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UNIVERSIT A DEGLI STUDI DI GENOVA

FACOLTA DI SCIENZE MATEMATICHE FISICHE E NATURALI

Dipartimento di Matematica

Cohomologicaland Combinatorial Methods in the Studyof Symbolic Powersand Equations defining Varieties

Tesi per il conseguimento delDottorato di Ricerca in Matematica e Applicazioni, ciclo XXIII

Sede amministrativa: Universita di Genova

Settore Scientifico Disciplinare: MAT 02 - Algebra

AutoreMatteo VARBARO

Supervisore CoordinatoreProf. Aldo CONCA, Prof. Michele PIANA

Universita di Genova Universita di Genova

http://de.arxiv.org/abs/1105.5507v1

Contents

Introduction v

Notation ix

A quick survey of local cohomology xi0.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi0.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xii0.3 Equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . . . .xiii

0.3.1 Via Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiii0.3.2 Via theCech complex . . . . . . . . . . . . . . . . . . . . .xiv

0.4 Why is local cohomology interesting? . . . . . . . . . . . . . . . .. xvi0.4.1 Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xvi0.4.2 Cohomological dimension and arithmetical rank . . . . .. . xvi0.4.3 The graded setting . . . . . . . . . . . . . . . . . . . . . . .xviii

1 Cohomological Dimension 11.1 Necessary conditions for the vanishing of local cohomology . . . . . 3

1.1.1 Cohomological dimension vs connectedness . . . . . . . . .. 41.1.2 Noncomplete case . . . . . . . . . . . . . . . . . . . . . . . 71.1.3 Cohomological dimension and connectedness of open subschemes

of projective schemes . . . . . . . . . . . . . . . . . . . . . . 81.1.4 Discussion about the sufficiency for the vanishing of local co-

homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Sufficient conditions for the vanishing of local cohomology . . . . . . 11

1.2.1 Open subschemes of the projective spaces over a field ofchar-acteristic 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.2 Etale cohomological dimension . . . . . . . . . . . . . . . .151.2.3 Two consequences . . . . . . . . . . . . . . . . . . . . . . .171.2.4 Translation into a more algebraic language . . . . . . . . .. 20

2 Properties preserved under Grobner deformations and arithmetical rank 232.1 Grobner deformations . . . . . . . . . . . . . . . . . . . . . . . . . .25

2.1.1 Connectedness preserves under Grobner deformations . . . . 252.1.2 The Eisenbud-Goto conjecture . . . . . . . . . . . . . . . . .272.1.3 The initial ideal of a Cohen-Macaulay ideal . . . . . . . . .. 30

2.2 The defining equations of certain varieties . . . . . . . . . . .. . . . 322.2.1 Notation and first remarks . . . . . . . . . . . . . . . . . . .322.2.2 Lower bounds for the number of defining equations . . . . .. 34

iv CONTENTS

2.2.3 Hypersurfaces cross a projective spaces . . . . . . . . . . .. 352.2.4 The diagonal of the product of two projective spaces . .. . . 42

3 Relations between Minors 433.1 Some relations for the defining ideal ofAt . . . . . . . . . . . . . . . 45

3.1.1 Make explicit the shape relations . . . . . . . . . . . . . . . .493.2 Upper bounds on the degrees of minimal relations among minors . . . 51

3.2.1 Tha Castelnuovo-Mumford regularity of the algebra ofminors 523.2.2 The independence onn for the minimal relations . . . . . . . 563.2.3 Relations between 2-minors of a 3×n and a 4×n matrix . . . 573.2.4 The uniqueness of the shape relations . . . . . . . . . . . . .58

3.3 The initial algebra and the algebra ofU-invariants ofAt . . . . . . . . 613.3.1 A finite Sagbi basis ofAt . . . . . . . . . . . . . . . . . . . . 613.3.2 A finite system of generators forAU

t . . . . . . . . . . . . . . 63

4 Symbolic powers and Matroids 654.1 Towards the proof of the main result . . . . . . . . . . . . . . . . . .66

4.1.1 The algebra of basick-covers and its dimension . . . . . . . . 684.1.2 If-part of Theorem4.1.6 . . . . . . . . . . . . . . . . . . . . 704.1.3 Only if-part of Theorem4.1.6 . . . . . . . . . . . . . . . . . 72

4.2 Two consequences . . . . . . . . . . . . . . . . . . . . . . . . . . .754.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76

A Other cohomology theories 79A.1 Sheaf cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . .79A.2 GAGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.3 Singular homology and cohomology . . . . . . . . . . . . . . . . . .83A.4 Algebraic De Rham cohomology . . . . . . . . . . . . . . . . . . . .84

B Connectedness in noetherian topological spaces 85B.1 Depth and connectedness . . . . . . . . . . . . . . . . . . . . . . . .88

C Grobner deformations 91C.1 Initial objects with respect to monomial orders . . . . . . .. . . . . . 91C.2 Initial objects with respect to weights . . . . . . . . . . . . . .. . . 92C.3 Some properties of the homogenization . . . . . . . . . . . . . . .. 93

D Some facts of representation theory 95D.1 General facts on representations of group . . . . . . . . . . . .. . . 95D.2 Representation theory of the general linear group . . . . .. . . . . . 97

D.2.1 Schur modules . . . . . . . . . . . . . . . . . . . . . . . . .99D.2.2 Plethysms . . . . . . . . . . . . . . . . . . . . . . . . . . . .102

D.3 Minors of a matrix and representations . . . . . . . . . . . . . . .. . 103

E Combinatorial commutative algebra 107E.1 Symbolic powers . . . . . . . . . . . . . . . . . . . . . . . . . . . .108E.2 Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109E.3 Polarization and distractions . . . . . . . . . . . . . . . . . . . . .. 109

Bibliography 115

Introduction

In this thesis we will discuss some aspects of Commutative Algebra, which have in-teractions with Algebraic Geometry, Representation Theory and Combinatorics. Inparticular, we will use local cohomology, combinatorial methods and the representa-tion theory of the general linear group to study symbolic powers of monomial ideals,connectedness properties, arithmetical rank and the defining equations of certain va-rieties. Furthermore, we will focus on understanding when certain local cohomologymodules vanish. The heart of the thesis is in Chapters1, 2, 3 and4. They are based onpapers published by the author, but each chapter contains some results never appearedanywhere.

Chapter1 is an outcome of our papers [103, 104]. The aim of this chapter is toinquire on a question raised by Grothendieck in his 1961 seminar on local cohomologyat Harvard (see the notes written by Hartshorne [51]). He asked to find conditions onan ideala of a ring R to locate the top-nonvanishing local cohomology module withsupport ina, that is to compute the cohomological dimension ofa. We show, undersome additional assumptions on the ringR, that the vanishing ofH i

a(R) for i > c forcessome connectedness properties, depending onc, of the topological space Spec(R/a),generalizing earlier results of Grothendieck [47] and of Hochster and Huneke [63].WhenR is a polynomial ring over a field of characteristic 0 anda is a homogeneousideal such thatR/a has an isolated singularity, we give a criterion to compute thecohomological dimension ofa, building ideas from Ogus [87]. Such a criterion leadsto the proof of:

(i) A relationship between cohomological dimension ofa and depth ofR/a, whichis a small-dimensional version of a result of Peskine and Szpiro [89] in positivecharacteristic.

(ii) A remarkable case of a conjecture stated in [77] by Lyubeznik, concerning aninequality about the cohomological dimension and the etale cohomological di-mension of an open subscheme ofPn.

In order to show these results we use also ideas involving Complex Analysis and Al-gebraic Topology.

In Chapter2 we show some applications of the results gotten in the first chapter, us-ing interactions between Commutative Algebra, Combinatorics and Algebraic Geom-etry. It shapes, once again, on our papers [103, 104]. In the first half of the chapter wegeneralize a theorem of Kalkbrener and Sturmfels obtained in [67]. In simple terms, weshow that Grobner deformations preserve connectedness properties: As a consequence,we get that ifS/I is Cohen-Macaulay, then∆(

√in(I)) is a strongly connected simpli-

cial complex. This fact induced us to investigate on the question whether∆(in(I)) isCohen-Macaulay provided thatS/I is Cohen-Macaulay and in(I) is square-free, sup-

vi Introduction

plying some evidences for a positive answer to this question. Another consequence isthe settlement of the Eisenbud-Goto conjecture, see [37], for a new class of ideals. Inthe second half of the chapter, we compute the number of equations needed to cut outset-theoretically some algebraic varieties. For an overview on this important problemwe refer the reader to Lyubeznik [74]. The needed lower bounds for this number usu-ally come from cohomological arguments: This is our case, infact we get them fromthe results of the first chapter. For what concerns upper bounds, there is no generalmethod, rather one has to invent ad hoc tricks, depending on the kind of varieties onedeals with. For the varieties we are interested in, we get ideas from the Theory ofAlgebras with Straightening Laws, exhibiting the necessary equations. Among otherthings, we extend a result of Singh and Walther [96].

Chapter3 is based on a preprint in preparation with Bruns and Conca, namely [21].This time, beyond Commutative Algebra, Algebraic Geometryand Combinatorics, oneof the primary roles is dedicated to Representation Theory.We investigate on a prob-lem raised by Bruns and Conca in [18]: To find out the minimal relations between2-minors of a genericm×n-matrix. Whenm= 2 (without loss of generality we canassumem≤ n), such relations define the Grassmannian of all the 2-dimensional sub-spaces of ann-dimensional vector space. It is well known that the demanded equationsare, in this case, the celebrated Plucker relations. In particular, they are quadratic. Dif-ferently, apart from the casem= n= 3, we show that there are always both quadraticand cubic minimal relations. We describe them in terms of Young tableux, exploit-ing the natural action of the group GL(W)×GL(V) on X, the variety defined by therelations (hereW andV are vector spaces, respectively, of dimensionm andn). Fur-thermore, we show that these are the only minimal relations whenm= 3,4, and wegive some reasons to believe that this is true also for higherm. Moreover, we computethe Castelnuovo-Mumford regularity ofX, we exhibit a finite Sagbi bases associatedto a toric deformation ofX (solving a problem left open by Bruns and Conca in [19])and we show a finite system of generators of the coordinate ring of the subvariety ofU-invariants ofX, settling the “first main problem” of invariant theory in this case. Themost of the results are proved more in general, consideringt-minors of anm×n-matrix.

Chapter4 concerns our paper [105]. The aim is to compare algebraic propertieswith combinatorial ones. The main result is quite surprising: We prove that the Cohen-Macaulayness of all the symbolic powers of a Stanley-Reisner ideal is equivalent to thefact that the related simplicial complex is a matroid! The beauty of this result is that theconcepts of “Cohen-Macaulay” and of “matroid”, a priori unrelated to each other, areboth fundamental, respectively, in Commutative Algebra and in Combinatorics. An-other interesting fact is that the proof is not direct, passing through the study of thesymbolic fiber cone of the Stanley-Reisner ideal. It is fair to say that the same resulthas been proven independently and with different methods byMinh and Trung [84].Actually, here we prove the above result for a class of idealsmore general than thesquare-free ones, generalizing the result of our paper and of [84]. As a consequence,we show that the Stanley-Reisner ideal of a matroid is a set-theoretic complete inter-section after localizing at the maximal irrelevant ideal. We end the chapter presenting astrategy, through an example, to produce non-Cohen-Macaulay square-free monomialideals of codimension 2 whose symbolic power’s depth is constant. Here the crucialfact is that in the codimension 2-case the structure of the mentioned fiber cone has beenunderstood better thanks to our work with Constantinescu [25].

vii

In addition to the four chapters described above, we decidedto include a prelimi-nary chapter entitled “A quick survey of local cohomology”.It is a collection of knownresults on local cohomology, the first topic studied at a certain level by the author. Lo-cal cohomology appears throughout the thesis in some more orless evident form, sowe think that such a preliminary chapter will be useful to thereader. We added alsofive appendixes at the end of the thesis, which should make thetask of searching forreferences less burdensome.

Acknowledgements

First of all, I wish to thank my advisor, Aldo Conca. Some years ago, I decided thatI wanted to do math in life. I started my PhD three years ago and, for reasons notonly related to mathematics, I did it in Genova. In retrospect it could not get any better.Aldo was always able to stimulate me with new problems, giving me, most of the times,lighting suggestions. He let me do all the experiences that might have been useful tomy intellectual growth. And he always granted me a certain freedom in choosing myresearch topics, something that not all the advisors are willing to do. So, I want to thankhim for all these reasons, hoping, and being confident of it, to have the opportunity tocollaborate with him again in the future.

During these three years (actually a little more) I attendedconferences, seminarsand summer schools in different institutes and universities around the world, where Imet many commutative algebraists. I learned that, even if I don’t know where, as longas I do math I’ll have a coffee with each of them at least once every year. I find thatthere is something poetic in this! Apart from the coffee, with some of these personsI have also had useful discussions about mathematics, and I am grateful to them forthis reason. Special thanks are due to Winfried Bruns, who I met in several occasions.Especially, I spent the spring of 2009 in Osnabruck, to learn from him. He taught me alot of mathematics, and from such a visit came a close collaboration, which led to theresults of Chapter3. Besides, he patiently helped me for my stay there, making suchan experience possible for me. Another professor who deserves a special mention, forhaving taken the trouble to proofread the present thesis, isAnurag Singh.

I want to thank also four guys that, besides sharing with me pleasant conversationsabout mathematics, which led to the works [6, 94, 7, 25, 85, 26], I consider dear friends.I am talking about Le Dinh Nam, Alexandru Constantinescu, Bruno Benedetti andLeila Sharifan.

I’d like to say thank you to my parents, Annalisa and Rocco. Not just becausethey are my parents, which of course would be enough by itself. Somehow, they reallyhelped me in my work, dealing with a lot of bureaucratic stufffor me: This way theyexempted me from facing my “personal enemy number 1”! In the same manner, thanksto my brother, Roberto. He has always instilled me a sort of serenity, which is anessential ingredient for doing research.

Eventually, let me thank all the diploma students, PhD students, PostDoc students,professors, secretaries and technical assistants of the Department of Mathematics of theUniversity of Genova, namely the DIMA, who came positively into my life. It wouldbe nice to thank them one by one, but it would be hard as well: Sofar, I have beenhaunting the DIMA for more than eight years, and in this time Imet a lot of people.Many of them belong to the so-called “bella gente”, whose meaning is “people whosecompany is good”. I am lucky to say that they are too many to be mentioned all.

viii Introduction

Notation

Throughout the thesis we will keep notation more or less faithful to some textbooks:(i) For what concerns Commutative Algebra: Matsumura [80], Atiyah and Macdon-

ald [3] and Bruns and Herzog [13].(ii) For what concerns Algebraic Geometry: Hartshorne [56].(iii) For what concerns Algebraic Combinatorics: Stanley [100], Miller and Sturm-

fels [81] and Bruns and Herzog [13, Chapter 5].(iv) For what concerns Representation Theory: Fulton and Harris [43], Fulton [42]

and Procesi [90].

• We will often say “a ringR” instead of “a commutative, unitary and Noetherianring R”. Whenever some of these assumptions fail, we will explicitly remark it.

• The set of natural numbers will include 0, i.e.N= 0,1,2,3, . . ..

• An ideala of Rwill always be different fromR.

• If R is a local ring with maximal idealm, we will write (R,m) or (R,m,κ), whereκ = R/m.

• Given a ringR, an R-moduleM and a multiplicative systemT ⊆ R, we willdenote byT−1M the localization ofM at T. If ℘ is a prime ideal ofR, we willwrite M℘ instead of(R\℘)−1M.

• Given a ringR, an ideala ⊆ R and anR-moduleM, we will denote byMa thecompletion ofM with respect to thea-adic topology. If(R,m) is local we willjust writeM instead ofMm.

• The spectrum of a ringR is the topological space

Spec(R) := ℘⊆ R : ℘ is a prime ideal

supplied with the Zariski topology. I.e. the closed sets areV (a) := ℘ ∈Spec(R) : ℘⊇ a with a varying among the ideals ofR andR itself.

• Given a ringR and anR-moduleM, the support ofM is the subset of Spec(R)

Supp(M) := ℘∈ Spec(R) : M℘ 6= 0.

