commuta-tive algebras and which contain all R-orders.

A trace map tr is a C-linear function A −→ A satisfying for all a, b ∈ A

tr(tr(a)b) = tr(a)tr(b) tr(ab) = tr(ba) and tr(a)b = btr(a)

216 LIEVEN LE BRUYN

so in particular, the image tr(A) is contained in the center of A.If M ∈Mn(R) where R is a commutative C-algebra, then its characteristic polynomial

χM = det(tn −M) = tn + a1tn−1 + a2t

n−2 + · · ·+ an

has coefficients ai which are polynomials with rational coefficients in traces of powers of M

ai = fi(tr(M), tr(M2), . . . , tr(Mn−1)

Hence, if we have an algebra A with a trace map tr we can define a formal characteristicpolynomial of degree n for every a ∈ A by taking

χa = tn + f1(tr(a), . . . , tr(an−1)tn−1 + · · ·+ fn(tr(a), . . . , tr(an−1)

which allows us to define the category alg@n of Cayley-Hamilton algebras of degree n.

Definition 2.18. An object A in alg@n is a Cayley-Hamilton algebra of degree n, that is,a C-algebra with trace map tr : A −→ A satisfying

tr(1) = n and ∀a ∈ A : χa(a) = 0

Morphisms A −→ B in alg@n are trace preserving C-algebra morphisms, that is,

is a commutative diagram.

Example 2.19. Azumaya algebras, reflexive Azumaya algebras and more generally everyR-order A in a central simple K-algebra of dimension n2 is a Cayley-Hamilton algebra ofdegree n. For, consider the inclusions

Here, tr : Mn(K) −→ K is the usual trace map. By Galois descent this induces a tracemap, the so called reduced trace, tr : Σ −→ K. Finally, because R is integrally closed in Kand A is a finitely generated R-module it follows that tr(a) ∈ R for every element a ∈ A.


If A is a finitely generated object in alg@n, we can define an affine PGLn-scheme,trepn A, classifying all trace preserving n-dimensional representations A

φ−→Mn(C) of A.The action of PGLn on trepn A is induced by conjugation in the target space, that is g.φis the trace preserving algebra map

Aφ−→Mn(C)

g.−.g−1

−→ Mn(C)

Orbits under this action correspond precisely to isomorphism classes of representations. Thescheme trepn A is a closed subscheme of repn A the more familiar PGLn-affine scheme ofall n-dimensional representations of A. In general, both schemes may be different.

Example 2.20. Let A be the quantum plane at −1, that is

A =C〈x, y〉

(xy + yx)

then A is an order with center R = C[x2, y2] in the quaternion algebra (x, y)2 = K1 ⊕Ku ⊕ Kυ ⊕ Kuυ over K = C(x, y) where u2 = x.v2 = y and uv = −vu. Observe thattr(x) = tr(y) = 0 as the embedding A → (x, y)2 →M2(C[u, y]) is given by

x →[u 00 −u

]and y →

[0 1y 0

]

Therefore, a trace preserving algebra map A −→ M2(C) is fully determined by the imagesof x and y which are trace zero 2× 2 matrices

φ(x) =[a bc −a

]and φ(y) =

[d ef −d

]satisfying bf + ce = 0

That is, trep2 A is the hypersurface V(bf+ce) ⊂ A6 which has a unique isolated singularityat the origin. However, rep2 A contains more points, for example

φ(x) =[a 00 b

]and φ(y) =

[0 00 0

]

is a point in rep2 A–trep2 A whenever b = −a.

A functorial description of trepnA is given by the following universal property provedby C. Procesi [39]

Theorem 2.21. Let A be a C-algebra with trace map trA, then there is a trace preservingalgebra morphism

jA : A −→Mn(C[trepn A])

218 LIEVEN LE BRUYN

satisfying the following universal property. If C is a commutative C-algebra and there is atrace preserving algebra map A

ψ−→Mn(C) (with the usual trace on Mn(C)), then there is a

unique algebra morphism C[trepn A]φ−→C such that the diagram

is commutative. Moreover, A is an object in alg@n if and only if jA is a monomorphism.

The PGLn-action on trepn A induces an action of PGLn by automorphisms onC[trepn A]. On the other hand, PGLn acts by conjugation on Mn(C) so we have acombined action on Mn(C[trepn A]) = Mn(C) ⊗ C[trepn A] and it follows from theuniversal property that the image of jA is contained in the ring of PGLn-invariants

AjA−→Mn(C[trepnA])PGLn

which is an inclusion if A is a Cayley-Hamilton algebra. In fact, C. Procesi proved in [39]the following important result which allows to reconstruct orders and their centers fromPGLn-equivariant geometry.

Theorem 2.22. The functor

trepn : alg@n −→ PGL(n)-affine

has a left inverse

A : PGL(n)-affine −→ alg@n

defined by Ay = Mn(C[Y ])PGLn . In particular, we have for any A in alg@n

A = Mn(C[trepn A])PGLn and tr(A) = C[trepn A]PGLn

That is the central subalgebra tr(A) is the coordinate ring of the algebraic quotient variety

trepn A//PGLn = tissn A

classifying isomorphism classes of trace preserving semi-simple n-dimensional representa-tions of A.

However, these functors do not give an equivalence between alg@n and PGLn-equivariantaffine geometry. There are plenty more PGLn-varieties than Cayley-Hamilton algebras.


Example 2.23. Conjugacy classes of nilpotent matrices in Mn(C) correspond bijective topartitions λ = (λ1 ≥ λ2 ≥ · · · ) of n (the λi determine the sizes of the Jordan blocks). Itfollows from the Gerstenhaber-Hesselink theorem that the closures of such orbits

Oλ = ∪µ≤λOµ

where ≤ is the dominance order relation. Each Oλ is an affine PGLn-variety and thecorresponding algebra is

AOλ= C[x]/(xλ1)

whence many orbit closures (all of which are affine PGLn-varieties) correspond to thesame algebra.

note: The category alg@n of Cayley-Hamilton algebras is to noncommutative geometry@nwhat commalg, the category of all commutative algebras is to commutative algebraicgeometry.

In fact, alg@1 commalg by taking as trace maps the identity on every commutativealgebra. Further we have a natural commutative diagram of functors

where the bottom map is the equivalence between affine algebras and affine schemes andthe top map is the correspondence between Cayley-Hamilton algebras and affine PGLn-schemes, which is not an equivalence of categories.

2.7. Smooth orders. To finish this talk let us motivate and define the notion of a Smoothorder properly. Among the many characterizations of commutative regular algebras is thefollowing due to A. Grothendieck.

Theorem 2.24. A commutative C-algebra A is regular if and only if it satisfies the followinglifting property: if (B,I) is a test-object such that B is a commutative algebra and I is anilpotent ideal of B, then for any algebra map φ, there exists a lifted algebra morphism φ

making the diagram commutative.

220 LIEVEN LE BRUYN

As the category commalg of all commutative C-algebras is just alg@l it makes sense todefine Smooth Cayley-Hamilton algebras by the same lifting property. This was done firstby W. Schelter [42] in the category of all algebras satisfying all polynomial identities ofn× n matrices and later by C. Procesi [39] in alg@n.

Definition 2.25. A Smooth Cayley-Hamilton algebra A is an object in alg@n satisfyingthe following lifting property. If (B, I) is a test-object in alg@n, that is, B is an object inalg@n, I is a nilpotent ideal in B such that B/I is an object in alg@n and such that thenatural map B

π−→−→B/I is trace preserving, then every trace preserving algebra map φ hasa lift φ

making the diagram commutative. If A is in addition an order, we say that A is aSmooth order.

Next talk we will give a large class of Smooth orders but again it should be stressed thatthere is no connection between this notion of non-commutative smoothness and the morehomological notion of Regular orders (except in dimension one when all notions coincide).

Still, in the context of PGLn-equivariant affine geometry this notion of non-commutativesmoothness is quite natural as illustrated by the following result due to C. Procesi [39].

Theorem 2.26. An object A in alg@n is Smooth if and only if the corresponding affinePGLn-scheme trepn A is smooth (and hence, in particular, reduced).