• We say that a ringR is graded if it isN-graded, i.e. if there is a decompositionR= ⊕i∈NRi such that eachRi is anR0-module andRiRj ⊆ Ri+ j . The irrelevantideal ofR is R+ := ⊕i>0Ri. An elementx of R has degreei if x∈ Ri . Therefore0 has degreei for anyi ∈ N.

x Notation

• The projective spectrum of a graded ringR is the topological space

Proj(R) := ℘⊆ R : ℘ is a homogeneous prime ideal not containingR+,

where the closed sets areV+(a) := ℘∈Proj(R) : ℘⊇ a with a varying amongthe homogeneous ideals ofR.

• When we speak about the grading on the polynomial ringS:= k[x1, . . . ,xn] in nvariables over a fieldk we always mean the standard grading, unless differentlyspecified: I.e. deg(x1) = deg(x2) = . . .= deg(xn) = 1.

• For a posetΠ := (P,≤) we mean a setP endowed with a partial order≤.

A quick survey of localcohomology

This is a preliminaries chapter, which contains no originalresults. Our aim is just tocollect some facts about local cohomology we will use throughout the thesis. This isa fascinating subject, essentially introduced by Grothendieck during his 1961 HarvardUniversity seminar, whose notes have been written by Hartshorne in [51]. Besides it,other references we will make use of are the book of Brodmann and Sharp [10], theone of Iyengar et al. [66] and the notes written by Huneke and Taylor [65].

0.1 Definition

Let R be a commutative, unitary and Noetherian ring. Given an ideal a ⊆ R, we canconsider thea-torsion functorΓa, from the category ofR-modules to itself, namely

Γa(M) :=⋃

n∈N(0 :M an) := m∈ M : there existsn∈ N for whichanm= 0,

for anyR-moduleM. ClearlyΓa(M) ⊆ M. Furthermore ifφ : M → N is a homomor-phism ofR-modules thenφ(Γa(M)) ⊆ Γa(N). SoΓa(φ) will simply be the restrictionof φ to Γa(M). It is easy to see that the functorΓa is left exact. Thus, for anyi ∈ N,we denote byH i

a theith derived functor ofΓa, which will be referred to as theith localcohomology functor with support ina. For anR-moduleM, we will refer toH i

a(M) asthe ith local cohomology module of M with support ina . The following remark is asimportant as elementary.

Remark 0.1.1. If a andb are ideals ofRsuch that√a=

√b, thenΓa = Γb. Therefore

H ia = H i

bfor everyi ∈ N.

Example 0.1.2.We want to compute the local cohomology modules ofZ as aZ-modulewith respect to the ideal(d)⊆ Z, where d is a nonzero integer. An injective resolutionofZ is:

0−→ Z ι−→Q π−→Q/Z−→ 0.

From the above resolution, we immediately realize that Hi(d)(Z) = 0 for any i≥ 2.

Furthermore H0(d)(Z) = Γ(d)(Z) = 0. So we have just to compute H1

(d)(Z). To this aimlet us applyΓ(d) to the above exact sequence, getting:

0−→ Γ(d)(Z)Γ(d)(ι)−−−−→ Γ(d)(Q)

Γ(d)(π)−−−−→ Γ(d)(Q/Z)−→ 0.

xii A quick survey of local cohomology

By definition H1(d)(Z) = Γ(d)(Q/Z)/Γ(d)(π)(Γ(d)(Q)). SinceΓ(d)(Q) = 0, we have

H1(d)(Z) = Γ(d)(Q/Z). Finally, Γ(d)(Q/Z) consists in all elementss/t ∈ Q/Z with

s, t ∈ Z and t dividing dn for some n∈ N. In particular, notice that H1(d)(Z) is not

finitely generated as aZ-module.

It is worthwhile to remark that from the definition of local cohomology we get thefollowing powerful instrument: Any short exact sequence ofR-modules

0−→ P−→ M −→ N −→ 0

yields the long exact sequence ofR-modules:

0→ H0a(P)−→ H0

a(M)−→ H0a(N)−→ H1

a(P)−→ H1a(M)→ . . .

. . .→ H i−1a (N)−→ H i

a(P)−→ H ia(M)−→ H i

a(N) −→ H i+1a (P)→ . . . (0.1)

In Example0.1.2we hadH i(d)(Z) = 0 for eachi > 1= dimZ. Actually Grothendi-

eck showed that this is a general fact (for instance see [10, Theorem 6.1.2]):

Theorem 0.1.3. (Grothendieck) Let R anda be as above. For any R-module M, wehave Hi

a(M) = 0 for all i > dimM.

Anyway, we should say that in Exemple0.1.2we were in a special case. In fact, anR-moduleM has a finite injective resolution if and only ifM is Gorenstein.

0.2 Basic properties

In addition to the long exact sequence (0.1), we want to give other three basic toolswhich are essential in working with local cohomology. The first property we want to listis that local cohomology commutes with direct limits (for example see [10, Theorem3.4.10]).

Lemma 0.2.1. Let R anda be as usual. LetΠ be a poset andMα : α ∈ Π a directsystem overΠ of R-modules. Then

H ia(lim−→Mα)∼= lim−→H i

a(Mα ) ∀ i ∈N.

As an immediate consequence of Lemma0.2.1, we have that

H ia(M⊕N)∼= H i

a(M)⊕H ia(N)

for any i ∈ N and for anyR-modulesM andN.The second property we would like to state allows us, in many cases, to control

local cohomology when the base ring changes. The proofs can be found, for instance,in [10, Theorem 4.3.2, Theorem 4.2.1].

Lemma 0.2.2. Let R anda be as above, M an R-module, Rφ−→ S a homomorphism of

Noetherian rings and N an S-module.(i) (Flat Base Change) Ifφ is flat, then Hi

φ(a)S(M⊗RS)∼=H ia(M)⊗RS for any i∈N.

The above isomorphism is functorial.(ii) (Independence of the Base) Hi

a(N) ∼= H iφ(a)S(N) for any i∈ N, where the first

local cohomology is computed over the base field R and the second over S. Theabove isomorphism is functorial.

0.3 Equivalent definitions xiii

Lemma0.2.2has some very useful consequences: Leta andb be ideals ofR, M anR-module andT ⊆ R a multiplicative system. Then, for anyi ∈ N, there are functorialisomorphisms

T−1H ia(M) ∼= H i

aT−1R(T−1M) (0.2)

H ia(M)⊗RRb ∼= H i

aRb(M⊗RRb) (0.3)

Furthermore, ifR is local, then its completionR is faithfully flat overR. So, for anyi ∈ N,

H ia(M) = 0 ⇐⇒ H i

aR(M⊗RR) = 0. (0.4)

The last basic tool we want to present in this section is theMayer-Vietories se-quence, for instance see [10, 3.2.3]. Sometimes, it allows us to handle with the supportof the local cohomology modules. Leta andb be ideals ofR andM an R-module.There is the following exact sequence:

0→ H0a+b

(M)→ H0a(M)⊕H0

b(M)→ H0

a∩b(M)→ H1a+b(M)→ . . .

. . .→ H ia+b

(M)→ H ia(M)⊕H i

b(M)→ H i

a∩b(M)→ H i+1a+b

(M)→ . . . (0.5)

0.3 Equivalent definitions

In this section we want to introduce two different, but equivalent, definitions of localcohomology. Depending on the issue one has to deal with, among these definitions onecan be more suitable than the other.

0.3.1 Via Ext

Let R anda be as in the previous section. Ifn andm are two positive integers suchthatn≥ m, then there is a well defined homomorphism ofR-modulesR/an −→ R/am.Given anR-moduleM, the above map yields a homomorphism HomR(R/am,M) −→HomR(R/an,M), and more generally, for alli ∈ N,

ExtiR(R/am,M)−→ ExtiR(R/a

n,M).

The above maps clearly yield a direct system overN, namelyExtiR(R/an,M) : n∈N.

So, we can form the direct limit lim−→ExtiR(R/an,M). Since HomR(R/an,M) ∼= 0 :M an,

we get the isomorphism:

H0a(M) = Γa(M) ∼= lim−→HomR(R/a

n,M).

More generally the following theorem holds true (for instance see [10, Theorem 1.3.8]):

Theorem 0.3.1.Let R anda be as above. For any R-module M and i∈ N, we have

H ia(M)∼= lim−→ExtiR(R/a

n,M).

Furthermore, the above isomorphism is functorial.

xiv A quick survey of local cohomology

Actually, Theorem0.3.1can be stated as

H ia(M) ∼= lim−→ExtiR(R/an,M), (0.6)

where(an)n∈N is an inverse system of ideals cofinal with(an)n∈N, i.e. for anyn∈ Nthere existk,m∈ N such thatak ⊆ an andam ⊆ an. Several inverse systems of idealscofinal with (an)n∈N have been used by many authors together with (0.6) to proveimportant theorems. Let us list three examples:

(i) If R is a complete local domain and℘ is a prime ideal such that dimR/℘= 1,one can show that the inverse system of symbolic powers(℘(n))n∈N is cofinalwith (℘n)n∈N. On this fact is based the proof of the Hartshorne-LichtenbaumVanishing Theorem [52], which essentially says the following:

Theorem 0.3.2. (Hartshorne-Lichtenbaum) Let R be a d-dimensional completelocal domain anda⊆ R an ideal. The following are equivalent:

(a) Hda(R) = 0;

(b) dimR/a> 0.

(ii) In the study of the local cohomology modules in characteristic p> 0 a very pow-erful tool is represented by the inverse system(a[p

n])n∈N of Frobenius powers ofa: The ideala[p

n] is generated by thepnth powers of the elements ofa. Theabove inverse system played a central role in the works of Peskine and Szpiro[89], Hartshorne and Speiser [57] and Lyubeznik [78]. For instance, it was cru-cial in the proof of the following result of Peskine and Szpiro:

Theorem 0.3.3. (Peskine-Szpiro) Let R be a regular local ring of positive char-acteristic anda⊆ R an ideal. Then

H ia(R) = 0 ∀ i > projdim(R/a).

(iii) The last example of inverse system we want to show was used by Lyubeznik in[73] when R is a polynomial ring anda is a monomial ideal. The (pretended)Frobeniusnth power ofa, denoted bya[n], is the ideal generated by thenth pow-ers of the monomials ina. The inverse system(a[n])n∈N is obviously cofinal with(an)n∈N. It was the fundamental ingredient in the proof of the following result:

Theorem 0.3.4.(Lyubeznik) Let R be a polynomial ring anda⊆ R be a square-free monomial ideal. Then

H ia(R) = 0 ∀ i > projdim(R/a) and Hprojdim(R/a)

a (R) 6= 0.

0.3.2 Via theCech complex

Let a= a1, . . . ,ak be elements ofR. TheCech complex of R with respect toa , C(a)•,is the complex ofR-modules

0d−1

−−→C(a)0 d0

−→C(a)1 → . . .→C(a)k−1 dk−1

−−→C(a)k dk

−→ 0,

where:(i) C(a)0 = R;

0.3 Equivalent definitions xv

(ii) For p= 1, . . . ,k, theR-moduleC(a)p is⊕

1≤i1<i2<...<ip≤k

Rai1···ai p

;

(iii) The composition

C(a)0 d0

−→C(a)1 −→ Rai

has to be the natural map fromR to Rai ;(iv) The mapsdp are defined, forp= 1, . . . ,k−1, as follows: The composition

Rai1···ai p

−→C(a)p dp

−→C(a)p+1 −→ Ra j1···a j p+1

has to be(−1)s−1 times the natural map fromRai1···ai p

to Ra j1···a j p+1

whenever

(i1, . . . , ip) = ( j1, . . . , js, . . . , jp+1); it has to be 0 otherwise.It is straightforward to check that theCech complex is actually a complex. Thus we

can define the cohomology groupsH i(C(a)•) = Ker(di)/Im(di−1). For anyR-moduleM we can also defineC(a,M)• =C(a)•⊗RM. We have the following result (see [10,Theorem 5.1.19]):

Theorem 0.3.5.Leta= (a1, . . . ,ak) be an ideal of R. For any R-module M and i∈ N,we have

H ia(M) ∼= H i(C(a,M)•).

Furthermore, the above isomorphism is functorial.

Probably theCech complex supplies the “easiest” way to compute the localcoho-mology modules. Anyway, even the computation of the local cohomology modules ofthe polynomial ring with respect to the maximal irrelevant ideal requires some effort,as the following example shows.

Example 0.3.6. Let S= k[x,y] be the polynomial ring on two variables over a fieldk, a1 = x and a2 = y. Obviously H0

(x,y)(S) = 0. We want to compute H1(x,y)(S) and

H2(x,y)(S). In fact, we know by Theorem0.1.3that Hi

(x,y)(S) = 0 for any i> 2= dimS.

TheCech complex C(a)• is

0→ Sd0−→ Sx⊕Sy

d1−→ Sxy

d2−→ 0.

where the maps are:(i) d0( f ) = ( f/1, f/1);(ii) d1( f/xn,g/ym) = xmg/(xy)m− yn f/(xy)n.

One can easily realize that d1( f/xn,g/ym) = 0 if and only if xng= ym f . Since xn,ym

is a regular sequence, d1( f/xn,g/xm) = 0 if and only if there exists h∈ S such thatf = xnh and g= ymh. This is the case if and only if d0(h) = ( f/xn,g/ym). ThereforeKer(d1) = Im(d0) and Theorem0.3.5implies H1

(x,y)(S) = H1(C(a)•) = 0.Notice that Sxy is generated by xnym as an S-module, where n and m vary inZ.

Furthermore, if n≥ 0, then xnym = d1(0,xnym). Analogously, if m≥ 0, then xnym =d1(−ymxn,0). Moreover, if both n and m are negative, then it is clear that xnym does notbelong to the image of d1. Therefore, using Theorem0.3.5, H2

(x,y)(S)∼= H2(C(a)•) =

Sxy/Im(d1) is the S-module generated by the elements x−ny−m with both n and m biggerthan0 supplied with the following multiplication:

(xpyq) · (x−ny−m) =

xp−nyq−m if p < n and q< m,

0 otherwise

xvi A quick survey of local cohomology

What done in Example0.3.6can be generalized to compute the local cohomologymodulesH i

(x1,...,xn)(k[x1, . . . ,xn]).

0.4 Why is local cohomology interesting?

Local cohomology is an interesting subject on its own. However, it is also a valuabletool to face many algebraic and geometric problems. In this section, we list somesubjects in which local cohomology can give a hand.

0.4.1 Depth

Let Randa be as usual andM be a finitely generatedR-module such thataM 6= M. Werecall that

grade(a,M) := mini : ExtiR(R/a,M) 6= 0.By the celebrated theorem of Rees (for instance see the book of Bruns and Herzog[13, Theorem 1.2.5]), grade(a,M) is the lenght of any maximal regular sequence onMconsisting of elements ofa. On the other hand, ifa1, . . . ,ak is a regular sequence onM,thenan

1, . . . ,ank is a regular sequence onM as well. So, (0.6) implies thatH i

a(M) = 0for any i < grade(a,M). Actually, it is not difficult to show the following (for instancesee [10, Theorem 6.2.7]):

grade(a,M) = mini : H ia(M) 6= 0. (0.7)

In particular, ifR is local with maximal idealm, then

depthM = mini : H im(M) 6= 0. (0.8)

Therefore, one can read off the depth of a module by the local cohomology.

0.4.2 Cohomological dimension and arithmetical rank

Thecohomological dimensionof an ideala⊆ R is the natural number

cd(R,a) := supi ∈ N : H ia(R) 6= 0

By Theorem0.1.3we have cd(R,a)≤ dimR.

Remark 0.4.1. If cd(R,a) = s then for anyR-moduleM we haveH ia(M) = 0 for all

i > s. To see this, by contradiction suppose that there existsc> s and anR-moduleMwith Hc

a(M) 6= 0. By Theorem0.1.3, we havec ≤ dimR, so we can assume thatc ismaximal (considering all theR-modules). Consider an exact sequence

0−→ K −→ F −→ M −→ 0,

whereF is a freeR-module. The above exact sequence yields the long exact sequence

. . .−→ Hca(F)−→ Hc

a(M) −→ Hc+1a (K)−→ . . . .