Proof. (One implication) Assume A is Smooth, then to prove that trepn A is smooth wehave to prove that C[trepn A] satisfies Grothendieck’s lifting property. So let (B, I) be atest-object in commalg and take an algebra morphism φ : C[trepn A] −→ B/I. Considerthe following diagram

the morphism (1) follows from Smoothness of A applied to the morphism Mn(φ) jA. Fromthe universal property of the map jA it follows that there is a morphism (2) which is of theform Mn(ψ) for some algebra morphism ψ : C[trepnA] −→ B. This ψ is the required lift.


Example 2.27. Trace rings Tm,n are the free algebras generated by m elements in alg@nand as such trivially satisfy the lifting property so are Smooth orders. Alternatively, because

trepn Tm,n Mn(C)⊕ · · · ⊕Mn(C) = Cmn2

is a smooth PGLn-variety, Tm,n is Smooth by the previous result.

Example 2.28. Any commutative algebra C can be viewed as an element of alg@n viathe diagonal embedding C →Mn(C). However, if C is a regular commutative algebra it isnot true that C is Smooth in alg@n. For example, take C = C[x1, . . . , xd] and consider the4-dimensional non-commutative local algebra

B =C〈x, y〉

(x2, y2, xy + yx)= C⊕ Cx ⊕ Cy ⊕ Cxy

with the obvious trace map so that B ∈ alg@2. B has a nilpotent ideal I = B(xy−yx) suchthat the quotient B/I is a 3-dimensional commutative algebra. Consider the algebra map

C[x1, . . . , xd]φ−→B

Idefined by x1 → x x2 → y and xi → 0 for i ≥ 3

This map has no lift as for any potential lifted morphism φ we have

[φ(x), φ(y)] = 0

whence C[x1, . . . , xd] is not Smooth in alg@2.

Example 2.29. Consider again the quantum plane at −1

A =C〈x, y〉

(xy + yx)

then we have seen that trep2A = V(bf + ce) ⊂ A6 has a unique isolated singularity at theorigin. Hence, A is not a Smooth order.

note: Under the correspondence between alg@n and PGL(n)-aff, Smooth Cayley-Hamilton algebras correspond to smooth PGLn-varieties.

3. Non-commutative geometry

Last time we introduced alg@n as a level n generalization of commalg, the variety ofall commutative algebras. Today we will associate to any A ∈ alg@n a non-commutativevariety max A and argue that this gives a non-commutative manifold if A is a Smoothorder. In particular we will show that for fixed n and central dimension d there are a finitenumber of etale types of such orders. This fact is the non-commutative analogon of thefact that every manifold is locally diffeomorphic to affine space or, in ringtheory terms,that the m-adic completion of a regular algebra C of dimension d has just one etale type:Cm C[[x1, . . . , xd]].

222 LIEVEN LE BRUYN

3.1. Why non-commutative geometry?

note: There is one new feature that non-commutative geometry has to offer compared tocommutative geometry: distinct points can lie infinitesimally close to each other. As desin-gularization is the process of separating bad tangents, this fact should be useful somehowin our project.

Recall that if X is an affine commutative variety with coordinate ring R, then toeach point P ∈ X corresponds a maximal ideal mP R and a one-dimensional simplerepresentation

Sp =R

mp

A basic tool in the study of Hilbert schemes is that finite closed subschemes of X can bedecomposed according to their support. In algebraic terms this means that there are noextensions between different points, that if P = Q then

Ext1R(SP , SQ) = 0 whereas Ext1R(SP , SP ) = TP X

In more plastic lingo: all infinitesimal information of X near P is contained in the self-extensions of SP and distinct points do not contribute. This is no longer the case fornon-commutative algebras.

Example 3.1. Take the path algebra A of the quiver ←− , that is

A [

C C0 C

]

Then A has two maximal ideals and two corresponding one-dimensional simple representa-tions

S1 =[

C0

]=[

C C0 C

]/[0 C0 C

]and S2 =

[0C

]=[

C C0 C

]/[C C0 0

]

Then, there is a non-split exact sequence with middle term the second column of A

0 −→ S1 =[

C0

]−→M =

[CC

]−→ S2 =

[0C

]−→ 0

Whence Ext1A(S2, S1) = 0 whereas Ext1A(S1, S2) = 0. It is no accident that these two factsare encoded into the quiver.

Definition 3.2. For A an algebra in alg@n, define its maximal ideal spectrum max A to bethe set of all maximal twosided ideals M of A equipped with the non-commutative Zariskitopology, that is, a typical open set of max A is of the form

X(I) = M ∈ max A|I ⊂M


Recall that for every M ∈ max A the quotient

A

MMk(C) for some k ≤ n

that is, M determines a unique k-dimensional simple representation SM of A.

As every maximal ideal M of A intersects the center R in a maximal ideal mP = M ∩Rwe get, in the case of an R-order A a continuous map

max Ac−→X defined by M → P where M ∩R = mP

Ringtheorists have studied the fibers c−1(P ) of this map in the seventies and eighties inconnection with localization theory. The oldest description is the Bergman-Small theorem,see for example [2]

Theorem 3.3. (Bergman-Small) If c−1(P ) = M1, . . . ,Mk then there are naturalnumbers ei ∈ N+ such that

n =k∑i=1

eidi where di = dimC SMi

In particular, c−1(P ) is finite for all P.Here is a modern proof of this result based on the results of the previous lecture. Because

X is the algebraic quotient trepnA//GLn, points of X correspond to closed GLn-orbits inrepn A. By a result of M. Artin [1] closed orbits are precisely the isomorphism classes ofsemi-simple n-dimensional representations, and therefore we denote the quotient variety

X = trepn A//GLn = tissn A

So, a point P determines a semi-simple n-dimensional A-representation

MP = S⊕e11 ⊕ . . .⊕ S⊕ek

k

with the Si the distinct simple components, say of dimension di =dimCSi and occurring inMp with multiplicity ei ≥ 1. This gives n = Σeidi and clearly the annihilator of Si is amaximal ideal Mi of A lying over mP .

Another interpretation of c−1(P ) follows from the work of A. V. Jategaonkar andB. Mullen Define a link diagram on the points of max A by the rule

M M ′ ⇐⇒ Ext1A(SM , SM ′) = 0

In fancier language, M M ′ if and only if M and M ′ lie infinitesimally close together inmax A. In fact, the definition of the link diagram in [20, Chp. 5] or [18, Chp. 11] is slightlydifferent but amounts to the same thing.

224 LIEVEN LE BRUYN

Theorem 3.4. (Jategaonkar-Muller) The connected components of the link diagram onmax A are all finite and are in one-to-one correspondence with P ∈ X. That is, if

M1, . . . ,Mk = c−1(P ) ⊂ max A

then this set is a connected component of the link diagram.

note: In max A there is a Zariski open set of Azumaya points, that is those M ∈ max A suchthat A/M Mn(C). It follows that each of these maximal ideals is a singleton connectedcomponent of the link diagram. So on this open set there is a one-to-one correspondencebetween points of X and maximal ideals of A so we can say that max A and X are birational.However, over the ramification locus there may be several maximal ideals of A lying over thesame central maximal ideal and these points should be thought of as lying infinitesimallyclose to each other.

One might hope that the cluster of infinitesimally points of max A lying over a centralsingularity P ∈ X can be used to separate tangent information in P rather than having toresort to the blowing-up process to achieve this.

3.2. What non-commutative geometry? As anR-order A in a central simpleK-algebraΣ of dimension n2 is a finite R-module, we can associate to A the sheaf OA of non-commutative OX -algebras using central localization. That is, the section over a basic affineopen piece X(f) ⊂ X are

Γ(X(f),OA) = Af = A⊗R Rfwhich is readily checked to be a sheaf with global sections Γ(X,OA) = A. As we willinvestigate Smooth orders via their (central) etale structure, that is information about Amp

,we will only need the structure sheaf OA over X.

In the ’70-ties F. Van Oystaeyen [47] and A. Verschoren [48] introduced genuine non-commutative structure sheaves associated to an R-order A. It is not my intention to promotenostalgia here but perhaps these non-commutative structure sheaves OncA on max A deserverenewed investigation.

Definition 3.5. OncA is defined by taking as the sections over the typical open set X(I)(for I a twosided ideal of A) in max A

Γ(X(I),OncA ) = δ ∈ Σ | ∃l ∈ N : I lδ ⊂ A

By [47] this defines a sheaf of non-commutative algebras over max A with global sectionsΓ(max A, OncA ) = A. The stalk of this sheaf at a point M ∈ max A is the symmetric local-ization

OncA,M = QA−M (A) = δ ∈ Σ | Iδ ⊂ A for some ideal I ⊂ P


Often, these stalks have no pleasant properties but in some examples, these non-commutative stalks are nicer than those of the central structure sheaf.

Example 3.6. Let X = A1, that is, R = C[x] and consider the order

A =[R Rm R

]

where m = (x) R. A is an hereditary order so is both a Regular order and a Smoothorder. The ramification locus of A is P0 = V(x) so over any P0 = P ∈ A1 there is a uniquemaximal ideal of A lying over mp and the corresponding quotient is M2(C). However, overm there are two maximal ideals of A

M1 =[m Rm R

]and M2 =

[R Rm m

]

Both M1 and M2 determine a one-dimensional simple representation of A, so the Bergman-Small number are e1 = e2 = 1 and d1 = d2 = 1. That is, we have the following picture

There is one non-singleton connected component in the link diagram of A namely

with the vertices corresponding to M1,M2. The stalk of OA at the central point P0

is clearly

OA,P0 =[Rm Rm

mm Rm

]

On the other hand the stalks of the non-commutative structure sheaf OncA in M1 resp. M2

can be computed to be

OncA,M1=[Rm Rm

Rm Rm

]and OncA,M2

=[Rm x−1Rm

xRm Rm

]

and hence both stalks are Azumaya algebras. Observe that we recover the central stalkOA,P0 as the intersection of these two rings in M2(K).

Hence, somewhat surprisingly, the non-commutative structure sheaf of the hereditarynon-Azumaya algebra A is a sheaf of Azumaya algebras over max A.

226 LIEVEN LE BRUYN

3.3. Marked quiver and Morita settings. Consider the continuous map for the Zariskitopology

max Ac−→ X

and let for a central point P ∈ X the fiber be M1, . . . ,Mk where the Mi are maximalideals of A with corresponding simple di-dimensional representation Si. In the previoussection we have introduced the Bergman-Small data, that is

α = (e1, . . . , ek) and β = (d1, . . . , dk) ∈ Nk+ satisfying α.β =k∑i=1

eidi = n

(recall that ei is the multiplicity of Si in the semi-simple n-dimensional representationcorresponding to P . Moreover, we have the Jategaonkar-Muller data which is a directedconnected graph on the vertices υ1, . . . , υk (corresponding to the Mi) with an arrow

υi υj iff Ext1A(Si, Sj) = 0

We now want to associate combinatorial objects to this local data.To start, introduce a quiver setting (Q,α) where Q is a quiver (that is, a directed graph)

on the vertices υ1, . . . , υk with the number of arrows from υi to υj equal to the dimensionof Ext1A(Si, Sj),

# (υi −→ υj) = dimC Ext1A(Si, Sj)

and where α = (e1, . . . , ek) is the dimension vector of the multiplicities ei.Recall that the representation space repα Q of a quiver-setting is ⊕aMei×ej

(C) wherethe sum is taken over all arrows a : υj −→ υi of Q. On this space there is a natural actionby the group

GL(α) = GLe1 × · · · ×GLek

by base-change in the vertex-spaces Vi = Cei (actually this is an action of PGL(α) whichis the quotient of GL(α) by the central subgroup C∗(1e1 , . . . , 1ek

)).The ringtheoretic relevance of the quiver-setting (Q,α) is that

repα Q Ext1A(MP ,MP ) as GL(α)-modules

where Mp is the semi-simple n-dimensional A-module corresponding to P

MP = S⊕e11 ⊕ · · · ⊕ S⊕ek

k

and because GL(α) is the automorphism group of Mp there is an induced actionon Ext1A(Mp,MP ).

Because Mp is n-dimensional, an element ψ ∈ Ext1A(Mp,Mp) defines an alge-bra morphism

Aρ−→ Mn(C[ε])


where C[ε] = C[x]/(x2) is the ring of dual numbers. As we are working in the categoryalg@n we need the stronger assumption that ρ is trace preserving. For this reason we haveto consider the GL(α)-subspace

tExt1A(MP ,MP ) ⊂ Ext1A(MP ,MP )

of trace preserving extensions. As traces only use blocks on the diagonal (correspondingto loops in Q) and as any subspace Mei

(C) of repα Q decomposes as a GL(α)-module insimple representations

Mei(C) = M0

ei(C)⊕ C

where M0ei

(C) is the subspace of trace zero matrices, we see that

repα Q∗ tExt1A(MP ,MP ) as GL(α)-modules

where Q∗ is a marked quiver that has the same number of arrows between distinct verticesas Q has, but may have fewer loops and some of these loops may acquire a marking meaningthat their corresponding component in repαQ∗ is M0

ei(C) instead of Mei

(C).

note: Let the local structure of the non-commutative variety max A near the fiber c−1(P )of a point P ∈ X be determined by the Bergman-Small data

α = (e1, . . . , ek) and β = (d1, . . . , dk)

and by the Jategoankar-Muller data which is encoded in the marked quiver Q∗ on k-vertices.Then, we associate to P the combinatorial data

type(P ) = (Q∗, α, β)

We call (Q∗, α) the marked quiver setting associated to A in P ∈ X. The dimension vectorβ = (d1, . . . , dk) will be called the Morita setting associated to A in P .

Example 3.7. If A is an Azumaya algebra over R. then for every maximal ideal m corre-sponding to a point P ∈ X we have that

A/mA = Mn(C)

so there is a unique maximal ideal M = mA lying over m whence the Jategaonkar-Mullerdata are α = (1) and β = (n). If Sp = R/m is the simple representation of R we have

Ext1A(MP ,MP ) Ext1R(Sp, SP ) = TP X

and as all the extensions come from the center, the corresponding algebra representationsA −→ Mn(C[ε]) are automatically trace preserving. That is, the marked quiver-settingassociated to A in P is

228 LIEVEN LE BRUYN

where the number of loops is equal to the dimension of the tangent space TP X in P atX and the Morita-setting associated to A in P is (n).

Example 3.8. Consider the order of example 3.6 which is generated as a C-algebra bythe elements

a =[1 00 0

]b =

[0 10 0

]c =

[0 0x 0

]d =

[0 00 1

]

and the 2-dimensional semi-simple representation MP0 determined by m is given by thealgebra morphism A −→ M2(C) sending a and d to themselves and b and c to the zeromatrix. A calculation shows that

Ext1A(MP0 ,MP ) = repαQ for (Q,α) = 1u

υ

1

and as the correspondence with algebra maps to M2(C[ε]) is given by

a →[1 00 0

]b →

[0 ευ0 0

]c →

[0 0εu 0

]d →

[0 00 1

]

each of these maps is trace preserving so the marked quiver setting is (Q,α) and the Morita-setting is (1,1).

3.4. Local classification.

note: Because the combinatorial data type(P ) = [Q∗, α, β) encodes the infinitesimal infor-mation of the cluster of maximal ideals of A lying over the central point P ∈ X, (repα Q

∗, β)should be viewed as analogous to the usual tangent space TP X.

If P ∈ X is a singular point, then the tangent space is too large so we have to imposeadditional relations to describe the variety X in a neighborhood of P , but if P is a smoothpoint we can recover the local structure of X from TPX.

Here we might expect a similar phenomenon: in general the data (repα Q∗, β) will betoo big to describe AmP

unless A is a Smooth order in P in which case we can recover AmP.

We begin by defining some algebras which can be described combinatorially from (Q∗, α, β).For every arrow a : υi −→ υj define a generic rectangular matrix of size ej × ei

Xa =

x11(a) . . . . . . x1ei

(a)...

...xej1(a) . . . . . . xejei

(a)

(and if a is a marked loop take xeiei(a) = −x11(a)− x22(a)− . . .− xei−1ei

−1(a)) then thecoordinate ring C[repα Q

∗] is the polynomial ring in the entries of all Xa. For an orientedpath p in the marked quiver Q∗ with starting vertex υi and terminating vertex υj

υip υj = υi

a1−→ υi1a2−→ . . .

al−1−→ υilal−→ υj

we can form the square ej × ei matrix

Xp = XalXal−1 · · ·Xa2Xa1


which has all its entries polynomials in C[repα Q∗]. In particular, if the path is an oriented

cycle c in Q∗ starting and ending in υi then Xc is a square ei × ei matrix and we can takeits trace tr(Xc) ∈ C[repα Q

∗] which is a polynomial invariant under the action of GL(α)on repα Q

∗.In fact, it was proved in [31] that these traces along oriented cycles generate the invari-

ant ring

RαQ∗ = C[repα Q∗]GL(α) ⊂ C[repα Q

∗]

Next we bring in the Morita-setting β = (d1, . . . , dk) and define a block-matrix ring

Aα,βQ∗ =

Md1×d1(P11) . . . Md1×dk

(P1k)...