By the maximality ofc it follows thatHc+1a (K) = 0. FurthermoreHc

a(F) = Hca(R) = 0

because local cohomology commutes with the direct limits (Lemma0.2.1). So, we geta contradiction.

0.4 Why is local cohomology interesting? xvii

Thearithmetical rankof an ideala⊆ R is the natural number

ara(a) := mink : ∃ r1, . . . , rk ∈ R such that√(r1, . . . , rk) =

√a.

If R= k[x1, . . . ,xn] is the polynomial ring over an algebraically closed fieldk, by Null-stellensatz ara(a) is the minimal natural numberk such that

Z (a) = H1∩H2∩ . . .∩Hk

set-theoretically, whereZ (·) means the zero-locus of an ideal and theH j ’s are hyper-surfaces ofAn

k. From now on, anyway, we will consider varieties just in the schematicsense: For instance,An

k = Spec(k[x1, . . . ,xn]), V (a) = ℘∈ Ank : ℘⊇ a and for

H ⊆ Ank to be a hypersurface we mean thatH = V (( f )), where f ∈ k[x1, . . . ,xn]. In

this sense we have that, even ifk is not algebraically closed, ara(a) is the minimalnatural numberk such that

V (a) = H1∩H2∩ . . .∩Hk,

where theH j ’s are hypersurfaces ofAnk. Actually, the same statement is true whenever

R is a UFD.If R is graded anda homogeneous we can also define thehomogeneous arithmetical

rank, namely:

arah(a) := mink : ∃ r1, . . . , rk ∈ R homogeneous such that√a=

√(r1, . . . , rk).

Obviously we haveara(a)≤ arah(a).

If R= k[x0, . . . ,xn] is the polynomial ring over a fieldk, then arah(a) is the minimalnatural numberk such that

V+(a) = H1∩H2∩ . . .∩Hk

set-theoretically, where theH j ’s are hypersurfaces ofPnk. It is an open problem whether

ara(a) = arah(a) in this case (see the survey article of Lyubeznik [74]).

Remark 0.4.2. The reader should be careful to the following: If the base field is notalgebraically closed, the number ara(a), wherea is an ideal of a polynomial ring, mightnot give the minimal number of polynomials whose zero-locusis Z (a). For instance,if a= ( f1, . . . , fm)⊆ R[x1, . . . ,xn], clearly

Z (a) = Z ( f 21 + . . .+ f 2

m).

However ara(a) can be bigger than 1 (for instance by (0.9)).

From what said up to now the arithmetical rank has a great geometrical interest.Intuitively, everybody would say that there is no hope to define a curve ofA4 as theintersection of 2 hypersurfaces. Actually this intuition is a consequence of the cele-brated Krull’s Hauptidealsatz (for instance see Matsumura[80, Theorem 13.5]), whichessentially says: Given an ideala in a ringR,

ara(a)≥ ht(a). (0.9)

If equality holds in (0.9), a is said to be aset-theoretic complete intersection. Not allthe ideals are set-theoretic complete intersections, but how can we decide it? One lowerbound for the arithmetical rank is supplied by the cohomological dimension, namely

ara(a)≥ cd(R,a). (0.10)

xviii A quick survey of local cohomology

To see inequality (0.10) recall that the local cohomology modulesH ia(R) are the same

of H i√a(R) (Remark0.1.1). Then inequality (0.10) is clear from the interpretation

of local cohomology by meaning of theCech complex (Theorem0.3.5). Actually,inequality (0.10) is better than (0.9) thanks to the following theorem of Grothendieck(for example see [10, Theorem 6.1.4]):

Theorem 0.4.3. (Grothendieck) Leta be an ideal of a ring R and M be a finitelygenerated R-module. Let℘ be a minimal prime ideal ofa anddimM℘ = q. Then

Hqa(M) 6= 0.

In particular cd(R,a)≥ ht(a).

Example 0.4.4. Inequality in Theorem0.4.3can be strict. Let R= k[[x,y,v,w]] bethe ring of formal series in4 variables over a fieldk anda= (xv,xw,yv,yw) = (x,y)∩(v,w). We have thatht(a)= 2. Let us consider the following piece of the Mayer-Vietorissequence(0.5)

H3a(R)−→ H4

(x,y,v,w)(R)−→ H4(x,y)(R)⊕H4

(v,w)(R).

By Theorem0.4.3H4(x,y,v,w)(R) 6= 0, whereas H4(x,y)(R)⊕H4

(v,w)(R) = 0 by (0.10). There-

fore H3a(R) has to be different from zero, so thatcd(R,a) ≥ 3> 2 = ht(a). In partic-

ular ara(a) ≥ 3 > ht(a), thusa is not a set-theoretic complete intersection. Actuallyara(a) = 3, in fact one can easily show thata=

√(xv, yw, xw+ yv).

In Chapters1and2, we will see many examples of ideals which are not set-theoreticcomplete intersections.

0.4.3 The graded setting

If R is a graded ring,a ⊆ R is a graded ideal andM is aZ-gradedR-module, thenthe local cohomology modulesH i

a(M) carry on a naturalZ-grading. For instance, wecan choose homogeneous generators ofa, saya = a1, . . . ,ak. Then we can give aZ-grading to theR-modules in theCech complexC(a,M)• in the obvious way. Moreover,it is straightforward to check that, this way, the maps inC(a,M)• are homogeneous.Therefore, the cohomology modules ofC(a,M)• areZ-graded, and soH i

a(M) is Z-graded for anyi ∈ N by Theorem0.3.5. For any integerd ∈ Z, we will denote by[H i

a(M)]d, or simply byH ia(M)d, the graded piece of degreed of H i

a(M). Of course onecould try to give aZ-grading toH i

a(M) also using Theorem0.3.1, or computing directlythe local cohomology functors as derived functors ofΓa(·) restricted to the categoryof Z-gradedR-modules. Well, it turns out that all the above possible definitions agree(see [10, Chapter 12]).

Suppose thatR is a standard graded ring (i.e.R is graded and it is generated byR1

as anR0-algebra). TheCastelnuovo-Mumford regularityof a Z-graded moduleM isdefined as

reg(M) := supd+ i : [H iR+

(M)]d 6= 0 andi ∈ N.UnlessH i

R+(M) = 0 for all i ∈N, it turns out that the Castelnuovo-Mumford regularity

is always an integer (Theorem0.1.3and [10, Proposition 15.1.5 (ii)]). In the case inwhich R0 is a field, if M is a finitely generated Cohen-MacaulayZ-graded module,then, using (0.8) and Theorem0.1.3we have that

reg(M) = supd+dimM : [HdimMR+

(M)]d 6= 0.

0.4 Why is local cohomology interesting? xix

At this point it is worthwhile to introduce an invariant related to the regularity. LetR be a Cohen-Macaulay graded algebra such thatR0 is a field. Thea-invariantof R isdefined to be

a(R) := supd ∈ Z : [HdimRR+

(R)]d 6= 0.Therefore, ifR is furthermore standard graded, then

reg(R) = a(R)+dimR. (0.11)

Example 0.4.5.Let S= k[x,y] be the polynomial ring in two variables over a fieldk.By Example0.3.6we immediately see that a(S) =−2 andreg(S) = 0. More generally,if S= k[x1, . . . ,xn] is the polynomial ring in n variables over a fieldk, then a(S) =−nandreg(S) = 0.

Up to now, we have not yet stated one of the fundamental theorems in the theory oflocal cohomology: TheGrothendieck local duality theorem. We decided to do so be-cause such a statement would require to introduce Matlis duality theory, which wouldtake too much space, for our purposes. However, if the ambient ring R is a gradedCohen-Macaulay ring such thatR0 is a field, we can state a weaker version of localduality theorem, which actually turns out to be useful in many situations and whichavoids the introduction of Matlis duality. So letR be as above. ByωR we will denotethe canonical module ofR (for the definition see [13, Definition 3.6.8], for the exis-tence and unicity in this case put together [13, Example 3.6.10, Proposition 3.6.12 andProposition 3.6.9]).

Theorem 0.4.6. (Grothendieck) Let R be an n-dimensional Cohen-Macaulay gradedring such that R0 = k is a field and letm= R+ be its maximal irrelevant ideal. For anyfinitely generatedZ-graded R-module M, we have

dimk(Him(M)−d) = dimk(Extn−i

R (M,ωR)d) ∀ d ∈ Z.

In particular, if R= S= k[x1, . . . ,xn] is the polynomial ring in n variables over a fieldk, we have

dimk(Him(M)−d) = dimk(Extn−i

S (M,S)d−n) ∀ d ∈ Z.

Proof. The theorem follows at once by [13, Theorem 3.6.19]. For the part concerningthe polynomial ringS, simply notice thatωS= S(−n).

A consequence of Theorem0.4.6is that

dimk(Hnm(R)−d) = dimk(HomR(R,ωR)d) = dimk([ωR]d) ∀ d ∈ Z.

So we get another interpretation of thea-invariant of a Cohen-Macaulay graded algebrasuch thatR0 is a field, namely

a(R) =−mind ∈ Z : [ωR]d 6= 0. (0.12)

If R= S is the polynomial ring inn variables over a fieldk, local duality suppliesanother interpretation of the Castelnuovo-Mumford regularity. For any nonzero finitelygeneratedZ-gradedS-moduleM there is a graded minimal free resolution ofM:

0→⊕

j∈ZS(− j)βp, j →

⊕

j∈ZS(− j)βp−1, j → . . .→

⊕

j∈ZS(− j)β0, j → M → 0

xx A quick survey of local cohomology

with p≤ n. Actually the natural numbersβi, j are numerical invariants ofM, and theyare called thegraded Betti numbersof M. Let us define

r := max j − i : βi, j 6= 0, i ∈ 0, . . . , p.

Then, for anyi = 1, . . . , p, the S-moduleFi :=⊕

j∈ZS(− j)βi, j has not minimal gen-erators in degree bigger thanr + i. This implies that HomS(Fi ,S)d = 0 wheneverd < −r − i. So, since ExtiS(M,S) is a quotient of a submodule of HomS(Fi,S), weget that

d <−r − i =⇒ ExtiS(M,S)d = 0.

Actually, playing more attention, we can find an indexi ∈ 0, . . . , p such that

ExtiS(M,S)−r−i 6= 0

(for the details see Eisenbud [34, Proposition 20.16]). Therefore, by Theorem0.4.6,we getHn−i

m (M)−d−n = 0 for anyd <−r − i, which means thatH jm(M)k = 0 whenever

j + k> r. Moreover there exists aj such thatH jm(M)r− j 6= 0. Eventually, we get:

reg(M) = max j − i : βi, j 6= 0, i ∈ 0, . . . , p. (0.13)

From (0.13) it is immediate to check that a minimal generator ofM has at most degreereg(M). This fact is true over much more general rings thanS.

Theorem 0.4.7. Let R be a standard graded ring and M be a finitely generatedZ-graded module. A minimal generator of M has at most degreereg(M).

We end this chapter remarking a relationship between local cohomology modulesand Hilbert polynomials. LetR be a graded ring such thatR0 = k is a field and letMbe a finitely generatedZ-gradedR-module. We denote by HFM : Z→ N the Hilbertfunction of M, i.e.

HFM(d) = dimk(Md).

A result of Serre (see [13, Theorem 4.4.3 (a)]) states that there exists a unique quasi-polynomial functionP : Z→ N such thatP(m) = HFM(m) for anym≫ 0. Recall that“P quasi-polynomial function” means that there exists a positive integerg and poly-nomialsPi ∈ Q[X] for i = 0, . . . ,g−1 such thatP(m) = Pi(m) for all m= kg+ i withk ∈ Z. Such a quasi-polynomial function is called theHilbert quasi-polynomial of M,and denoted by HPM. If R is standard graded, then HPM is actually a polynomial func-tion of degree dimM−1 by a previous theorem by Hilbert (see [13, Theorem 4.1.3]),and in this case HPM is simply called theHilbert polynomial of M. The promised re-lationship between local cohomology modules and Hilbert polynomials is once againwork of Serre ([13, Theorem 4.4.3 (b)]).

Theorem 0.4.8. (Serre) Let R be a graded ring such that R0 = k is a field and letm= R+. If M is a nonzero finitely generatedZ-graded R-module of dimension d, then

HFM(m)−HPM(m) =d

∑i=0

(−1)i dimkH im(M)m ∀ m∈ Z.

Eventually, notice that Theorem0.4.8yields a third interpretation of thea-invariant.If R is a Cohen-Macaulay graded ring such thatR0 is a field, then:

a(R) = supi ∈ Z : HFR(i) 6= HPR(i). (0.14)

Chapter 1

Cohomological Dimension

In his seminair on local cohomology (see the notes written byHartshorne [51, p. 79]),Grothendieck raised the problem of finding conditions underwhich, fixed a positiveintegerc, the local cohomology modulesH i

a(R) = 0 for everyi > c, wherea is an idealin a ring R. In other words, looking for terms under which cd(R,a) ≤ c. Ever sincemany mathematicians worked on this question: For instance,see Hartshorne [52, 53],Peskine and Szpiro [89], Ogus [87], Hartshorne and Speiser [57], Faltings [39], Hunekeand Lyubeznik [64], Lyubeznik [78], Singh and Walther [97].

This chapter is dedicated to discuss the question raised by Grothendieck. Essen-tially, we will divide the chapter in two parts. In the first one we will treat of necessaryconditions for cd(R,a) being smaller than a given integer, whereas in the second onewe will discuss about sufficient conditions for it. However,before describing the con-tents and results of the chapter, we want to give a brief summary of the classical resultsexisting in the literature around this problem.

First of all, the reader might object: Why do not ask conditions about the lowestnonvanishing local cohomology module, rather than the highest one? The reason is thatthe smallest integeri such thatH i

a(R) 6= 0 is well understood: It is the maximal lengthof a regular sequence onR consisting of elements ofa, see (0.7). The highest non-vanishing local cohomology module, instead, is much more subtle to locate. Besides,the number cd(R,a) is charge of informations. For instance, it supplies one of the fewknown obstructions for an ideal being generated up to radical by a certain number ofelements, see (0.10). The two starting results about “the problem of the cohomologicaldimension” are both due to Grothendieck: They are essential, as they fix the range inwhich we must look for the natural number cd(R,a) (Theorems0.4.3and0.1.3):

ht(a)≤ cd(R,a)≤ dimR.

Let us setn := dimR. In light of the results of Grothendieck, as a first step it wasnatural to try to characterize when cd(R,a)≤ n−1. In continuing the discussion let ussuppose the ringR is local and complete. Such an assumption is reasonable, since forany ringR:

(i) cd(R,a) = supcd(Rm,aRm) : m is a maximal ideal by (0.2).(ii) cd(Rm,aRm) = cd(Rm,aRm) by (0.4).

First Lichtenbaum and then, more generally, Hartshorne [52, Theorem 3.1], settled theproblem of characterizing when cd(R,a) ≤ n− 1. Essentially, they showed that thenecessary and sufficient condition for this to happen is thatdimR/a> 0, see Theorem

2 Cohomological Dimension

0.3.2. By then, the next step should have been to describe when cd(R,a) ≤ n−2. Ingeneral this case is still not understood well as the first one. However, ifR is regular, anecessary and sufficient condition, besides being known, isvery nice. Such a condition,which has been proven under different assumptions in [52, 89, 64, 97], essentially isthat the punctured spectrum ofR/a is connected. Actually, if the “ambient ring”R isregular, cd(R,a) can always be characterized in terms of the ringR/a ([87, 57, 78]).However, in any of these papers, the described conditions are quite difficult to verify.Our task in this chapter will be to give some necessary and/orsufficient conditions, aseasier as possible to verify, for cd(R,a) being smaller than a fixed integer.