...Mdk×d1(Pk1) . . . Mdk×dk

(Pkk)

⊂Mn(C[repα Q

∗])

where Pij is the RαQ∗-submodule of Mej×ei(C[repα Q

∗]) generated by all Xp where p is anoriented path in Q∗ starting in υi and ending in υk.

Observe that for triples (Q∗, α, β1) and (Q∗, α, β2) we have that

Aα,β1Q∗ is Morita-equivalent to Aα,β2

Q∗

whence the name Morita-setting for β.Before we can state the next result we need the Euler-form of the underlying quiver Q

of Q∗ (that is, forgetting the markings of some loops) which is the bilinear form χQ on Zk

determined by the matrix having as its (i,j)-entry δij − #a : υia−→υj. The statements

below can be deduced from those of [31]

Theorem 3.9. For a triple (Q∗, α, β) with α.β = n we have

1. Aα,βQ∗ is an RαQ-order in alg@n if and only if α is the dimension vector of a simplerepresentation of Q∗, that is, for all vertex-dimensions δi we have

χQ(α, δi) ≤ 0 and χQ(δi, α) ≤ 0

unless Q∗ is an oriented cycle of type Ak−1 then α must be (1, . . . , 1).2. If this condition is satisfied, the dimension of the center RαQ∗ is equal to

dim RαQ∗ = 1− χQ(α, α)−#marked loops in Q∗

These combinatorial algebras determine the etale local structure of Smooth orders as wasproved in [29]. The principal technical ingredient in the proof is the Luna slice theorem, seefor example [45] or [34].

Theorem 3.10. Let A be a Smooth order over R in alg@n and let P ∈ X with correspondingmaximal ideal m. If the marked quiver setting and the Morita-setting associated to A in P is

230 LIEVEN LE BRUYN

given by the triple (Q∗, α, β), then there is a Zariski open subset X(fi) containing P and anetale extension S of both Rfi

and the algebra RαQ∗ such that we have the following diagram

In particular, we haveRm RαQ∗ and Am Aα,βQ∗

where the completions at the right hand sides are with respect to the maximal (graded) idealof RαQ∗ corresponding to the zero representation.

Example 3.11. From example 3.7 we recall that the triple (Q∗, α, β) associated to anAzumaya algebra in a point P ∈ X is given by

and β = (n)

where the number of arrows is equal to dimC TPX. In case P is a smooth point of X thisnumber is equal to d = dim X. Observe that GL(α) = C∗ acts trivially on repα Q∗ = Cd

in this case. Therefore we have that

RαQ∗ C[x1, . . . , xd] and AαβQ∗ = Mn(C[x1, . . . , xd])

Because A is a Smooth order in such points we get that

AmB Mn(C[[x1, . . . , xd]])

consistent with our etale local knowledge of Azumaya algebras.

note: Because α.β = n, the number of vertices of Q∗ is bounded by n and as

d = 1− χQ(α, α)−#marked loops

the number of arrows and (marked) loops is also bounded. This means that for a particulardimension d of the central variety X there are only a finite number of etale local types ofSmooth orders in alg@n.

This fact might be seen as a non-commutative version of the fact that there is just oneetale type of a smooth variety in dimension d namely C[[x1, . . . , xd]]. At this moment asimilar result for Regular orders seems to be far out of reach.


3.5. A two-person game. Starting with a marked quiver setting (Q∗, α) we will play atwo-person game. Left will be allowed to make one of the reduction steps to be definedbelow if the condition on Leaving arrows is satisfied, Red on the other hand if the conditionon aRRiving arrows is satisfied. Although we will not use combinatorial game theory in anyway, it is a very pleasant topic and the interested reader is referred to [12] or [3].

The reduction steps below were discovered by R. Bocklandt in his Ph.D. thesis [4] (seealso [5]) in which he classifies quiver settings having a regular ring of invariants. These stepswere slightly extended in [6] in order to classify central singularities of Smooth orders. Allreductions are made locally around a vertex in the marked quiver. There are three types ofallowed moves

Vertex removalAssume we have a marked quiver setting (Q∗, α) and a vertex υ such that the local struc-

ture of (Q∗, α) near υ is indicated by the picture on the left below, that is, inside the verticeswe have written the components of the dimension vector and the subscripts of an arrow indi-cate how many such arrows there are in Q∗ between the indicated vertices. Define the newmarked quiver setting (Q∗R, αR) obtained by the operation RυV which removes the vertex υand composes all arrows through υ, the dimensions of the other vertices are unchanged:

where cij = aibj (observe that some of the incoming and outgoing vertices may be the sameso that one obtains loops in the corresponding vertex). Left (resp. Right) is allowed to makethis reduction step provided the following condition is met

(Left) χQ(α, ευ)≥ 0 ⇔ αυ ≥l∑

j=1

ajij

(Right) χQ(ευ, α)≥ 0 ⇔ αυ ≥l∑

j=1

bjuj

(observe that if we started off from a marked quiver setting (Q∗, α) coming from an order,then these inequalities must actually be equalities).

loop removalIf υ is a vertex with vertex-dimension αυ = 1 and having k ≥ 1 loops. Let (Q∗R, αR) be

the marked quiver setting obtained by the loop removal operation Rυl

232 LIEVEN LE BRUYN

removing one loop in υ and keeping the same dimension vector. Both Left and Right areallowed to make this reduction step.

Loop removalIf the local situation in υ is such that there is exactly one (marked) loop in υ, the

dimension vector in υ is k ≥ 2 and there is exactly one arrow Leaving υ and this to a vertexwith dimension vector 1, then Left is allowed to make the reduction RυL indicated below

Similarly, if there is one (marked) loop in υ and αυ = k ≥ 2 and there is only one arrowaRRiving at υ coming from a vertex of dimension vector 1, then Right is allowed to makethe reduction RυL

In accordance with combinatorial game theory we call a marked quiver setting (Q∗, α) azero setting if neither Left nor Right has a legal reduction step. The relevance of this gameon marked quiver settings is that if

(Q∗1, α1) (Q∗2, α2)

is a sequence of legal moves (both Left and Right are allowed to pass), then

Rα1Q∗

1 Rα2

Q∗2[y1, . . . , yz]


where z is the sum of all loops removed in Rυl reductions plus the sum of αυ for eachreduction step RυL involving a genuine loop and the sum of αυ − 1 for each reduction stepRυL involving a marked loop. That is, marked quiver settings which below to the same gametree have smooth equivalent invariant rings.

In general games, a position can reduce to several zero-positions depending on the chosenmoves. For this reason the next result, proved in [6] is somewhat surprising

Theorem 3.12. Let (Q∗, α) be a marked quiver setting, then there is a unique zero-setting(Q∗0, α0) far which there exists a reduction procedure

(Q∗, α) (Q∗0, α0)

We will denote this unique zero-setting by Z(Q∗, α).

note: Therefore it is sufficient to classify the zero-positions if we want to characterize allcentral singularities of a Smooth order in a given central dimension d.

3.6. Central singularities. Let A be a Smooth R-order in alg@n and P a point in thecentral variety X with corresponding maximal ideal m R. We now want to classify thetypes of singularities of X in P , that is to classify Rm.

To start, can we decide when P is a smooth point of X? In the case that A is an Azumayaalgebra in P , we know already that A can only be a Smooth if R is regular in P . Moreoverwe have seen for A a Regular reflexive Azumaya algebra that the non-Azumaya points inX are precisely the singularities of X.

For Smooth orders the situation is more delicate but as mentioned before we have acomplete solution in terms of the two-person game by a slight adaptation of Bocklandt’smain result [5].

Theorem 3.13. If A is a Smooth R-order and (Q∗, α, β) is the combinatorial data asso-ciated to A in P ∈ X. Then, P is a smooth point of X if and only if the unique associ-ated zero-setting

The Azumaya points are such that Z(Q∗, α) = 1 hence the singular locus of X iscontained in the ramification locus ram A but may be strictly smaller.

To classify the central singularities of Smooth orders we may reduce to zero-settings(Q∗, α) = Z(Q∗, α). For such a setting we have for all vertices υi the inequalities

χQ(α, δi) < 0 and χQ(δi, α) < 0

and the dimension of the central variety can be computed from the Euler-form χQ. Thisgives us an estimate of d = dimX which is very efficient to classify the singularities inlow dimensions.

234 LIEVEN LE BRUYN

Theorem 3.14. Let (Q∗, α) = Z(Q∗, α) be a zero-setting on k ≥ 2 vertices. Then,

In this sum the contribution of a vertex υ with αυ = a is determined by the numberof (marked) loops in υ. By the reduction steps (marked) loops only occur at verticeswhere αυ > 1.

Let us illustrate this result by classifying the central singularities in low dimensions

Example 3.15. (dimension 2) When dim X = 2 no zero-position on at least two verticessatisfies the inequality of theorem 3.14, so the only zero-position possible to be obtainedfrom a marked quiver-setting (Q∗, α) in dimension two is

Z(Q∗, α) = 1

and therefore the central two-dimensional variety X of a Smooth order is smooth.

Example 3.16. (dimension 3) If (Q∗, α) is a zero-setting for dimension ≤ 3 then Q∗ canhave at most two vertices. If there is just one vertex it must have dimension 1 (reducingagain to 1 whence smooth) or must be

which is again a smooth setting. If there are two vertices both must have dimension 1and both must have at least two incoming and two outgoing arrows (for otherwise wecould perform an additional vertex-removal reduction). As there are no loops possible inthese vertices for zero-settings, it follows from the formula d = 1 − χQ(α, α) that the onlypossibility is

The ring of polynomial invariants RαQ∗ is generated by traces along oriented cycles in Q∗ soin this case it is generated by the invariants

x = ac, y = ad, u = bc and υ = bd


and there is one relation between these generators, so

RαQ∗ C(x, y, u, υ)(xy − uυ)

Therefore, the only etale type of central singularity in dimension three is the conifold singularity.

Example 3.17. (dimension 4) If (Q∗, α) is a zero-setting for dimension 4 then Q∗ canhave at most three vertices. If there is just one, its dimension must be 1 (smooth setting)or 2 in which case the only new type is

which is again a smooth setting.If there are two vertices, both must have dimension 1 and have at least two incoming

and outgoing arrows as in the previous example. The only new type that occurs is

for which one calculates as before the ring of invariants to be

RαQ∗ =C[a, b, c, d, e, f ]

(ae− bd, af − cd, bf − ce)

If there are three vertices all must have dimension 1 and each vertex must have at least twoincoming and two outgoing vertices. There are just two such possibilities in dimension 4

The corresponding rings of polynomial invariants are

RαQ∗ =C[x1, x2, x3, x4, x5](x4x5 − x1x2x3)

resp. RαQ∗ =C[x1, x2, x3, x4, y1, y2, y3, y4]

R2

where R2 is the ideal generated by all 2× 2 minors of the matrix

[x1 x2 x3 x4

y1 y2 y3 y4

]

In [6] it was proved that there are exactly ten types of Smooth order central singularitiesin dimension d = 5 and 53 in dimension d = 6. The strategy to prove such a result isas follows.

236 LIEVEN LE BRUYN

First one makes a full list of all zero-settings (Q∗, α) = Z(Q∗, α) such that d = 1 −χQ(α, α)−# marked loops, using theorem 3.14.

Next, one has to weed out zero-settings having isomorphic rings of polynomial invari-ants (or rather, having the same m-adic completion where m RαQ∗ is the unique gradedmaximal ideal generated by all generators). There are two invariants to separate two ringsof invariants.

One is the sequence of numbers

dimCmn

mn+1

which can sometimes be computed easily (for example if all dimension vector componentsare equal to 1).

The other invariant is what we call the fingerprint of the singularity. In most cases, therewill be other types of singularities (necessarily also of Smooth order type) in the varietycorresponding to RαQ∗ and the methods of [29] allow us to determine their associated markedquiver settings as well as the dimensions of these strata.

In most cases these two methods allow to separate the different types of singularities. Inthe few remaining cases it is then easy to write down an explicit isomorphism. We refer to(the published version of) [6] for the full classification of these singularities in dimension 5and 6.

note: In low dimensions there is a full classification of all central singularities Rm of aSmooth order in alg@n. However, at this moment no such classification exists for Am. Thatis, under the game rules it is not clear what structural results of the ordersAαQ∗ are preserved.

3.7. Isolated singularities. In the classification of central singularities of Smooth orders,isolated singularities stand out as the fingerprinting method to separate them clearly fails.Fortunately, we do have by [7] a complete classification of these (in all dimensions).

Theorem 3.18. Let A be a Smooth order over R and let (Q∗, α, β) be the combinatorialdata associated to a A in a point P ∈ X. Then, P is an isolated singularity if and only ifZ(Q∗, α) = T (k1, . . . , kl) where

with d = dim X =∑i ki − l + 1.

Moreover, two such singularities, corresponding to T (k1, . . . , kl) and T (k′1, . . . , k′l′), are

isomorphic if and only if

l = l′ and k′i = kσ(i)

for some permutation σ ∈ Sl.


The results we outlined in this talk are good as well as bad news.

note: On the positive side we have very precise information on the types of singularitieswhich can occur in the central variety of a Smooth order (certainly in low dimensions) insharp contrast to the case of Regular orders.

However, because of the scarcity of such types most interesting quotient singularitiesCd/G will not have a Smooth order over their coordinate ring R = C[Cd/G].

So, after all this hard work we seem to have come to a dead end with respect to thedesingularization problem as there are no Smooth orders with center C[Cd/G]. Fortunately,we have one remaining trick available: to bring in a stability structure.

4. Non-commutative desingularizations

In the first talk I claimed that in order to find good desingularizations of quotient singu-larities Cd/G we had to find Smooth orders in alg@n with center R = C[Cd/G]. Last timewe have seen that Smooth orders can be described and classified locally in a combinatorialway but also that there can be no Smooth order with center C[Cd/G]. What we will seetoday is that there are orders A over R which may not be Smooth but are Smooth on asufficiently large Zariski open subset of repα A. Here ‘sufficiently large’ means determinedby a stability structure. Whenever this is the case we can apply the results of last time toconstruct nice (partial) desingularizations of Cd/G and if you are in for non-commutativegeometry, even a genuine non-commutative desingularization.

4.1. Quotient singularities. Last time we associated to a combinatorial triple (Q∗, α, β)a Smooth order Aα,βQ∗ with center the ring of polynomial quiver-invariants RαQ∗ . As we wereable to classify the quiver-invariants it followed that there is no triple such that the centerof Aα,βQ∗ is the coordinate ring R = C[Cd/G] of the quotient singularity. However, there arenice orders of the form

A =Aα,βQ∗

I

for some ideal I of relations which do have center R are have been used in studyingquotient singularities.

Example 4.1. (Kleinian singularities) For a Kleinian singularity, that is, a quotientsingularity C2/G with G ⊂ SL2(C) there is an extended Dynkin diagram D associated.

Let Q be the double quiver of D, that is to each arrow x−→ in D we adjoin an arrow

x∗←− in Q in the opposite direction and let α be the unique minimal dimension vector

such that χD(α, α) = 0. Further, consider the moment element

m =∑x∈D

[x, x∗]

in the order AαQ then

A =AαQ(m)

238 LIEVEN LE BRUYN

is an order with center R = C[C2/G] which is isomorphic to the skew-group algebraC[x, y]#G. Moreover, A is Morita equivalent to the preprojective algebra which is thequotient of the path algebra of Q by the ideal generated by the moment element

∏0

= CQ/(∑

[x, x∗])

For more details we refer to the lecture notes by W. Crawley-Boevey [14].

Example 4.2. Consider a quotient singularity X = Cd/G with G ⊂ SLd(C) and Q be theMcKay quiver of G acting on V = Cd.

That is, the vertices υ1, . . . , υk of Q are in one-to-one correspondence with the irre-ducible representations R1, . . . , Rk of G such that R1 = Ctriv is the trivial representation.Decompose the tensorproduct in irreducibles

V ⊗C Rj = R⊗j11 ⊗ . . .⊗R⊗jkk

then the number of arrows in Q from υi to υj

#(υi −→ υj) = ji

is the multiplicity of Ri in V ⊗Rj . Let α = (e1, . . . , ek) be the dimension vector where ei =dimC Ri.