In the first section we will focus on giving necessary conditions forH ia(R) vanishing

for anyi bigger than a fixed integer. The framework of this section comes from the firstpart of our paper [103]. As we remarked in0.1.1, the local cohomology functorsH i

a

depend just on the radical ofa. Therefore, a way to face the Grothendieck’s problemcould be to study the topological properties of Spec(R/a). In fact, such a topologicalspace and Spec(R/

√a) are obviously homeomorphic. Actually in this section we will

act in this way; particularly, we will study the connectedness properties of Spec(R/a),in the sense explained in AppendixB. In [63], Hochster and Huneke proved that ifais an ideal of ann-dimensional,(n− 1)-connected , local, complete ringR such thatcd(R,a)≤ n−2, thenR/a is 1-connected, generalizing a result previously obtainedbyFaltings in [40]. We prove, under the same hypotheses on the ringR, that

cd(R,a)≤ n− s =⇒ R/a is (s−1)-connected.

More generally, we prove that ifR is r-connected, withr < n, then

cd(R,a)≤ r − s =⇒ R/a is s-connected, (1.1)

see Theorem1.1.5. This result also strengthens a connectedness theorem due to Gro-thendieck [47, Expose XIII, Theoreme 2.1], who got the same thesis under the assump-tion thata could be generated byr−selements. To prove (1.1) we drew on the proof ofGrothendieck’s theorem given in the book of Brodmann and Sharp [10, 19.2.11]. (The-orem1.1.5has been proved independently in the paper of Divaani-Aazar, Naghipourand Tousi [33, Theorem 2.8]. We illustrate a relevant mistake in [33, Theorem 3.4] inRemark1.1.8).

We prove (1.1) also for certain noncomplete ringsR, such as local rings satisfyingtheS2 Serre’s condition (Proposition1.1.10), or graded rings over a field (in this casethe ideala must be homogeneous), see Theorem1.1.11. This last version of (1.1)allows us to translate the result into a geometric point of view, dicscussing about thecohomological dimension of an open subschemeU of a schemeX projective over afield (Theorem1.1.13). Particularly, we give topological necessary conditionson theclosed subsetX \U for U being affine (Corollary1.1.14).

We end the section discussing whether the implication of (1.1) can be reversed. Ingeneral the answer is no, and we give explicit counterexamples.

The aim of the second section, whose results are part of our paper [104], is to ex-plore the case in whichR := k[x1, . . . ,xn] is a polynomial ring over a fieldk, usually ofcharacteristic 0, anda is a homogeneous ideal. In this setting we have a better under-standing of the cohomological dimension. The characteristic 0 assumption allows usto reduce the issues, the most of the times, to the casek = C: This way we can bor-row results from Algebraic Topology and Complex Analysis. Akey result we prove is

1.1 Necessary conditions for the vanishing of local cohomology 3

Theorem1.2.4. An essential ingredient in its proof is the work of Ogus [87], combinedwith a comparison theorem of Grothendieck obtained in [48] and classical results fromHodge theory. Theorem1.2.4gives a criterion to compute the cohomological dimen-sion of a homogeneous ideala in a polynomial ringR over a field of characteristic 0,provided thatR/a has an isolated singularity (see also Theorem1.2.21for a more al-gebraic interpretation). Roughly speaking, such a criterion relates the cohomologicaldimension cd(R,a) with the dimensions of the finitek-vector spaces

[H in(Λ

jΩA/k)]0,

whereA := R/a, n is the irrelevant maximal ideal ofA and ΩA/k is the module ofKahler differentials ofA overk. In what follows, let us denote theA-moduleΛ jΩA/k

by Ω j . The advantages of such a characterization are essentiallytwo:(i) The cohomological dimension cd(R,a), in this case, is an intrinsic invariant of

A= R/a, which is not obvious a priori.(ii) The dimensions dimk[H i

n(Ω j)]0 are moderately easy to compute, thanks to theGrothendieck’s local duality (see Theorem0.4.6):

dimk[Hin(Ω j)]0 = dimk[Extn−i

R (Ω j ,R)]−n.

Even if Theorem1.2.4will essentially be obtained putting together earlier results with-out upsetting ideas, it has at least two amazing consequences. The first one regards arelationship between depth and cohomological dimension. Before describing it, let usremind the result of Peskine and Szpiro mentioned in Theorem0.3.3. It implies that, ifchar(k)> 0, then

depth(R/a)≥ t =⇒ cd(R,a)≤ n− t. (1.2)

In characteristic 0, the above fact does not hold true already for t = 4 (see Example1.2.14). However, fort ≤ 2, (1.2) holds true also in characteristic 0 from [52] (moregenerally see Proposition1.2.15). Well, Theorem1.2.4enables us to settle the caset = 3 of (1.2) also in characteristic 0, provided thatR/a has an isolated singularity (seeTheorem1.2.16). From this fact it is natural to raise the following problem:

Question1.2.17. Suppose thatR is a regular local ring, and thata⊆ R is an ideal suchthat depth(R/a)≥ 3. Is it true that cd(R,a)≤ dimR−3?

The second consequence of Theorem1.2.4consists in the solution of a remarkablecase of a conjecture done by Lyubeznik in [77]. It concerns a relationship betweencohomological dimension and etale cohomological dimension of a scheme, alreadywondered by Hartshorne in [53]. The truth of Lyubeznik’s guess would imply thatetale cohomological dimension provides a better lower bound for the arithmetical rankthan the one supplied by cohomological dimension (see (A.7) and (0.10)). In Theorem1.2.20we give a positive answer to the conjecture in characteristic 0, under a smooth-ness assumption. On the other hand, Lyubeznik informed us that recently he found acounterexample in positive characteristic.

1.1 Necessary conditions for the vanishing of local co-homology

Let a be an ideal of an ringR, andc be a positive integer such that ht(a)≤ c< dimR.As we anticipated in the introduction, this section is dedicated in finding necessaryconditions forH i

a(R) vanishing for anyi > c. Mainly, we will care about the topologicalproperties which Spec(R/a) must have to this aim.


1.1.1 Cohomological dimension vs connectedness

The purpose of this subsection is to prove Theorem1.1.5. It fixes the connectednessproperties that Spec(R/a) must own in order toH i

a(R) vanish for alli > c. Theorem1.1.5has as consequences many previously known theorems. Especially, we remark atheorem by Grothendieck (Theorem1.1.6) and one by Hochster and Huneke (Theorem1.1.7).

Let us start with a lemma which shows a sub-additive propertyof the cohomologi-cal dimension. It will be crucial to prove Proposition1.1.2.

Lemma 1.1.1. Let R be any ring, anda, b ideals of R. Then

cd(R,a+b)≤ cd(R,a)+ cd(R,b).

Proof. By [10, Proposition 2.1.4],Γb maps injectiveR-modules into injective ones.Furthermore notice thatΓa Γb = Γa+b. So the statement follows at once by the spec-tral sequence

Ei, j2 = H i

a(Hjb(R)) =⇒ H i+ j

a+b(R),

for instance see the book of Gelfand and Manin [44, Theorem III.7.7].

The next two propositions strengthen [10, Proposition 19.2.7] and [10, Lemma19.2.8].

Proposition 1.1.2. Let (R,m) be a complete local domain and leta andb be ideals ofR such thatdimR/a> dimR/(a+b) anddimR/b> dimR/(a+b). Then

cd(R,a∩b)≥ dimR−dimR/(a+b)−1

Proof. Setn := dimR andd := dimR/(a+ b). We make an induction ond. If d = 0we consider the Mayer-Vietoris sequence (0.5)

. . .−→ Hn−1a∩b(R)−→ Hn

a+b(R)−→ Hna(R)⊕Hn

b(R)−→ . . . .

Hartshorne-Lichtenbaum Vanishing Theorem0.3.2 implies thatHna(R) = Hn

b(R) = 0

andHna+b

(R) 6= 0. So the above exact sequence implies cd(R,a∩b)≥ n−1.Consider the case in whichd > 0. By prime avoidance we can choose an element

x ∈ m such thatx does not belong to any minimal prime ofa, b anda+ b. Thenlet a′ = a+ (x) and b′ = b+ (x). From the choice ofx and by the Hauptidealsatzof Krull it follows that dimR/(a′+ b′) = d−1, dimR/a′ = dimR/a−1> d−1 anddimR/b′ = dimR/b− 1> d− 1; hence by induction we have cd(R,a′ ∩ b′) ≥ n− d.Because

√a∩b+(x) =

√a′∩b′, thenH i

a′∩b′(R) = H ia∩b+(x)(R) for all i ∈ N. From

Lemma1.1.1we have

n−d ≤ cd(R,a′∩b′) = cd(R,a∩b+(x))≤ cd(R,a∩b)+1.

Therefore the proof is completed.

The next result is a generalization of Proposition1.1.2 to complete local ringswhich are not necessarily domains.

Proposition 1.1.3. Let R be an r-connected complete local ring with r< dimR. Con-sider two idealsa, b of R such thatdimR/a> dimR/(a+b)< dimR/b. Then

cd(R,a∩b)≥ r −dimR/(a+b). (1.3)


Proof. Setd := dimR/(a+b), and let℘1, . . . ,℘m be the minimal prime ideals ofR.First we consider the following case:

∃ i ∈ 1, . . . ,m such that dimR/(a+℘i)> d and dimR/(b+℘i)> d.

In this case we can use Proposition1.1.2, consideringR/℘i asR, (a+℘i)/℘i asa and(b+℘i)/℘i asb. Then

cd(R/℘i,((a+℘i)∩ (b+℘i))/℘i)≥ dimR/℘i −d−1.

Notice that

cd(R/℘i,((a+℘i)∩ (b+℘i))/℘i) = cd(R/℘i,((a∩b)+℘i)/℘i)≤ cd(R,a∩b)

by Lemma0.2.2(ii). Moreover dimR/℘i > r: If dim R/℘i = dimR, then this followsby the assumptions. OtherwiseR is a reducible ring, thus LemmaB.0.5 (ii) impliesdimR/℘i > r. So we get the thesis.

Thus we can suppose that

∀ i ∈ 1, . . . ,m either dimR/(a+℘i)≤ d or dimR/(b+℘i)≤ d.

After a rearrangement we can picks∈ 1, . . . ,m−1 such that for alli ≤ s we havedimR/(a+℘i)≤ d and for alli > swe have dimR/(a+℘i)> d. The existence of suchans is guaranteed from the fact that dimR/a> d and dimR/b> d. Let us consider theideal ofR

q= (℘1∩ . . .∩℘s)+ (℘s+1∩ . . .∩℘m)

and let℘be a minimal prime ofq such that dimR/℘= dimR/q. LemmaB.0.5(i) saysthat dimR/℘≥ r. Moreover, since there existi ∈ 1, . . . ,s and j ∈ s+1, . . . ,m suchthat℘i ⊆℘ e℘j ⊆℘, we have

dimR/(a+℘)≤ dimR/(a+℘i)≤ d and

dimR/(b+℘)≤ dimR/(b+℘j)≤ d

Therefore we deduce that dimR/((a∩ b)+℘) ≤ d. But R/℘ is catenary, (see Mat-sumura [80, Theorem 29.4 (ii)]), then

ht(((a∩b)+℘)/℘)= dimR/℘−dimR/((a∩b)+℘).

and hence ht(((a∩b)+℘)/℘)≥ r −d. So Lemma0.2.2(ii) and Theorem0.4.3

cd(R,a∩b)≥ cd(R/℘,((a∩b)+℘)/℘)≥ r −d.

The hypothesisr < dimR in Proposition1.1.3is crucial, as we are going to showin the following example.

Example 1.1.4. Let R= k[[x,y]] the ring of formal power series over a fieldk, a =(x) and b = (y). Being a domain, R is2-connected by RemarkB.0.3 (dimR= 2).Moreovercd(R,a∩b) = cd(R,(xy))≤ 1 by(0.10). Finally dimR/(a+b) = 0, thereforeProposition1.1.3does not hold if we chose r= 2. However in this case, and in generalwhen R is irreducible, we can use Proposition1.1.3choosing r= dimR−1.


Eventually, we are ready to prove the main theorem of this section.

Theorem 1.1.5. Let R be an r-connected complete local ring with r< dimR. Givenan ideala⊆ R, we have

cd(R,a)≤ r − s =⇒ R/a is s-connected.

Proof. Let ℘1, . . . ,℘m be the minimal primes ofa. If m= 1, thenR/a is (dimR/a)-connected. Let℘ be a minimal prime ofR such that℘⊆℘1. Using Lemma0.2.2(ii)and Theorem0.1.3we have cd(R,a) = cd(R,℘1)≥ cd(R/℘,℘1/℘)≥ ht(℘1/℘). So,sinceR/℘ is catenary, we have

dimR/a= dimR/℘1 = dimR/℘−ht(℘1/℘)≥ dimR/℘− cd(R,a).

By LemmaB.0.5(ii) we get the thesis. Ifm> 1, letA andB be a pair of disjoint subsetsof 1, . . . ,m such thatA∪B= 1, . . . ,m and, setting

c := dim

(R

(∩i∈A℘i)+ (∩ j∈B℘j)

),

R/a is c-connected (the existence ofA andB is ensured by LemmaB.0.5 (i)). SetJ :=∩i∈A℘i andK :=∩ j∈B℘j . Since dimR/J > c and dimR/K > c, Proposition1.1.3implies

c= dimR/(J +K )≥ r − cd(R,J ∩K ).

Since√a= J ∩K , the theorem is proved.

As we have already mentioned Theorem1.1.5unifies many previous results con-cerning relationships between connectedness properties and the cohomological dimen-sion. For instance, since cd(R,a) bounds from below the number of generators ofa

by (0.10), we immediately get a theorem obtained in [47, Expose XIII, Theoreme 2.1](see also [10, Theorem 19.2.11]).

Theorem 1.1.6. (Grothendieck) Let(R,m) be a complete local r-connected ring withr < dimR. If an ideala⊆ R is generated by r− s elements, then R/a is s-connected.

Theorem1.1.5implies also [63, Theorem 3.3], which is in turn a generalization ofa result of [40]. See also Schenzel [92, Corollary 5.10].

Theorem 1.1.7. (Hochster-Huneke) Let(R,m) be a complete equidimensional localring of dimension d such that Hdm(R) is an indecomposable R-module. Ifcd(R,a) ≤d−2, then the punctured spectrumSpec(R/a)\ m is connected.

Proof. SinceHdm(R) is indecomposable, [63, Theorem 3.6] implies thatR is connected

in codimension 1. Because Spec(R/a) \ m is connected if and only ifR/a is 1-connected, the statement follows at once from Theorem1.1.5.

Remark 1.1.8. In [33, Theorem 3.4] the authors claim that Theorem1.1.7holds with-out the assumption thatHd

m(R) is indecomposable. This is not correct, as we are goingto show: SetR := k[[x,y,u,v]]/(xu,xv,yu,yv) and leta be the zero ideal ofR. The min-imal prime ideals ofR are(x,y) and(u,v), soR is a complete equidimensional localring of dimensiond = 2, and cd(R,a) = 0 = d− 2. From LemmaB.0.6 (i) one cansee thatR/a = R is not 1-connected. Therefore the punctured spectrum ofR/a is notconnected.


1.1.2 Noncomplete case

So far, we have obtained a certain understanding of the problem assuming the ambientring R to be complete. Of course, in the results of the previous subsection we couldhave avoided the completeness assumption. The inconvenient would have been that weshould have complicated the statements asking for properties of the rings after com-pleting, e.g.:Let R be a local ring such thatR is r-connected.The aim of the presentsubsection is to introduce rings for which, even if noncomplete, the results of the pre-vious section still hold true, without using “completing-hypothesis”. We start with alemma.

Lemma 1.1.9. Let (R,m) be a d-dimensional local analytically irreducible ring (i.e.R is irreducible). Given an ideala⊆ R, we have

cd(R,a)≤ d− s−1 =⇒ R/a is s-connected.

Proof. SinceR is irreducible, it isd-connected. In particular it is(d−1)-connected.Moreover, cd(R,a) = cd(R,aR) by (0.4). Thus Theorem1.1.5yields thatR/aR is s-connected. Eventually, LemmaB.0.8 (i) implies thatR/a is s-connected as well, andwe conclude.

The following proposition gives a class of rings for which wedo not have to careabout their properties after completion. Let us remark thatsuch a class comprehendsCohen-Macaulay local rings.