The relevance of this quiver-setting is that

repαQ = HomG(R,R⊗ V )

where R is the regular representation, see for example [13]. Consider Y ⊂ repαQ theaffine subvariety of all α-dimensional representations of Q for which the correspondingG-equivariant map B ∈ HomG(R, V ⊗R) satisfies

B ∧B = 0 ∈ HomG(R,∧2V ⊗R)

Y is called the variety of commuting matrices and its defining relations can be expressed aslinear equations between paths in Q evaluated in repαQ, say (l1, . . . , lz). Then,

A =AαQ

(l1, . . . , lz)

is an order with center R = C[Cd/G]. In fact, A is just the skew group algebra

A = C[x1, . . . , xd]#G

Let us give one explicit example illustrating both approaches to the Kleinian singular-ity C2/Z3.


Example 4.3. Consider the natural action of Z3 on C2 via its embedding in SL2(C) sendingthe generator to the matrix [

ρ 00 ρ−1

]

where ρ is a primitive 3-rd root of unity. Z3 has three one-dimensional simples R1 =Ctriv, R2 = Cρ and R2 = Cρ2 . As V = C2 = R2 ⊗ R3 it follows that the McKay quiversetting (Q,α) is

Consider the matrices

X =

0 0 x3

x1 0 00 x2 0

and Y =

0 y1 0

0 0 y2y3 0 0

then the variety of commuting matrices is determined by the matrix-entries of [X, Y] that is

I = (x3y3 − y1x1, x1y1 − y2x2, x2y2 − y3x3)

so the skew-group algebra is the quotient of the Smooth order AαQ (which incidentally isone of our zero-settings for dimension 4)

C[x, y]#Z3 AαQ

(x3y3 − y1x1, x1y1 − y2x2, x2y2 − y3x3)

Taking yi = x∗i this coincides with the description via preprojective algebras as the momentelement is

m =3∑i=1

[xi, x∗i ] = (x3y3 − y1x1)e1 + (x1y1 − y2x2)e2 + (x2y2 − y3x3)e3

where the ei are the vertex-idempotents.

note: Many interesting examples of orders are of the following form:

A =AαQ∗

I

240 LIEVEN LE BRUYN

satisfying the following conditions:

• α = (e1, . . . , ek) is the dimension vector of a simple representation of A, and• the center R = Z(A) is an integrally closed domain.

These requirements (which are often hard to verify!) imply that A is an order over R inalg@n where n is the total dimension of the simple representation, that is |α| = Σiei.

Observe that such orders occur in the study of quotient singularities (see above) or asthe etale local structure of (almost all) orders. From now on, this will be the setting we willwork in.

4.2. Stability structures. ForA = AαQ∗/I we define the affine variety of α-dimensional repre-sentations

repαA = V ∈ repαQ∗|r(V = 0∀r ∈ I

The action of GL(α) =∏iGLei

by basechange on repαQ∗ induces an action (actually ofPGL(α)) on repα A. Usually, repα A will have singularities but it may be smooth on theZariski open subset of θ-semistable representations which we will now define.

A character of GL(α) is determined by an integral k-tuple θ = (t1, . . . , tk) ∈ Zk

χθ : GL(α) −→ C∗ (g1, . . . , gk) → det(g1)t1 · · · det(gk)tk

Characters define stability structures on A-representations but as the acting group on repα Ais really PGL(α) = GL(α)/C∗(1e1 , . . . , 1ek

) we only consider characters θ satisfying θ.α =∑i tiei = 0.If V ∈ repα A and V ′ ⊂ V is an A-subrepresentation, that is V ′ ⊂ V as representations

of Q∗ and in addition I(V ′) = 0, we denote the dimension vector of V ′ by dimV ′.

Definition 4.4. For θ satisfying θ.α = 0, a representation V ∈ repα A is said to be

• θ-semistable if and only if for every proper A-subrepresentation 0 = V ′ ⊂ V wehave θ.dimV ′ ≥ 0.

• θ-stable if and only if for every proper A-subrepresentation 0 = V ′ ⊂ V wehave θ.dimV ′ > 0.

For any setting θ.α = 0 we have the following inclusions of Zariski open GL(α)-stablesubsets of repαA

repsimpleα A ⊂ repθ−stableα A ⊂ repθ−semistα A ⊂ repα A

but one should note that some of these open subsets may actually be empty!Recall that a point of the algebraic quotient variety issαA = repα//GL(α) represents

the orbit of an α-dimensional semi-simple representation V and such representations canbe separated by the values f(V ) where f is a polynomial invariant on repαA. This followsbecause the coordinate ring of the quotient variety

C[issα A] = C[repα A]GL(α)

and points correspond to maximal ideals of this ring. Recall from [31] that the invariant ringis generated by taking traces along oriented cycles in the marked quiver-setting (Q∗, α).


note: For θ-stable and θ-semistable representations there are similar results and morallyone should view θ-stable representations as corresponding to simple representations whereasθ-semistables are arbitrary representations.

For this reason we will only be able to classify direct sums of θ-stable representations bycertain algebraic varieties which are called the moduli spaces of semistables representations.

The notion corresponding to a polynomial invariant in this more general setting is thatof a polynomial semi-invariant. A polynomial function f ∈ C[repα A] is said to be a θ-semi-invariant of weight l if for all g ∈ GL(α) we have

g ·f = χθ(g)lf

where χθ is the character of GL(α) corresponding to θ. A representation V ∈ repα A isθ-semistable if and only if there is a θ-semi-invariant f of some weight l such that f(V ) = 0.

Clearly, θ-semi-invariants of weight zero are just polynomial invariants and the multipli-cation of θ-semi-invariants of weight l resp. l′ has weight l+ l′. Hence, the ring of all θ-semi-invariants

C[repα A]GL(α),θ = ⊕∞l=0f ∈ C[repα A]|∀g ∈ GL(α) : g ·f = χθ(g)lf

is a graded algebra with part of degree zero C[issαA]. But then we have a projec-tive morphism

proj C[repαA]GL(α),θ π−→−→ issα A

such that all fibers of π are projective varieties. The main properties of π can be deducedfrom [22]

Theorem 4.5. Points in proj C[repαA]GL(α),θ are in one-to-one correspondence withisomorphism classes of direct sums of θ-stable representations of total dimension α.

If α is such that there are α-dimensional simple A-representations, then π is a bira-tional map.

Definition 4.6. We call proj C[repαA]GL(α),θ the moduli space of θ-semistable represen-tations of A and denote it with moduliθα A.

Example 4.7. In the case of Kleinian singularities, see example 4.1, if we take θ to be ageneric character such that θ.α = 0, then the projective map

moduliθαA X = C2/G

is a minimal resolution of singularities. Note that the map is birational as α is the dimensionvector of a simple representation of A =

∏0, see [14].

Example 4.8. For general quotient singularities, see example 4.2, assume that the firstvertex in the McKay quiver corresponds to the trivial representation. Take a characterθ ∈ Zk such that t1 < 0 and all ti > 0 for i ≥ 2, for example take

θ = (−k∑i=2

dimRi, 1, . . . , 1)

242 LIEVEN LE BRUYN

Then, the corresponding moduli space is isomorphic to

moduliθα A G−Hilb Cd

the G-equivariant Hilbert scheme which classifies all #G-codimensional ideals I C[x1, . . . , xd] where

C[x1, . . . , xd]I

CG

as G-modules, hence in particular I must be stable under the action of G. It is well knownthat the natural map

G−Hilb Cd X = Cd/G

is a minimal resolution if d = 2 and if d = 3 it is often a crepant resolution, for examplewhenever G is Abelian. In non-Abelian cases it may have remaining singularities thoughwhich often are of conifold type. See [13] for more details.

note: My motivation for this series of talks was to look for a non-commutative explanationfor the omnipresence of conifold singularities in partial resolutions of three dimensionalquotient singularities as well as to have a conjectural list of possible remaining singularitiesfor higher dimensional quotient singularities.

Example 4.9. In the C2/Z3-example one can take θ = (−2, 1, 1). The following represen-tations

are all nilpotent and are θ-stable. In fact if bc = 0 they are representants of the exceptionalfiber of the desingularization

moduliθα A issα A = C2/Z3

4.3. Partial resolutions. It is about time we state the main result of these notes whichwas proved in [32].

Theorem 4.10. Let A = AαQ∗/(R) be an R-order in alg@n. Assume that there exists astability structure θ ∈ Zk such that the Zariski open subset repθ−semistα A of all θ-semistableα-dimensional representations of A is a smooth variety.

Then there exists a sheaf A of Smooth orders over moduliθα A such that the diagrambelow is commutative


Here, spec A is a non-commutative variety obtained by gluing affine non-commutative vari-eties spec Ai together and c is the map which intersects locally a maximal ideal with thecenter. As A is a sheaf of Smooth orders, φ can be viewed as a non-commutative desingu-larization of X.

If you are only interested in commutative desingularizations, π is a partial resolutionof X and we have full control over the remaining singularities in moduliθα A, that is, allremaining singularities are of the form classified in the previous lecture.

Moreover, if θ is such that all θ-semistable A-representations are actually θ-stable, thenA is a sheaf of Azumaya algebras over moduliθα A and in this case π is a commutative desin-gularization of X. If, in addition, also gcd(α) = 1, then A End P for some vectorbundleof rank n over moduliθα A.

note: It should be stressed that the condition that repθ−semistα A is a smooth variety isvery strong and is usually hard to verify in concrete situations.

Example 4.11. In the case of Kleinian singularities, see example 4.1, there exists a suitablestability structure θ such that repθ−semistα Π0 is smooth. For consider the moment map

repαQµ−→ lie GL(α) = Mα(C) = Me1(C)⊗ . . .⊗M∈k

(C)

defined by sending V = (Va, Va∗) to

The differential dµ can be verified to be surjective in any representation V ∈ repα Qwhich has stabilizer subgroup C∗(1e1 , . . . , 1ek

) (a so called Schur representation) see forexample [15, lemma 6.5].

Further, any θ-stable representation is Schurian. Moreover, for a generic stability struc-ture θ ∈ Zk we have that every θ-semistable α-dimensional representation is θ-stable asthe gcd(α) = 1. Combining these facts it follows that µ−1(0) = repα Π0 is smooth in allθ-stable representations.

Example 4.12. Another case where smoothness of repθ−semistα A is evident is whenA = AαQ∗ is a Smooth order as then repα A itself is smooth. This observation can be usedto resolve the remaining singularities in the partial resolution.

If gcd(α) = 1 then for a sufficiently general θ all θ-semistable representations are actuallyθ-stable whence the quotient map

repθ−semistα A moduliθα A

is a principal PGL(α)-fibration and as the total space is smooth, so is moduliθα A. Therefore,the projective map

moduliθαAπ issα A

is a resolution of singularities in this case.

244 LIEVEN LE BRUYN

However, if l = gcd(α), then moduliθα A will usually contain singularities which are as badas the quotient variety singularity of tuples of l×l matrices under simultaneous conjugation.

Fortunately, the proof of the theorem will follow from the hard work we did in last lectureprovided we can solve two problems.

A minor problem is that we classified central singularities of Smooth orders in alg@nbut here we are working with α-dimensional representations and with the action of GL(α)rather than GLn. This problem we will address immediately.

A more serious problem is that repθ−semistα A is not an affine variety in general so wewill have to cover it with affine varieties Xi and consider associated orders Ai. But then wehave to clarify why θ-semistable representations of A correspond to all representations ofthe Ai. This may not be clear at first sight.

4.4. Going from. alg@n to alg@αIf Q∗ is a marked quiver on k vertices, then the subalgebra generated by the vertex-idempotents Ck is a subalgebra of A = AαQ∗/(R) hence we have a morphism

repnA −→ repnCk =⊔|α|=n

GLn/GL(α)

where the last decomposition follows from the fact that Ck is semi-simple whence everyn-dimensional representation is fully determined by the multiplicities of the simple1-dimensional components.

Further, we should consider trepn A the subvariety of trace preserving A-representationsbut a trace map on A fixes the trace on Ck and hence determines the componentGLn/GL(α). That is, we have that

trepnA = GLn ×GL(α) repα A

the variety is a principal fiber bundle.That is, if V is any n-dimensional trace preserving A-representation A

φ−→Mn(C) thenthe images φ(υi) of the vertex-idempotents are a full set of orthogonal idempotents so theycan be conjugated to a set of matrices

φ′(υi) =

. . .1

. . .1

. . .

with only 1’s from place∑i−1j=1 ej + 1 to place

∑ij=1 ej . But using these idempotents we see

that the representation φ′ : A −→Mn(C) has block-matrices coming from a representationin repαA.

As is the case for any principal fiber bundle, this gives a natural one-to-one correspon-dence between

• GLn-orbits in trepnA, and• GL(α)-orbits in repαA.


Moreover the corresponding quotient varieties tissnA = trepnA//GLn and issαA =repαA//GL(α) are isomorphic so we can apply all our (P )GLn-results to this setting.

note: Alternatively, we can define alg@α to be the subcategory of alg@n with objects thealgebras A ∈ alg@n which are Ck-algebras via the embedding given by the matrices φ′(υi)above and with morphism the Ck-algebra morphisms in alg@n.

It is then clear that a Smooth order in alg@α (that is, having the lifting propertywith respect to nilpotent ideals in alg@α) is a Smooth order in alg@n which is an objectin alg@α.

4.5. The affine opens XD. To solve the second problem, we claim that we can cover themoduli space

moduliθαA =⋃D

XD

where XD is an affine open subset such that under the canonical quotient map

repθ−semistα Aπ moduliθαA

we have thatπ−1(XD) = repα AD

for some C[XD]-order AD in [email protected] in addition repθ−semistα A is a smooth variety, each of the repα AD are smooth affine

GL(α)-varieties whence the orders AD are all Smooth and the result will follow from theresults of last lecture.

Because moduliθα A = projC[repα A]GL(α),θ we need control on the generators of allθ-semi-invariants. Such a generating set was found by Aidan Schofield and Michel Van denBergh in [44]: determinantal semi-invariants. In order to define them we have to introducesome notation first.

Reorder the vertices in Q∗ such that the entries of θ are separated in three strings

θ = (t1, . . . , ti)︸︷︷︸>0

, ti+1, . . . , tj︸︷︷︸=0

, tj+1, . . . , tk︸︷︷︸<0

and let θ be such that θ.α = 0. Fix a weight l ∈ N+ and take arbitrary naturalnumbers li+1, . . . , lj.

Consider a rectangular matrix L with

• lt1 + · · ·+ lti + li+1 + · · ·+ lj rows and• li+1 + · · ·+ lj − ltj+1 − · · · − ltk columns

L =

li+1︷︸︸︷ . . . lj︷︸︸︷ −ltj+1︷︸︸︷ . . . −ltk︷︸︸︷lt1 L1,i+1 L1,j L1,j+1 L1,k

...lti Li,i+1 Li,j Li,j+1 Li,kli+1 Li+1,i+1 Li+1,j Li+1,j+1 Li+1,k

...lj Lj,i+1 Lj,j Lj,j+1 Lj,k

246 LIEVEN LE BRUYN

in which each entry of Lr,c is a linear combination of oriented paths in the marked quiverQ∗ with starting vertex υc and ending vertex υr.

The relevance of this is that we can evaluate L at any representation V ∈ repα A andobtain a square matrix L(V) as θ.α = 0. More precisely, if Vi is the vertex-space of V atvertex υi (that is, Vi has dimension ei), then evaluating L at V gives a linear map

V⊕li+1i+1 ⊕ · · · ⊕ V ⊕ljj ⊕ V ⊕−ltj+1

j+1 ⊕ · · · ⊕ V ⊕−ltkk

L(V ) ↓V ⊕lt11 ⊕ · · · ⊕ V ⊕ltii ⊗ V ⊕li+1

i+1 ⊕ · · · ⊕ V ⊕ljj

and L(V) is a square N ×N matrix where

li+1 + · · ·+ lj − ltj+1 − · · · − ltk = N = lt1 + · · ·+ lti + li+1 + · · ·+ lj

So we can consider D(V ) = detL(V ) and verify that D is a GL(α)-semi-invariant poly-nomial on repα A of weight χlθ. The result of [44] asserts that these determinantal semi-invariants are algebra generators of the graded algebra

C[repα A]GL(α),θ

Observe that this result is to semi-invariants what the result of [31] is to invariants. In fact,one can deduce the latter from the first.