Proposition 1.1.10. Let R be a d-dimensional local ring satisfying Serre’s conditionS2, which is a quotient of a Cohen-Macaulay local ring. Given anideala⊆ R, we have

cd(R,a)≤ d− s−1 =⇒ R/a is s-connected.

Proof. The completion ofR satisfiesS2 as well asR (see [80, Exercise 23.2]). ThenRis connected in codimension 1 by CorollaryB.1.3. Arguing as in the proof of Lemma1.1.9, we conclude.

Now we prove a version of Theorem1.1.5in the case whenR is a graded algebraover a field. Such a version will be useful also in the next chapter.

Theorem 1.1.11.Let R be a graded r-connected ring such that R0 is a field, withr < dimR. Given a homogeneous ideala⊆ R, we have

cd(R,a)≤ r − s =⇒ R/a is s-connected.

Proof. Let m be the irrelevant maximal ideal ofR. Using LemmaB.0.5(i), since theminimal prime ideals ofRare inm, we have thatRm is r-connected. Furthermore, let usnotice thatRm

∼= Rm. Since the minimal prime ideals ofRm come from homogeneousprime ideals ofR, by LemmaB.0.9 their extension℘Rm belongs to Spec(Rm). ThenLemmaB.0.8(ii) yields thatRm is r-connected. Notice that, by (0.4) and (0.2) we have

cd(Rm,aRm)≤ cd(R,a),

soRm/aRm is s-connected by Theorem1.1.5. LemmaB.0.8(i) implies thatRm/aRm iss-connected as well. Eventually we conclude by LemmaB.0.5(i): In fact the minimalprimes ofa, being homogeneous, are inm, soR/a is s-connected as well asRm/aRm.


1.1.3 Cohomological dimension and connectedness of open sub-schemes of projective schemes

In this subsection we give a geometric interpretation of theresults obtained in Subsec-tion 1.1.2, using (A.6). More precisely, given a projective schemeX over a fieldk andan open subschemeU ⊆ X, our purpose is to find necessary conditions for which thecohomological dimension ofU is less than a given integer. By a well known result ofSerre, there is a characterization of noetherian affine schemes in terms of the cohomo-logical dimension, namely: A noetherian schemeX is affine if and only if cd(X) = 0(see Hartshorne [56, Theorem 3.7]). Hence, as a particular case, in this subsectionwe give necessary conditions for the affineness of an open subscheme of a projectivescheme overk. This is an interesting theme in algebraic geometry, and it was studiedfrom several mathematicians (see for example Goodman [45] or Brenner [9]). For in-stance, it is well known that ifX is a noetherian separated scheme andU ⊆ X is anaffine open subscheme, then every irreducible component ofZ = X \U has codimen-sion less than or equal to 1. Here it follows a quick proof.

Proposition 1.1.12.Let X a noetherian separated scheme, U⊆ X an affine open sub-scheme and Z= X \U. Then every irreducible component of Z has codimension lessthan or equal to 1.

Proof. Let Z1 be an irreducible component ofZ and letV be an open affine such thatV ∩ Z1 6= /0. SinceX is separated,U ∩V is also affine, and since codim(Z1,X) =codim(Z1∩V,V), we can suppose thatX is a notherian affine scheme. So letR anda ⊆ R be such thatX = Spec(R) andZ = V (a). By the affineness criterion of Serre,we haveH i(U,OX) = 0 for all i > 0. Then (A.5) implies cd(R,a) ≤ 1. This impliesht(℘)≤ 1 for all℘ minimal prime ofa (Theorem0.4.3).

In light of this result it is natural to ask:What can we say about the codimension ofthe intersection of the various components of Z?To answer this question we investigateon the connectedness ofZ.

Theorem 1.1.13.Let X be an r-connected projective scheme over a fieldk with r <dimX, U ⊆ X an open subscheme and Z= X \U. Then

cd(U)≤ r − s−1 =⇒ Z is s-connected.

Proof. Let X =Proj(R), with Ra graded finitely generatedk-algebra, and leta⊆Rbe agraded ideal definingZ. Then, from (A.6) it follows that cd(R,a)≤ r −s. By ExampleB.0.4(ii) we have thatR is (r +1)-connected. Therefore, from Theorem1.1.11we getthatR/a is (s+1)-connected. SoZ = V+(a) is s-connected, using Example (B.0.4) (ii)once again.

From Theorem1.1.13we can immediately obtain the following corollary.

Corollary 1.1.14. Let X be an r-connected projective scheme over a fieldk with r <dimX, U ⊆ X an open subscheme and Z= X \U. If U is affine, then Z is(r − 1)-connected. In particular, if X is connected in codimension 1andcodimX Z = 1, then Zis connected in codimension 1.

Proof. By the affineness criterion of Serre cd(U) = 0, so we conclude by Theorem1.1.13.


Example 1.1.15.Let R:= k[X0, . . . ,X4]/(X0X1X4−X2X3X4) and denote by xi the residueclass of Xi modulo R. Consider the ideala = (x0x1,x0x3,x1x2,x2x3) =℘1∩℘2 ⊆ R,where℘1 = (x0,x2) and℘2 = (x1,x3). Clearly℘1 and℘2 are prime ideals of height1,therefore U= Proj(R)\V+(a) might be affine. SinceProj(R) is a complete intersectionof P4, it is connected in codimension 1 (PropositionB.1.2). ButdimV+(℘1+℘2) = 0,thereforeV+(a) is not1-connected by LemmaB.0.5(i). Thus, by Corollary1.1.14weconclude that U is not affine.

1.1.4 Discussion about the sufficiency for the vanishing of local co-homology

Theorem1.1.5 gives a necessary condition for the cohomological dimension to besmaller than a given integer. In order to make the below discussion easier, supposethat the local complete ring(R,m) is an n-dimensional domain. In this case Theo-rem1.1.5says the following:If a is an ideal of R such thatcd(R,a) < c, then R/a is(n− c)-connected.One question which might come in mind is if the conditionR/a iss-connected is also sufficient to cd(R,a) being smaller thann− s. If s= 1, with somefurther assumptions onR, Theorem1.1.5can actually be reversed. This fact has beenproven by various authors, with different assumptions on the ringRand different tools,see [52], [87], [89] and [97]. The more general version of the result has been settled in[64, Theorem 2.9]. First recall that a local ring(A,m) is 1-connected if and only if itspunctured spectrum Spec(A)\ m is connected (ExampleB.0.4(i)).

Theorem 1.1.16.(Huneke and Lyubeznik) Let R be an n-dimensional local, complete,regular ring containing a separably closed field. For an ideal a⊆ R,

cd(R,a)≤ n−2 ⇐⇒ dimR/a≥ 2 and R/a is 1-connected.

The next step we might consider is the case in whichR/a is 2-connected. FromTheorem1.1.5we know the following implication:

cd(R,a)≤ n−3 =⇒ dimR/a≥ 3 and R/a is 2-connected. (1.4)

Unfortunately, this time, the converse does not hold, neither assumingR to be regular.We show this in the following example.

Example 1.1.17.Let R:= k[[x1,x2,x3,x4,x5]] be the (5-dimensional) ring of formalpower series in5 variables over a fieldk. Let us consider the ideal of R

a := (x1x2x4,x1x3x4,x1x3x5,x2x3x5,x2x4) = (x1,x2)∩ (x2,x3)∩ (x3,x4)∩ (x4,x5).

One can see, for instance using LemmaB.0.7, that A= R/a is 2-connected. Howeverwe are going to show thatcd(R,a) ≥ 3 = 5− 2. To this aim, let us localize A at theprime ideal℘= (x1,x2,x4,x5). Since x3 is invertible in R℘, we have that

A℘ ∼= R℘/(x1x4,x1x5,x2x4,x2x5).

Moreover,b = aR℘ = (x1x4,x1x5,x2x4,x2x5) = (x1,x2)∩ (x3,x4). By LemmaB.0.6itfollows that A℘ is not1-connected. Moreover notice that R℘ is analytically irreducible,sinceR℘ is a4-dimensional regular domain as well as R℘. Therefore, Proposition1.1.9implies thatcd(R℘,b)≥ 3. So(0.2) implies thatcd(R,a)≥ 3.


In Example1.1.17, we exploited the fact that the connectedness properties ofaring, in general, do not pass to its localizations. More precisely, even if a ringA is r-connected, in general it might happen thatA℘ is nots-connected for some prime ideal℘of A, wheres= ht(℘)+ r−dimA is the expected number. This fact was the peculiarityof Example1.1.17which allowed us to use Theorem1.1.5. So it is understandable ifthe reader is not appeased from the above example, especially because there are ringsin which such a pathology cannot happen. For instance if the ring A is a domain, andtherefore dimA-connected, thenA℘ is a domain for any prime℘, and therefore ht℘-connected. So, we decided to give a counterexample to the converse of (1.4) also inthis situation, i.e. whena is a prime ideal.

Example 1.1.18.Consider the polynomial ring R′′′ :=C[x1,x2,x3,x4] and theC-algebrahomomorphism

φ : R′′′ → C[t,u]x1 7→ tx2 7→ tux3 7→ u(u−1)x4 7→ u2(u−1)

The image ofφ is the coordinate ring of an irreducible surface inA4, and it has beenconsidered by Hartshorne in [50, Example 3.4.2]. It is not difficult to see that

a′′′ := Ker(φ) = (x1x4− x2x3,x21x3+ x1x2− x2

2,x33+ x3x4− x2

4).

Obviouslya′′′ is a prime ideal of R′′′. Let us denote by R′′ the ring of formal powerseriesC[[x1,x2,x3,x4]]. Doing some elementary calculations, one can show that thereexist g1,h1,g2,h2 ∈ R′′ such that(g1,h1,g2,h2) = (x1,x2,x3,x4) (we are consideringideals in R′′) and √

a′′ =√(g1,h1)∩

√(g2,h2),

wherea′′ := a′′′R′′ (see [10, Example 4.3.7]). Using LemmaB.0.6, we infer that R′′/a′′

is not1-connected.Now let us homogenizea′′′ to a′, i.e. set

a′ = (x1x4− x2x3,x21x3+ x0x1x2− x0x2

2,x33+ x0x3x4− x0x2

4)⊆ R′ = R′′′[x0].

The ideala′ is prime as well asa′′′ (for instance see the book of Bruns and Vetter [15,Proposition 16.26 (c)]). FurthermoredimR′/a′ = dimR′′′/a′′′+1= 3. Eventually, setR := C[[x0, . . . ,x5]] and a := a′R. The ideala is prime by LemmaB.0.9. Moreover,dimR/a = dimR′/a′ = 3. Thus R/a is 3-connected. In particular it is2-connected!Let us consider the prime ideal℘= (x1,x2,x3,x4) ⊆ R. Since R℘ is a 4-dimensionalregular local ring containingC, then Cohen’s structure theorem impliesR℘ ∼= R′′.Furthermore, since x0 is invertible in R℘, the idealaR℘ ⊆ R℘ is generated by x1x4−x2x3, x2

1x3+x1x2−x22 and x33+x3x4−x2

4. Therefore, the ideala read in R′′ is the same

ideal ofa′′. So,R℘/aR℘ is not1-connected as R′′/a′′ is not. At this point, Theorem1.1.5implies

cd(R℘,aR℘)≥ 3.

So, using(0.4) and(0.2), we have

cd(R,a)≥ 3> 5−3.

This shows that the converse of(1.4) does not hold, neither if we assume that R is thering of formal power series over the complex numbers and thata⊆ R is a prime ideal.

1.2 Sufficient conditions for the vanishing of local cohomology 11

As the reader can notice, also in the above example we used Theorem1.1.5. Infact, with that notation, the ideala⊆ R, even if prime, was still a bit pathological. Thatis, we found a prime ideal℘ of A= R/a such thatA℘ was not ht℘-connected, at thecontrary ofA℘. Once again, the reader might demand an example in which all thecompletion of the localizations ofA are domain. To this aim, for instance, it would beenough to consider an ideala⊆Rsuch thatA=R/a has an isolated singularity, i.e suchthat A℘ is a regular ring for any non maximal prime ideal℘. In fact the completionof a regular ring is regular, thus a domain. Also in this situation, counterexamples to(1.4) still exist. However to construct them are needed techniques different from thosedeveloped in this section. How to establish these examples will be clear from the resultsof the next section.

1.2 Sufficient conditions for the vanishing of local co-homology

In this section, we will focus on giving sufficient conditions for cd(R,a)≤ c. Howeverthe conditions we find will often be also necessary. To this aim we have to move to aregular ambient ring. Actually, we will even suppose that the ringR is a polynomialring over a field of characteristic 0. Moreover the ideala will be homogeneous and suchthatR/a has an isolated singularity, which is a case of great interest by the discussion ofSubsection1.1.4. Let us recall that the assumption “R/a has an isolated singularity” isequivalent to “X = Proj(R/a) is a regular scheme”, i.e. the local ringsOX,P are regularfor all pointsP of X. If the field is algebraically closed, this is in turn equivalent toXbeing nonsingular.

1.2.1 Open subschemes of the projective spaces over a field ofchar-acteristic 0

In this subsection we prove the key-theorem of the section (Theorem1.2.4), which willlet us achieve considerable consequences. We start with a remark which allows us touse complex coefficients in many situations.

Remark 1.2.1. Let A be a ring,B a flatA-algebra,RanA-algebra andM anR-module.SetRB :=R⊗AB andMB :=M⊗AB. ClearlyRB is a flatR-algebra. Therefore Theorem0.2.2(i) implies that, for any ideala⊆ Revery j ∈ N:

H ja(M)⊗A B∼= H j

a(M)⊗RRB∼= H j

aRB(M⊗RRB)∼= H j

aRB(MB) (1.5)

We want to use (1.5) in the following particular case. LetS:= K[x0, . . . ,xn] be thepolynomial ring inn+1 variables over a fieldK of characteristic 0, andI ⊆ San ideal.SinceI is finitely generated we can find a fieldk of characteristic 0 such that, settingSk := k[x0, . . . ,xn], the following properties hold:

k⊆K, Q⊆ k⊆C, (I ∩Sk)S= I

(to this aim we have simply to add toQ the coefficients of a set of generators ofI ).SinceK andC are faithfully flatk-algebras, (1.5) implies that

cd(S, I) = cd(Sk, I ∩Sk) = cd(SC,(I ∩Sk)SC), (1.6)

whereSC := C[x1, . . . ,xn].


In the above situation, assume thatI is homogeneous and thatX := Proj(S/I) isa regular scheme. Then setXk := Proj(Sk/(I ∩Sk)) andXC := Proj(SC/((I ∩Sk)C)).Notice thatX ∼= Xk×k Spec(K) andXC ∼= Xk×k Spec(C). FurthermoreXk andXCare regular schemes. SinceK andC are both flatk-algebras, we get, for all naturalnumbersi, j,

H i(X,Ω jX/K)

∼= H i(Xk,ΩjXk/k

)⊗kK

andH i(XC,Ω

jXC/C

)∼= H i(Xk,ΩjXk/k

)⊗kC

(for example see the book of Liu [79, Chapter 6, Proposition 1.24 (a) and Chapter 5,Proposition 2.27]). Particularly we have

dimK(Hi(X,Ω j

X/K)) = dimC(Hi(XC,Ω

jXC/C

)). (1.7)

We will call the dimension of theK-vector spaceH i(X,Ω jX/K) the(i,j)-Hodge number

of X, denoting it byhi j (X).

In the next remark, for the convenience of the reader, we collect some well knownfacts which we will use throughout the paper.

Remark 1.2.2. Let X be a projective scheme overC: Let us recall thatXh meansXregarded as an analytic space, as explained in SectionA.2, andC denotes the constantsheaf associated toC. We will denote byβi(X) thetopological Betti number

βi(X) := rankZ(HSingi (Xh,Z)) = rankZ(H

iSing(X

h,Z)) =

= dimC(HiSing(X

h,C)) = dimC(Hi(Xh,C)).

The equalities above hold true from the universal coefficient TheoremsA.3.1andA.3.2,and by the isomorphismA.9. Pick another projective scheme overC, sayY, and denoteby Z the Segre productX ×Y. The Kunneth formula for singular cohomology (forinstance see Hatcher [58, Theorem 3.16]) yields

H iSing(Z

h,C)∼=⊕

p+q=i

H pSing(X

h,C)⊗CHqSing(Y

h,C).