We have seen that a representation V ∈ repαA is θ-semistable if and only if somesemi-invariant of weight χlθ for some l is non-zero on it. This proves

Theorem 4.13. The Zariski open subset of θ-semistable α-dimensional A-representationscan be covered by affine GL(α)-stable open subsets

repθ−semistα A =⊔D

V |D(V ) = detL(V ) = 0

and hence the moduli space can also be covered by affine open subsets

moduliθα A =⋃D

XD

where XD = [V ] ∈ moduliθαA|D(V ) = detL(V ) = 0.

Example 4.14. In the C2/Z3 example, the θ-semistable representations


with θ = (−2, 1, 1) all lie in the affine open subset XD where L is a matrix of the form

L =[x1 0∗ y3

]

where ∗ is any path in Q starting in x1 and ending in x3.

4.6. The C[XD]- orders AD. Analogous to the rectangular matrix L we define a rectan-gular matrix N with

• lt1 + · · ·+ lti + li+1 + · · ·+ lj columns and• li+1 + · · ·+ lj − ltj+1 − · · · − ltk rows

N =

lt1︷︸︸︷ . . . lti︷︸︸︷ li+1︷︸︸︷ . . . lj︷︸︸︷li+1 Ni+1,1 Ni+1,i Ni+1,i+1 Ni+1,j

...lj Nj,1 Nj,i Nj,i+1 Nj,j

−ltj+1 Nj+1,1 Nj+1,iNj+1,i+1 Nj+1,j

...−ltk Nk,1 Nk,i Nk,i+1 Nk,j

filled with new variables and define an extended marked quiver Q∗D where we adjoin foreach entry in Nr,c an additional arrow from υc to υr and denote it with the correspondingvariable from N .

Let I1 (resp. I2 be the set of relations in CQ∗D determined from the matrix-equations

respectively

where (υi)njis the square nj × nj matrix with υi on the diagonal and zeroes elsewhere.

248 LIEVEN LE BRUYN

Define a new non-commutative order

AD =AαQ∗

D

I, I1, I2

then AD is a C[XD]-order in alg@n.

Example 4.15. In the setting of example 4.14 with ∗ = y3, the extended quiver-setting(QD, α) is

Hence, with

L =[x1 0y3 y3

]N =

[n1 n3

n2 n4

]

the defining equations of the order AD become

I = (x3y3 − y1x1, x1y1 − y2x2, x2y2 − y3x3)I1 = (n1x1 + n3y3 − υ1, n3y3, n2x1 + n4y3, n4y3 − υ1)I2 = (x1n1 − υ2, x1n3, y3n1 + y2n2, y3n3 + y3n4 − υ3)

note: This construction may seem a bit mysterious at first but what we are really doing isto construct the universal localization as in for example [43] associated to the map betweenprojective A-modules determined by L, but this time not in the category alg of all algebrasbut in alg@α.

That is, take Pi = υi A be the projective right ideal associated to vertex υi, then Ldetermines an A-module morphism

P = P⊕li+1i+1 ⊕ · · · ⊕ P⊕−ltkk

L−→ P⊕lt11 ⊕ · · · ⊕ P⊕ljj = Q

The algebra map Aφ−→AD is universal in alg@α with respect to L⊗φ being invertible, that

is, if Aψ−→S is a morphism in alg@α such that L⊗ψ is an isomorphism of right S-modules,

then there is a unique map in alg@α ADu−→S such that ψ = u φ.

The proof of the main result follows from the following result:


Theorem 4.16. The following statements are equivalent

1. V ∈ repθ−semistα A lies in π−1(XD), and2. There is a unique extension V of V such that V ∈ repα AD.

Proof. 1⇒ 2: Because L(V ) is invertible we can take N(V ) to be its inverse and decomposeit into blocks corresponding to the new arrows in Q∗D. This then defines the unique extensionV ∈ repα Q∗D of V . As V satisfies R (because V does) and R1 and R2 (because N(V ) =L(V )−1) we have that V ∈ repα AD.

2⇒ 1: Restrict V to the arrows of Q to get a V ∈ repα Q. As V (and hence V ) satisfiesR, V ∈ repα A. Moreover, V is such that L(V ) is invertible (this follows because V satisfiesR1 and R2). Hence, D(V ) = 0 and because D is a θ-semiinvariant it follows that V isan α-dimensional θ-semistable representation of A. An alternative method to see this is asfollows. Assume that V is not θ-semistable and let V ′ ⊂ V be a subrepresentation such thatθ.dimV ′ < 0. Consider the restriction of the linear map L(V ) to the subrepresentation V ′

and look at the commuting diagram

As θ.dimV ′ < 0 the top-map must have a kernel which is clearly absurd as we know thatL(V ) is invertible.

Example 4.17. In the setting of example 4.14 with ∗ = y3 we have that the uniquelydetermined extension of the A-representation

Observe that this extension is a simple AD-representation for every b, c ∈ C.

4.7. Non-commutative desingularizations. There is just one more thing to clarify: howare the different AD’s glued together to form a sheaf A of non-commutative algebras overmoduliθαA and how can we construct the non-commutative variety spec A? The solutionto both problems follows from the universal property of AD.

Let AD1 (resp. AD2) be the algebra constructed from a rectangular matrix L1 (resp.L2), then we can construct the direct sum map L = L1 ⊕ L2 for which the corresponding

250 LIEVEN LE BRUYN

semi-invariantD = D1D2. As A −→ AD makes the projective module morphisms associatedto L1 and L2 an isomorphism we have uniquely determined maps in alg@α

As repαAD = π−1(XD) (and similarly for Di) we have that i∗j are embeddings as arethe ij . This way we can glue the sections Γ(XD1 ,A) = AD1 with Γ(XD2 ,A) = AD2 overtheir intersection XD = XD1 ∩XD2 via the inclusions ij . Hence we get a coherent sheaf ofnon-commutative algebras A over moduliθαA.

Observe that many of the orders AD are isomorphic. In example 4.14 all matrices L withfixed diagonal entries x1 and y3 but with varying ∗-entry have isomorphic orders AD (usethe universal property).

In a similar way we would like to glue max AD1 with max AD2 over max AD using thealgebra maps ij to form a non-commutative variety spec A. However, the construction ofmax A and the non-commutative structure sheaf is not functorial in general.

Example 4.18. Consider the inclusion map map in alg@2

A =[R RI R

]→[R RR R

]= A′

then all twosided maximal ideals of A′ are of the form M2(m) where m is a maximal idealof R. If I ⊂ m then the intersection

[m mm m

]∩[R RI R

]=[m mI m

]

which is not a maximal ideal of A as[m RI R

] [R RI m

]=[m mI m

]

and so there is no natural map max A′ −→ max A, let alone a continuous one.

note: Associating to a non-commutative algebra A its prime ideal spectrum spec A is onlyfunctorial for extensions A

f−→B, that is, satisfying

B = f(A)ZB(A) with ZB(A) = b ∈ B|bf(a) = f(a)b∀a ∈ A

In [38] it was proved that if Af−→ B is an extension then the map

spec B −→ spec R P −→ f−1(P )

is well-defined and continuous for the Zariski topology.


Fortunately, in the case of interest to us, that is for the maps ij : ADj−→ AD this

presents no problem as they are even central extensions, that is

AD = ADjZ(AD)

which follows again from the universal property by localizing ADjat the central element

D. Hence, we can define a genuine non-commutative variety spec A with central schememoduliθαA, finishing the proof of the main result and these talks.

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Bartels, Hans-Jochen ...........................................1Burban, Igor.......................................................23

Detomi, Eloisa ...................................................47Drozd, Yuriy.......................................................23

Elder, G. Griffith................................................63Eriksen, Eivind ..................................................90

Hazewinkel, Michiel........................................126

Le Bruyn, Lieven.............................................203Lucchini, Andrea ...............................................47

Malinin, D.A........................................................1

Ndirahisha, Janvière ........................................147

Petit, Toukaiddine............................................162

Rashkova, Tsetska Grigoriva ...........................175Rump, Wolfgang..............................................184

Van Oystaeyen, Freddy....................................147

253

Author Index

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DK3043_BM 6/17/05 8:21 AM Page 254

[corrado de concini, freddy van oystaeyen, nikolai(bookfi.org)

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