Thus at the level of topological Betti numbers we have

βi(Z) = ∑p+q=i

βp(X)βq(Y). (1.8)

Now assume thatX is a regular scheme projective overC. It is well known thatXh

is a Kahler manifold (see [56, Appendix B.4]), so the Hodge decomposition (see thenotes of Arapura [2, Theorem 10.2.4]) is available. Therefore, using TheoremA.2.1,we have

H iSing(X

h,C)∼=⊕

p+q=i

H p(Xh,ΩqXh)∼=

⊕

p+q=i

H p(X,ΩqX/C).

Thusβi(X) = ∑

p+q=ihpq(X) (1.9)

Finally, note that the restriction map on singular cohomology

H iSing(P(C

n+1),C)−→ H iSing(X

h,C) (1.10)


is injective provided thati = 0, . . . ,2dimX (see Shafarevich [93, pp. 121-122]). SoExampleA.3.3 implies that

β2i(X)≥ 1 provided thati ≤ dimX (1.11)

The contents of the next remark do not concern the cohomological dimension. Wejust want to let the reader notice a nice consequence of Remark 1.2.2when combinedwith a result of Barth.

Remark 1.2.3. Let X andY be two positive dimensional regular scheme projectiveoverC. Chose any embedding ofZ := X×Y in PN. Then the dimension of the secantvariety ofZ in PN is at least 2dimZ−1.

To show this notice that (1.11) yields β0(X) ≥ 1, β2(X) ≥ 1, β0(Y) ≥ 1 andβ2(Y)≥ 1. Therefore using (1.8) we have

β2(Z)≥ β2(X)β0(Y)+β2(Y)β0(X)≥ 2.

By a theorem of Barth (see Lazarsfeld [71, Theorem 3.2.1]), it follows thatZ cannotbe embedded in anyPM with M < 2dimX−1. If the dimension of the secant varietywere less than 2dimX−1, it would be possible to project down in a biregular wayXfrom PN in P2dimX−2, and this would be a contradiction.

Note that the above lower bound is the best possible. For instance consider theclassical Segre embeddingZ := Pr ×Ps⊆ Prs+r+s. The defining ideal ofZ is generatedby the 2-minors of a generic(r +1)× (s+1)-matrix. The secant variety ofZ is wellknown to be generated by the 3-minors of the same matrix, therefore its dimension is2(r + s)−1.

The following theorem provides some necessary and sufficient conditions for thecohomological dimension of the complement of a smooth variety in a projective spacebeing smaller than a given integer. For the proof we recall the concept ofDeRham-depthintroduced in [87, Definition 2.12]. The general definition requires the notion oflocal algebraic DeRham cohomology (see Hartshorne [55, Chapter III.1]). Howeverfor the case we are interested in there is an equivalent definition in terms of localcohomology ([87, proof of Theorem 4.1]). LetX ⊆ Pn be a projective scheme overa fieldk of characteristic 0. LetS := k[x0, . . . ,xn] be the polynomial ring anda ⊆ Sbe the defining ideal ofX. Then the DeRham-depth ofX is equal tos if and onlyif Supp(H i

a(S)) ⊆ m for all i > n− s and Supp(Hn−sa (S)) * m, wherem is the

maximal irrelevant ideal ofS.

Theorem 1.2.4. Let X⊆ Pn be a regular scheme projective over a fieldk of charac-teristic 0. Moreover, let r be an integer such thatcodimPn X ≤ r ≤ n and U := Pn \X.Thencd(U)< r if and only if

hpq(X) =

0 if p 6= q, p+q< n− r1 if p = q, p+q< n− r

Moreover, ifk= C, the above conditions are equivalent to:

βi(X) =

1 if i < n− r and i is even0 if i < n− r and i is odd

Proof. SetXC ⊆ PnC as in Remark1.2.1andUC := Pn

C \XC. Then cd(U) = cd(UC) by(1.6) and (A.6). Moreover, notice thathi j (X) = hi j (XC) by (1.7). Finally, sinceXC is


regular, we can reduce to the case in whichk = C. If cd(U) < r thenr ≥ codimPn Xby Theorem0.4.3and (A.6). Thus the “only if”-part follows by [53, Corollary 7.5, p.148].

So, it remains to prove the “if”-part. By (A.11) algebraic De Rham cohomologyagrees with singular cohomology. Therefore by (1.10) the restriction maps

H iDR(P

n)−→ H iDR(X) (1.12)

are injective for alli ≤ 2dimX. By the assumptions, equation (1.9) yieldsβi(X) = 1 ifi is even andi < n− r, 0 otherwise. Moreover by ExampleA.3.3βi(Pn) = 1 if i is evenand i ≤ 2n, 0 otherwise. So the injections in (1.12) are actually isomorphisms for alli < n− r.

We want to use [87, Theorem 4.4], and to this aim we will show that the De Rham-depth ofX is greater than or equal ton− r. This means that we have to show thatSupp(H i

a(S))⊆m for all i > r, whereS=C[x0, . . . ,xn], a⊆Sis the ideal definingX andm is the maximal irrelevant ideal ofS. But this is easy to see, because if℘ is a gradedprime ideal containinga and different fromm, beingX regular,aS℘ is a completeintersection inS℘: Therefore(H i

a(S))℘ ∼= H iaS℘

(S℘) = 0 for all i > r (≥ ht(aS℘)).Hence [87, Theorem 4.4] yields the conclusion.

Finally, if k = C, the last condition in the statement is a consequence of the firstone by (1.9). Moreover, the restriction maps on singular cohomology

H iSing(P(C

n+1),C)−→ H iSing(X

h,C)

are compatible with the Hodge decomposition (see [2, Corollary 11.2.5]). The Hodgenumbers of the projective space are well known (for instancesee [53, Exercise 7.8, p.150]), namely

hpq(Pn) =

0 if p 6= q,1 if p= q, p≤ n

Since the restriction maps on singular cohomology are injective for i < n− r by (1.10),the last condition in the statement implies the first one. Thetheorem is proved.

Remark 1.2.5. Notice that Theorem1.2.4 implies the following surprising fact:IfX ⊆ Pn is a regular scheme projective over a field of characteristic0, then the naturalnumber n− cd(Pn \X) is an invariant of X, i.e. it does not depend on the embedding.Actually the same statement is true also in positive characteristic, using [78, Theorems5.1 and 5.2].

Theorem1.2.4is peculiar of the characteristic-zero-case. For instancepick an el-liptic curveE over a field of positive characteristic, whose Hasse invariant is 0. Thismeans that the Frobenius morphism acts as 0 onH1(E,OE) (for a more explanatorydefinition see [56, p. 332]). Such a curve exists in any positive characteristic by [56,Corollary 4.22]. Then setX := E × P1 ⊆ P5 andU := P5 \X. From the Kunnethformula for coherent sheaves (see Sampson and Washnitzer [91, Section 6, Theorem1]), one can deduce that the Frobenius morphism acts as 0 onH1(X,OX) as well ason H1(E,OE). Therefore [57, Theorem 2.5] implies cd(U) = 2 (see also [78, The-orem 5.2]). However notice that, using again [91, Section 6, Theorem 1], we haveh10(X) = h10(E) = 1.

We want to state a consequence of Theorem1.2.4which, roughly speaking, saysthat in characteristic 0 the cohomological dimension of thecomplement of the Segreproduct of two projective schemes is always large. We will also show, in Remark1.2.7,


that in positive caharacteristic, at the contrary, such a cohomological dimension can beas small as possible.

Proposition 1.2.6.Let X andY be two positive dimensional regular schemes projectiveover a fieldk of characteristic0. Choose an embedding of Z:= X×Y in somePN andset U:= PN \Z. Thencd(U)≥N−3. In particular if dimZ≥ 3, Z is not a set-theoreticcomplete intersection.

Proof. We can reduce the problem to the casek= C by (1.6). From (1.8) we have

β2(Z)≥ β2(X)β0(Y)+β2(Y)β0(X). (1.13)

Furthermore, (1.11) yields β0(X) ≥ 1, β2(X) ≥ 1, β0(Y) ≥ 1 andβ2(Y) ≥ 1. Ifcd(U) < N− 3, then Theorem1.2.4would imply thatβ2(Z) = 1. But, by what saidabove, one reads from (1.13) thatβ2(Z)≥ 2, so we get the thesis.

In the next remark we will show that the characteristic-0-assumption is crucial inProposition1.2.6. In fact in positive characteristic can happen that, using the notationof Proposition1.2.6, cd(U) is as small as possible, i.e. codimPN(Z)−1.

Remark 1.2.7. Let k be a field of positive characteristic. Let us consider two Cohen-Macaulay gradedk-algebrasA andB of negativea-invariant (for instance two poly-nomial rings overk). SetR := A#B := ⊕i∈NAi ⊗k Bi their Segre product (with thenotation of the paper of Goto and Watanabe [46]). By [46, Theorem 4.2.3],R isCohen-Macaulay. So, presentingR as a quotient of a polynomial ring ofN+ 1 vari-ables, sayR ∼= P/I , a theorem in [89] (see Theorem0.3.3) implies that cd(P, I) =N+1−dimR. Translating in the language of Proposition1.2.6we haveX := Proj(A),Y := Proj(B), Z := Proj(R) ⊆ PN = Proj(P) and cd(PN \ Z) = cd(P, I)− 1 = N −dimZ−1= codimPN(Z)−1.

1.2.2 Etale cohomological dimension

If the characteristic of the base field is 0 we have seen in the previous subsection that wecan, usually, reduce the problem tok= C. In this context, thanks to the work of Serredescribed in SectionA.2, is available the complex topology, so we can use methodsfrom algebraic topology and from complex analysis.

Unfortunately, when the characteristic ofk is positive, the above techniques arenot available. Moreover some of the results obtained in Subsection1.2.1are not truein positive characteristic, as we have shown just below Theorem1.2.4and in Remark1.2.7. In this subsection we want to let the reader notice that, in order to have, inpositive characteristic, cohomological results similar to those gettable in characteristic0, sometimes it is better to use the etale site rather than the Zariski one. Moreover,as well as local cohomology, etale cohomology gives a lowerbound for the numberof set-theoretically defining equations of a projective scheme by (A.7), therefore theresults of this subsection will be useful also in the next chapter.

We recall the following result of Lyubeznik [76, Proposition 9.1, (iii)], that can beseen as an etale version of [53, Theorem 8.6, p. 160].

Theorem 1.2.8. (Lyubeznik) Letk be a separably closed field of arbitrary character-istic, Y ⊆ X two projective schemes overk such that U:= X \Y is nonsingular. SetN := dimX and letℓ ∈ Z be coprime with the characteristic ofk. If ecd(U)< 2N− r,then the restriction maps

H i(Xet,Z/ℓZ)−→ H i(Yet,Z/ℓZ)


are isomorphism for i< r and injective for i= r.

Remark 1.2.9. An etale version of Theorem1.2.4cannot exist. In fact, ifX ⊆ Pn

is a regular scheme projective over a fieldk, the natural number ecd(Pn \X) is not aninvariant ofX andn, as instead is for cd(Pn\X), see Remark1.2.5. For instance, we canconsiderP2 ⊆ P5 embedded as a linear subspace andv2(P2)⊆ P5 (wherev2(P2) is the2nd Veronese embedding): the first one is a complete intersection, so ecd(P5\P2)≤ 7by (A.7). Instead, ecd(P5\ v2(P2)) = 8 by a result of Barile [5].

Notice that the above argument shows that the number of defining equations of aprojective schemeX ⊆ Pn depends on the embedding, and not only onX and onn. Thissuggests the limits of the use of local cohomology on certainproblems regarding thearithmetical rank.

We want to end this subsection stating a result similar to Proposition1.2.6. Firstwe need a remark, which is analogous to the last part of Remark1.2.2.

Remark 1.2.10. Let X be a regular scheme projective over a fieldk andℓ an integercoprime to char(k). The cycle map is a graded homomorphism

⊕

i∈NCHi(X)⊗Qℓ −→

⊕

i∈NH2i(Xet,Qℓ(i)),

where by CH∗(X) we denote the Chow ring ofX (for the definition of the cycle map seethe book of Milne [82, Chapter VI]). Let Ni(X) be the group CHi(X)modulo numericalequivalence. To prove [82, Chapter VI, Theorem 11.7], it is shown that the degreei-partof the kernel of the cycle map is contained in the kernel of

CHi(X)⊗Qℓ −→ Ni(X)⊗Qℓ.

As X is projective, Ni(X) is nonzero for anyi = 0, . . . ,dimX. Therefore,H2i(Xet,Zℓ(i))is nonzero for anyi = 0, . . . ,dimX. Finally, by the definition ofH2i(Xet,Zℓ(i)), thereexists an integermsuch that

H2i(Xet,Z/ℓmZ(i)) 6= 0

for any i = 0, . . . ,dimX.

Proposition 1.2.11. Let k a separably closed field of arbitrary characteristic. Let Xand Y be two positive dimensional regular schemes projective overk. Let us choosean embedding Z:= X ×Y ⊆ PN, and set U:= PN \ Z. Thenecd(U) ≥ 2N− 3. Inparticular if dimZ ≥ 3, Z is not a set-theoretic complete intersection.

Proof. By Remark1.2.10there is an integerℓ coprime with char(k) such that the mod-ules

H2i(Xet,Z/ℓZ(i)) 6= 0 and H2i(Yet,Z/ℓZ(i)) 6= 0

for i = 0,1. Therefore, using Kunneth formula for etale cohomology, [82, Chapter VI,Corollary 8.13], one can easily show thatH2(Zet,Z/ℓZ(1)) cannot be isomorphic to

Z/ℓZ. HoweverH2(PNet,Z/ℓZ(1)) ∼= Z/ℓZ (see [82, p. 245]). Now Theorem1.2.8

implies the conclusion.

Remark 1.2.12. With the notation of Proposition1.2.11, if dim Z ≥ 3, then it is nota set-theoretic complete intersection even ifk is not separably closed. In fact, if itwere,Z×k Spec(ks) would be a set-theoretic complete intersection (ks stands for theseparable closure ofk). This would contradict Proposition1.2.11.


1.2.3 Two consequences

In this subsection we draw two nice consequences of the investigations we made in thefirst part of the section.

The first fact we want to present is a consequence of Theorem1.2.4, and regards arelationship between cohomological dimension of an ideal in a polynomial ring and thedepth of the relative quotient ring. Let us recall a result byPeskine and Szpiro, alreadymentioned in the Preliminaries’ chapter.

Theorem 1.2.13.(Peskine and Szpiro) Let R be an n-dimensional regular localring ofpositive characteristic anda⊆ R an ideal. Ifdepth(R/a)≥ t, thencd(R,a)≤ n− t.

The same assertion does not hold in characteristic 0. In the following example weshow how, if the characteristic is not positive, Theorem1.2.13can fail already witht = 4.

Example 1.2.14.Let A := k[x,y,z], B := k[u,v] and T := A♯B their Segre product.Clearly T is a quotient of S:= k[X0, . . . ,X5]. Let I be the kernel of the surjectionS→ T. Arguing like in Remark1.2.7, T is Cohen-Macaulay. SincedimT = 4, wehave thatdepth(S/I) = depth(T) = 4. If char(k) = 0, Proposition1.2.6implies thatcd(P5 \Proj(T)) ≥ 2. Using(A.6), we getcd(S, I) ≥ 3. Now we can easily carry thisexample in the local case. Letm := (X0, . . . ,X5) denote the maximal irrelevant ideal ofS, and set R:= Sm. Clearly R is a6-dimensional regular local ring. Ifa := IR, we havedepth(R/a) = depth(S/I) = 4. Furthermore Hi

a(R)∼= (H iI (S))m for any i∈N by (0.2).

Since HiI (S) = 0 if and only if (H i

I (S))m = 0 (see Bruns and Herzog [13, Proposition1.5.15 c)]),

cd(R,a) = cd(S, I)≥ 3> 6−4.

When t = 2, Theorem1.2.13holds true also for regular local rings containing afield of characteristic 0. This fact easily follows from [64, Theorem 2.9], as we aregoing to show below.

Proposition 1.2.15. Let R be an n-dimensional regular local ring containing a fieldanda⊆ R an ideal. Ifdepth(R/a)≥ 2, thencd(R,a)≤ n−2.

Proof. Let A := ((R)sh)∧ the completion of the strict Henselization ofR. SinceA isfaithfully flat over R, we have cd(A,aA) = cd(R,a) using Lemma0.2.2 (i). More-over, sinceA/aA is faithfully flat over R/a, by Lemma0.2.2 (i) and (0.8) we getdepth(A/aA)= depth(R/a)≥ 2. Therefore, by PropositionB.1.2, A/aA is 1-connected.At this point the thesis follows by [64], see Theorem1.1.16.

Notice that Theorem1.2.13and Proposition1.2.15can be stated also ifR is apolynomial ring over a field anda is a homogeneous ideal. In fact it would be enoughto localize at the maximal irrelevant ideal to reduce the problem to the local case. FromTheorem1.2.4we are able to settle another case, in characteristic 0, of Theorem1.2.13.

Theorem 1.2.16.Let S:= k[x1, . . . ,xn] be a polynomial ring over a fieldk. Let I ⊆ Sbe a homogeneous prime ideal such that(S/I)℘ is a regular local ring for any homo-geneous prime ideal℘ 6=m := (x1, . . . ,xn). If depth(S/I)≥ 3, thencd(S, I)≤ n−3.

Proof. If the characteristic ofk is positive, the statement easily follows from Theorem1.2.13. Therefore we can assume char(k) = 0. Suppose by contradiction that cd(S, I)≥n−2. SetX := Proj(S/I)⊆ Pn−1 = Proj(S). So we are supposing that cd(Pn−1\X)≥


n−3 by (A.6). By the assumptions,X is a regular scheme projective overk, thereforeTheorem1.2.4implies thath10(X) 6= 0 or thath01(X) 6= 0. But, with the notation ofRemark1.2.1, we have thath10(X) = h10(XC) andh01(X) = h01(XC). SinceXh

C is acompact Kahler manifold, it is a well known fact of Hodge theory that

H1(XhC,OXh

C)∼= H0(Xh

C,ΩXhC)

(for instance see [2, Theorem 10.2.4]). Therefore TheoremA.2.1 implies h10(XC) =h01(XC). So

h10(X) 6= 0.

By (A.4) we haveH1(X,OX) = [H2m(S/I)]0 ⊆ H2

m(S/I). But therefore (0.8) impliesdepth(S/I)≤ 2, which is a contradiction.

Together with Theorem1.2.13and Proposition1.2.15, Theorem1.2.16raises thefollowing question:

Question 1.2.17.Suppose thatR is a regular local ring, and thata⊆ R is an ideal suchthat depth(R/a)≥ 3. Is it true that cd(R,a)≤ dimR−3?

But Theorem1.2.16, another supporting fact for a positive answer to Question1.2.17is provided by the following theorem.

Theorem 1.2.18.Let (R,m) be an n-dimensional regular local ring containing a field,and leta be an ideal of R. Ifdepth(R/a)≥ 3, then:

(i) cd(R,a)≤ n−2;(ii) Supp(Hn−2

a (R))⊆ m.

Proof. By Proposition1.2.15we have cd(R,a)≤ n−2. So we have just to show that,given a prime ideal℘ different fromm, we have(Hn−2

a (R))℘ = 0. Using (0.2) this isequivalent to show that

Hn−2aR℘

(R℘) = 0 ∀℘∈ Spec(R)\ m. (1.14)

Let us denote byh the height of℘. Thenh≤ n−1 because℘ 6=m. We can furthermoresuppose thath≥ n−2, since otherwise dimR℘ < n−2 andHn−2

aR℘(R℘) would be zero

from Theorem0.1.3. In such a case℘ cannot be a minimal prime ideal ofa becausedepth(R/a)≥ 3 (for instance, see [80, Theorem 17.2]). Particularly, dimR℘/aR℘ > 0.

First let us suppose thath = n− 2. Notice that bothR℘ and R℘ are (n− 2)-dimensional regular local domains. Moreover,

dimR℘/aR℘ = dimR℘/aR℘ > 0.

Therefore Hartshorne-Lichtenbaum Theorem0.3.2implies thatHn−2aR℘

(R℘) = 0. So, by

(0.4) we get (1.14).So we can supposeh= n−1. A theorem of Ishebeck ([80, Theorem 17.1]) yields

Ext0R(R/℘,R/a) = Ext1R(R/℘,R/a) = 0.

This means that grade(℘,R/a) > 1 and so thatH0℘(R/a) = H1

℘(R/a) = 0 by (0.7).Using (0.2) this implies

H0℘R℘(R℘/aR℘) = H1

℘R℘(R℘/aR℘) = 0.


This means that depth(R℘/aR℘) ≥ 2 by (0.8). SinceR℘ is an (n− 1)-dimensionalregular local ring containing a field, Proposition1.2.15yields

Hn−2aR℘

(R℘) = 0.

This concludes the proof.

The second fact we want to present in this subsection concerns a possible relation-ship between ordinary and etale cohomological dimension.Such a kind of questionwas already done by Hartshorne in [53, p. 185, Problem 4.1]. In [77, Conjecture, p.147] Lyubeznik made a precise conjecture about this topic:

Conjecture 1.2.19. (Lyubeznik). IfU is ann-dimensional scheme of finite type overa separably closed field, then ecd(U)≥ n+ cd(U).

Using Theorems1.2.4and1.2.8, we are able to solve the conjecture in a specialcase.

Theorem 1.2.20.Let X ⊆ Pn be a regular scheme projective over an algebraicallyclosed fieldK of characteristic0, and U := Pn \X. Then

ecd(U)≥ n+ cd(U)

Proof. Let k be the field obtained fromQ adding the coefficients of a set of generatorsof the defining ideal ofX, like in Remark1.2.1, and letk denote the algebraic closureof k (recall that in characteristic 0 separable and algebraic closures are the same thing).So we have

Q⊆ k⊆ C and Q⊆ k⊆K.Therefore, from [82, Chapter VI, Corollary 4.3], we have

ecd(U) = ecd(Uk) = ecd(UC).

Moreover, (1.6) yields cd(U) = cd(UC). So, from now on, we supposeK= C.Set cd(U) := s, and define an integerρs to be 0 (resp. 1) ifn− s− 1 is odd

(resp. ifn− s−1 is even). By Theorem1.2.4and by equation (1.11), it follows thatβn−s−1(X)> ρs. Consider, for a prime numberp, theZ/pZ-vector space

HomZ(HSingi (Xh,Z),Z/pZ).

SinceHSingi (Xh,Z) is of rank bigger thanρs, the aboveZ/pZ-vector space has dimen-

sion greater thanρs. Therefore by the surjection given by TheoremA.3.2

Hn−s−1Sing (Xh,Z/pZ)−→ HomZ(H

Singn−s−1(X

h,Z),Z/pZ),

we infer that dimZ/pZHn−s−1Sing (Xh,Z/pZ) > ρs. Now a comparison theorem due to

Grothendieck (see TheoremA.2.2) yields

dimZ/pZHn−s−1(Xet,Z/pZ)> ρs.

Since dimZ/pZ(Hn−s−1(Pn

et,Z/pZ)) = ρs by [82, p. 245], Theorem1.2.8implies that

ecd(U)≥ 2n− (n− s) = n+ s.


Theorem1.2.20might look like a very special case of Conjecture1.2.19. Howeverthe case whenU is the complement of a projective variety in a projective space is a veryimportant case. In fact the truth of Conjecture1.2.19would ensure that to bound thehomogeneous arithmetical rank from below it would be enoughto work just with theetale site, and not with the Zariski one. Since usually one is interested in computing thenumber of (set-theoretically) defining equations of a projective scheme in the projectivespace, in some sense the most interesting case of Conjecture1.2.19is whenU = Pn\Xfor some projective schemeX. From this point of view, one can look at Theorem1.2.20in the following way: In order to give a lower bound for the minimal numberof hypersurfaces ofPn

C cutting out set-theoretically a regular scheme projectiveovera field of characteristic0, say X⊆ Pn, it is better to computeecd(Pn \X) rather thancd(Pn \X).

Recently Lyubeznik, who informed us by a personal communication, found a coun-terexample to Conjecture1.2.19when the characteristic of the base field is positive: Hiscounterexample, not yet published, consists in a schemeU which is the complement inPn of a reducible projective scheme.

1.2.4 Translation into a more algebraic language

We want to end this chapter translating Theorem1.2.4, for the convenience of somereaders, in a more algebraic language. So, we leave the geometric notation, settingS:= k[x1, . . . ,xn] the polynomial ring inn variables over a fieldk of characteristic 0.

Theorem 1.2.21.Let I ⊆ S= k[x1, . . . ,xn] be a nonzero homogeneous ideal such thatA℘ is a regular ring for any homogeneous prime ideal℘ of A := S/I different fromthe irrelevant maximal idealn := A+. If r is an integer such thatht(I)≤ r ≤ n−1, thenthe following are equivalent:

(i) Local cohomology with support in I vanish beyond r, that is cd(S, I)≤ r.(ii) Denoting byΩA/k the module of Kahler differentials of A overk and byΩ j its

jth exterior powerΛ jΩA/k,

dimk(Hin(Ω

j )0) =

0 if i ≥ 2, i 6= j +1, i+ j < n− r1 if i ≥ 2, i = j +1, i+ j < n− rdimk(H0

n(Ω j )0)−1 if i = 1, j 6= 0, j < n− r −1dimk(H0

n(A)0) if i = 1, j = 0, r < n−1

(iii) Viewing A andΩ j as S-modules,

dimk(ExtiS(Ωj ,S)−n) =

0 if i ≤ n−2, i 6= n− j −1, i− j > r1 if i ≤ n−2, i = n− j −1, i− j > rdimk(ExtnS(Ω j ,S)−n)−1 if i = n−1, j 6= 0, j < n− r −1dimk(ExtnS(A,S)−n) if i = n−1, j = 0,r < n−1

Proof. SetX := V+(I)⊆ Pn−1 andU := Pn−1\X. Then cd(S, I)≤ r ⇐⇒ cd(U)< rby (A.6). By the assumptionsX is a regular scheme, therefore we can use the charac-terization of Theorem1.2.4as follows: First of all notice thatΩq

X/k∼= Ωq. So, (A.4)

implies:hpq(X) = dimk(H

p+1n (Ωq)0) ∀ p≥ 1.

Moreover, by (A.3), we have:

h0q(X) = dimk(H1n(Ω

q)0)−dimk([H0n(Ω

q)]0)+1.

Thus, using Theorem1.2.4we get the equivalence between (i) and (ii). The equivalencebetween (ii) and (iii) follows by local duality, see Theorem0.4.6. In fact, viewing


Ωq both asA- andS-modules, point (ii) of Theorem0.2.2guarantees thatH pn(Ωq) ∼=

H pm(Ωq), wherem is the irrelevant maximal ideal ofS. Therefore Theorem0.4.6yields

dimk(Hpn(Ωq)0) = dimk(Extn−p

S (Ωq,S)−n).

At first blush, conditions (ii) and (iii) of Theorem1.2.21may seem even moredifficult than understanding directly whetherH i

I (S) = 0 for eachi > r. However, oneshould focus on the fact that the modulesH i

n(Ω j) depend only on the ringA, and not bythe chosen presentation as a quotient of a polynomial ring. Furthermore, theS-modulesExtiS(Ω j ,S), being finitely generated, are much more wieldy with respectto the localcohomology modulesH i

I (S).

Chapter 2

Properties preserved underGr obner deformations andarithmetical rank

This chapter is devoted to some applications of the results of Chapter1. Essentially,it is structureted in two different parts, which we are goingto describe. Most of theresults appearing in this chapter are borrowed from our works [103, 104].

In Section2.1, we study the connectedness behavior under Grobner deformations.That is, if I is an ideal of the polynomial ringS:= k[x1, . . . ,xn] and≺ is a monomialorder onS, we study whether the ringS/ in≺(I) inherits some connectedness propertiesof S/I . The answer is yes, in fact we prove that, ifr is an integer less than dimS/I , thenS/ in≺(I) is r-connected wheneverS/I is r-connected (Corollary2.1.3). Actually, moregenerally, we prove the analog version for initial objects with respect to weights (The-orem2.1.2). This fact generalizes the result of Kalkbrener and Sturmfels [67, Theorem1], which says that, ifI is a geometrically prime ideal, in the sense that it is prime and itremains prime once we tensor with the algebraic closure ofk, then the simplicial com-plex ∆(

√in≺(I)) is pure and strongly connected (Theorem2.1.4). As a consequence

they settled a fact conjectured by Kredel and Weispfenning [68]. However, our proofis different from the one of Kalkbrener and Sturmfels: In fact, it leads to their resultjust assuming the primeness ofI , and not the geometrically primeness. If anything,the proof of Theorem2.1.2is inspired to the approach used in the lectures of Hunekeand Taylor [65] to show the theorem of Kalkbrener and Sturmfels. Another immediateconsequence of Corollary2.1.3is that, if I is homogeneous andV+(I) is a positive di-mensional connected projective scheme, thenV+(in≺(I)) is connected too (Corollary2.1.5). Eventually, we prove an even more general version of Corollary 2.1.3, replacingSwith anyk-subalgebraA⊆ Ssuch that in≺(A) is finitely generated (Corollary2.1.7).

Another nice consequence of Corollary2.1.3 is the solution of the conjecture ofEisenbud and Goto, formulated in their paper [37], for a new class of ideals. Namely,we prove the conjecture for idealsI which have an initial ideal with no embeddedprimes (Theorem2.1.11). In particular, we show that the Eisenbud-Goto conjectureistrue for ideals defining ASL, see Corollary2.1.12.

In Subsection2.1.3, we noticed that Corollary2.1.3also implies that the simpli-cial complex∆ := ∆(

√in≺(I)) is pure and strongly connected wheneverI is a ho-

24 Properties preserved under Grobner deformations and arithmetical rank

mogeneous ideal such thatS/I is Cohen-Macaulay (Theorem2.1.13). At first blush,the reader might ask whetherk[∆] is even Cohen-Macaulay wheneverS/I is Cohen-Macaulay. However, Conca showed, using the computer algebra system Cocoa [23],that this is not the case (see Example2.1.14). A more reasonable question could be thefollowing:

Question 2.1.16 If S/I is Cohen-Macaulay and in≺(I) is square-free, isS/ in≺(I)Cohen-Macaulay as well asS/I?

Maybe the answer to this question is negative, and the reasonwhy we cannot find acounterexample to it is that the property of in≺(I) being square-free is very rare. How-ever, in some cases we can give an affirmative answer to the above question, namely:

(i) If dim(S/I)≤ 2 (Proposition2.1.17).(ii) If S/I is Cohen-Macaulay with minimal multiplicity (Proposition2.1.18).(iii) For certain ASL (Proposition2.1.19).

Section2.2is dedicated to the study of the arithmetical rank of certainalgebraic va-rieties. The beauty of finding the number of defining equations of a variety is expressedby Lyubeznik in [75] as follows:

Part of what makes the problem about the number of defining equations so inter-esting is that it can be very easily stated, yet a solution, inthose rare cases when it isknown, usually is highly nontrivial and involves a fascinating interplay of Algebra andGeometry.

The varieties whose we study the number of defining equationsare certain Segreproducts of two projective varieties. In Subsection2.2.2we prove that, ifX andYare two smooth projective schemes over a fieldk, then, to define set-theoretically theSegre productX×Y embedded in some projective spacePN, are needed at leastN−2equations (Proposition2.2.2). Furthermore, ifX is a curve of positive genus, thenthe needed equations are at leastN− 1 (Proposition2.2.3). Both these results comefrom cohomological considerations, and they are consequences of the work done inChapter1. On the other hand, the celebrated theorem of Eisenbud and Evans [36,Theorem 2] tells us that, in any case,N homogeneous equations are enough to defineset-theoretically any projective scheme inPN. Unfortunately, to decide, once fixedXandY, whether the minimum number of equations isN, N−1 orN−2, is a very hardproblem. We will solve this issue in some special cases. Let us list some works thatalready exist in this direction.

(i) In their paper [14], Bruns and Schwanzl studied the number of defining equationsof a determinantal variety. In particular they proved that the Segre product

Pn×Pm ⊆ PN where N := nm+n+m

can be defined set-theoretically byN−2 homogeneous equations and not less.In particular, it is a set-theoretic complete intersectionif and only if n= m= 1.

(ii) In their work [96], Singh and Walther gave a solution in the case of

E×P1 ⊆ P5,

whereE is a smooth elliptic plane curve: The authors proved that thearithmeticalrank of this Segre product is 4. Later, in [98], Song proved that the arithmeticalrank ofC×P1, whereC ⊆ P2 is a Fermat curve (i.e. a curve defined by theequationxd

0 + xd1 + xd

2), is 4 provided thatd ≥ 3. In particular bothE×P1 andC×P1 are not set-theoretic complete intersections.

2.1 Grobner deformations 25

In light of these results (especially we have been inspired to [96]), it is natural to studythe following problem.

Let n,m,d be natural numbers such that n≥ 2 and m,d ≥ 1, and let X⊆ Pn bea smooth hypersurface of degree d. Consider the Segre product Z := X ×Pm ⊆ PN,where N:= nm+n+m. What can we say about the number of defining equations ofZ?

Notice that the arithmetical rank ofZ can depend, at least a priori, by invariantdifferent fromn,m,d: In fact we will need other conditions onX. However, for certainn,m,d, we can provide some answers to this question.

In the casen= 2 andm= 1, we introduce, for everyd, a locus of special smoothprojective plane curves of degreed, that we will denote byVd (see Remark2.2.13):This locus consists in those smooth projective curvesX of degreed which have ad-flex, i.e. a pointp at which the multiplicity intersection ofX and the tangent line inp isequal tod. Using methods coming from “ASL theory” (see De Concini, Eisenbud andProcesi [29] or Bruns and Vetter [15]), we prove that for such a curveX the arithmeticalrank of the Segre productX×P1 ⊆ P5 is 4, provided thatd≥ 3 (see Theorem2.2.7). Itis easy to show that every smooth elliptic curve belongs toV3 and that every Fermat’scurve of degreed belongs toVd, so we recover the results of [96] and of [98] (seeCorollaries2.2.9and2.2.11). However, the equations that we will find are differentfrom the ones found in those papers, and our result is characteristic free. Moreover, aresult of Casnati and Del Centina [22] shows that the codimension ofVd in the locus ofall the smooth projective plane curves of degreed is d−3, provided thatd≥ 3 (Remark2.2.13). So we compute the arithmetical rank ofX ×P1 ⊆ P5 for a lot of new planecurvesX.

For a generaln, we can prove that ifX ⊆ Pn is a general smooth hypersurface ofdegree not bigger than 2n−1, then the arithmetical rank ofX×P1 ⊆ P2n+1 is at most2n (Corollary 2.2.15). To establish this we need a higher-dimensional version ofVd

and Lemma2.2.14, suggested us by Ciliberto. This result is somehow in the directionof the open question whether any connected projective scheme of positive dimensionin PN can be defined set-theoretically byN−1 equations.

With some similar tools we can show that, ifF := xdn +∑n−3

i=0 xiGi(x0, . . . ,xn) andX := V+(F) is smooth, then the arithmetical rank ofX×P1 ⊆ P2n+1 is exactly 2n−1(Theorem2.2.16).

Eventually, using techniques similar to those of [96], we are able to show the fol-lowing: The arithmetical rank of the Segre productX ×Pm ⊆ P3m+2, whereX is asmooth conic ofP2, is equal to 3m, provided that char(k) 6= 2 (Theorem2.2.18). Inparticular,X×Pm is a set-theoretic complete intersection if and only ifm= 1.

2.1 Grobner deformations

2.1.1 Connectedness preserves under Grobner deformations

The main result of this subsection is Theorem2.1.2, which will launch the results ofthe two next subsections.

Lemma 2.1.1.Let I be an ideal of S andω ∈Nn≥1. For all r ∈ Z, if S/I is r-connected,

then S[t]/homω (I) is (r +1)-connected.

Proof. Let ℘1, . . . ,℘m be the minimal prime ideals ofI . Then, by LemmaC.3.2(v),it follows that homω(℘1), . . . ,homω(℘m) are the minimal prime ideals of homω (I).


Suppose by contradiction thatS[t]/homω (I) is not(r+1)-connected. Then, by LemmaB.0.5 (i), we can chooseA,B ⊆ 1, . . . ,m disjoint, such thatA∪B = 1, . . . ,m andsuch that, settingJ := ∩i∈Ahomω (℘i) andK := ∩ j∈Bhomω(℘j),

dimS[t]/(J +K )≤ r.

SetJ := ∩i∈A℘i andK := ∩ j∈B℘j . LemmaC.3.2(i) implies that homω(J) = J andhomω (K) = K . Obviously,J +K ⊆ homω(J+K). So, using LemmaC.3.2(vi),we get

r ≥ dimS[t]/(J +K )≥ dimS[t]/homω(J+K) = dimS/(J+K)+1.

At this point, LemmaB.0.5(i) implies thatS/I is not r-connected, which is a contra-diction.

Theorem 2.1.2.Let I be an ideal of S,ω ∈ Nn≥1 and r an integer less thandimS/I. If

S/I is r-connected, then S/ inω (I) is r-connected.

Proof. Note thatS[t]/homω (I) is a graded ring withk as degree 0-part (consideringthe ω-graduation). Moreover,(t) := (homω(I)+ (t))/homω (I) ⊆ S[t]/homω (I) is ahomogeneous ideal ofS[t]/homω(I). The ringS[t]/homω (I) is (r +1)-connected byLemma2.1.1, and

cd(S[t]/homω (I),(t))≤ 1= (r +1)− r

by (0.10). So Theorem1.1.11yields thatS[t]/(homω(I)+(t)) is r-connected. Eventu-ally, we get the conclusion, because PropositionC.2.2says that

S[t]/(homω(I)+ (t))∼= S/ inω (I).

Using TheoremC.2.3, we immediately get the following Corollary.

Corollary 2.1.3. Let I be an ideal of S,≺ a monomial order on S and r an integer lessthandimS/I. If S/I is r-connected, then S/ in≺(I) is r-connected.

Given a monomial order≺ on S, for any idealI let ∆≺(I) denote the simplicialcomplex∆(

√in≺(I)) on [n] (seeE). Corollary2.1.3gets immediately [67, Theorem

1].

Theorem 2.1.4. (Kalkbrener and Sturmfels). Let≺ a monomial order on S. If I is aprime ideal, then∆≺(I) is pure of dimensiondimS/I −1 and strongly connected.

Proof. Since I is prime, S/I is d-connected, whered := dimS/I . So S/I is (d −1)-connected, and Corollary2.1.3 implies thatS/ in≺(I) is (d − 1)-connected. SoS/√

in≺(I) is connected in codimension 1, which is equivalent to∆≺(I) being stronglyconnected by RemarkE.0.3.

The following is another nice consequence of Theorem2.1.2.

Corollary 2.1.5. Let I be a homogeneous ideal of S andω ∈ Nn≥1. If V+(I) ⊆ Pn−1 is

a positive dimensional connected projective scheme, thenV+(inω(I)) ⊆ Pn−1 is con-nected as well.


Proof. We recall that the fact thatV+(I) is connected is equivalent toV+(I) being 0-connected. Moreover, RemarkB.0.4, sinceV+(I) is homeomorphic to Proj(S/I), yieldsthatS/I is 1-connected. BecauseV+(I) is positive dimensional, we have dimS/I ≥ 2.So, Theorem2.1.2 implies thatS/ inω (I) is 1-connected. At this point, we can gobackwards getting the connectedness ofV+(inω(I)).

Remark 2.1.6. Once fixed an idealI ⊆ S, an integerr < dimS/I =: d and a monomialorder≺, Corollary2.1.3yields

S/I r -connected=⇒ S/ in≺(I) r-connected.

In general, the reverse of such an implication does not hold true. To see this, let us recallthat there exists a nonempty Zariski open setU ⊆ GL(V), whereV is ak-vector spaceof dimensionn, and a Borel-fixed idealJ ⊆ S, i.e an ideal fixed under the action of thesubgroup of all upper triangular matricesB+(V) ⊆ GL(V), such that in≺(gI) = J forall g∈U . The idealJ is called the generic initial ideal ofI , for instance see the book ofEisenbud [34, Theorem 15.18, Theorem 15.20]. It is known that, sinceJ is Borel-fixed,√

J = (x1, . . . ,xc) wherec= ht(I), see [34, Corollary 15.25]). HenceS/J= S/ in≺(gI)(for g∈U) is d-connected. On the other hand, for allg∈ GL(V), S/gI is r-connectedif and only if S/I is r-connected, and clearly, for anyr ≥ 1, we could have started froma homogeneous ideal notr-connected.

The last corollary of Theorem2.1.2we present in this subsection actually strength-ens Theorem2.1.2.

Corollary 2.1.7. Let A be ak-subalgebra of S andω ∈ Nn≥1 be such thatinω (A) is

finitely generated. If J is an ideal of A and r is an integer lessthan dimA/J, theninω(A)/ inω(J) is r-connected whenever A/J is r-connected.

Proof. We can reduce the situation to Theorem2.1.2using the result [20, Lemma 2.2]of the notes of Bruns and Conca.

2.1.2 The Eisenbud-Goto conjecture

Theorem2.1.2gives also a new class of ideals for which the Eisenbud-Goto conjectureholds true. First of all we recall what the conjecture claims. Let I be a homogeneousideal ofS. TheHilbert seriesof S/I is:

HSS/I (z) := ∑k∈N

HFS/I (k)zd ∈N[[z]],

where HF is the Hilbert function, see0.4.3. It is well known that

HSS/I (z) =hS/I (z)

(1− z)d

whered is the dimension ofS/I andhS/I ∈ Z[z], usually called theh-vectorof S/I , issuch thathS/I (1) 6= 0 (for instance see the book of Bruns and Herzog [13, Corollary4.1.8]). Themultiplicity of S/I is:

e(S/I) = hS/I (1).

Eisenbud and Goto, in [37], conjectured an inequality involving the multiplicity, theCastelnuovo-Mumford regularity and the height of a homogeneous ideal, namely:


Conjecture 2.1.8. (Eisenbud-Goto). LetI ⊆ S be a homogeneous radical ideal con-tained inm2, wherem := (x1, . . . ,xn). If S/I is connected in codimension 1, then

reg(S/I)≤ e(S/I)−ht(I).

Remark 2.1.9. Conjecture2.1.8can be easily shown whenS/I is Cohen-Macaulay.For an account of other known cases see the book of Eisenbud [35].

We recall that, for a monomial idealI ⊆ S, we denoteI ⊆ S the polarization ofI(seeE.3).

Lemma 2.1.10.Let I be a monomial ideal of S with no embedded prime ideals. Then,fixed c> 0, S/I is connected in codimension c if and only ifS/I is connected in codi-mension c.

Proof. SinceI has not embedded prime ideals, it has a unique primary decomposition,which is of the form:

I =⋂

F∈F (∆)IF ,

where∆ = ∆(√

I) and for any facetF ∈F (∆), the idealIF is a primary monomial idealwith

√IF =℘F = (xi : i ∈ F). Because polarization commutes with intersections, see

(E.6), we have that:I =

⋂

F∈F (∆)IF ⊆ S.

Being eachIF a primary monomial ideal, it turns out thatS/IF is Cohen-Macaulayfor anyF ∈ F (∆). ThereforeS/IF is Cohen-Macaulay for allF ∈ F (∆) by TheoremE.3.4. In particular, for all facetsF ∈F (∆), the ringS/IF is connected in codimension1 by PropositionB.1.2. So, using LemmaB.0.7, for any two minimal prime ideals℘ and℘′ of IF , there is a sequence℘=℘0, . . . ,℘s =℘′ of minimal prime ideals ofIF such that ht(℘i +℘i−1) ≤ ht(IF) + 1 = ht(IF) + 1 = |F |+ 1 for any i = 1, . . . ,s.Furthermore, one can easily show that the prime ideal

℘F = (xi1,1,xi2,1, . . . ,xi|F|,1)⊆ S,

whereF = i1, . . . , i|F |, is a minimal prime ideal ofIF , and therefore ofI . Seth :=ht(I). Using LemmaB.0.5(ii) and the fact thatc> 0, one can easily show:

h≤ |F |< h+ c.

Suppose thatS/I is connected in codimensionc. Let us consider two minimal primeideals℘and℘′ of I . To show thatS/I is connected in codimensionc, by LemmaB.0.7we have to exhibit a sequence℘=℘0, . . . ,℘s =℘′ of minimal prime ideals ofI suchthat ht(℘i +℘i−1)≤ h+c for anyi = 1, . . . ,s. Assume that℘ is a minimal prime ofIFand that℘′ is a minimal prime ofIG, whereF,G∈ F (∆). First of all, from what saidabove, there are two sequences℘=℘0, . . . ,℘t =℘F and℘′ =℘′

0, . . . ,℘′q =℘G such

that℘i ∈Min(IF), ℘′j ∈Min(IG), ht(℘i+℘i−1)≤ |F|+1≤ h+cand ht(℘′

j +℘′j−1)≤

|G|+1≤ h+c. Then, sinceS/I is connected in codimensionc, there exists a sequenceF = F0, . . . ,Fs = G of facets of∆ such that ht(℘Fi +℘Fi−1) ≤ h+ c. Eventually, thedesired sequence connecting℘ with ℘′ is:

℘=℘0,℘1, . . . ,℘t =℘F =℘F0,℘F1, . . . ,℘Fs =℘G =℘′q,℘

′q−1, . . . ,℘

′0 =℘′.

For the converse, the same argument works backwards.


Theorem 2.1.11.Let I ⊆ S be a homogeneous radical ideal contained inm2, andsuppose that S/I is connected in codimension1. If there exists a monomial order≺ onS such thatin≺(I) has no embedded prime ideals, then


Proof. Let≺ be a monomial order such thatJ := in≺(I) has no embedded prime ideals.The following are standard facts: reg(S/I) ≤ reg(S/J) ([20, Corollary 3.5]), ht(I) =ht(J) ([20, Theorem 3.9 (a)]) ande(S/I) = e(S/J) ([20, Proposition 1.4 (e)]). Also,obviously we have thatJ ⊆m2 as well asI . At last, Corollary2.1.3implies thatS/J isconnected in codimension 1.

BecauseS/J has no embedded prime ideals, Lemma2.1.10implies thatS/J is con-nected in codimension 1. From TheoremE.3.4, we have reg(S/J) = reg(S/J) (recallthe interpretation of the Castelnuovo-Mumford regularityin terms of the graded Bettinumbers (0.13)). Moreover, using RemarkE.3.3, we get ht(J) = ht(J) ande(S/J) =e(S/J). Besides, obviouslyJ ⊆ M2, where byM we denote the maximal irrelevantideal of S. Nevertheless, Eisenbud-Goto conjecture has been showed for square-freemonomial ideals by Terai in [102, Theorem 0.2], so that:

reg(S/J)≤ e(S/J)−ht(J).

Since, from what said above, reg(S/I) ≤ reg(S/J), ht(I) = ht(J) and e(S/I) =e(S/J), we eventually get the conclusion.

In view of Theorem2.1.11, it would be interesting to discoverclasses of ideals forwhich there exists a monomial order such that the initial ideal with respect to it has noembedded prime ideals. Actually, we already have for free a class like that: Namely,the homogeneous ideals defining an Algebra with Straightening Laws (ASL for short)overk. For the convenience of the reader we give the definition here: Let⊏ be a partialorder on[n], and let us denote byΠ the poset([n],⊏). To Π is associated a Stanley-Reisner ideal, namely the one of the order complex∆(Π) whose faces are the chains ofΠ:

IΠ := I∆(Π) = (xix j : i and j are incomparable elements ofΠ).

Given a homogeneous idealI ⊆ S, the standard graded algebraR := S/I is called aASLon Π overk if:

(i) The residue classes of the monomials not inIΠ are linearly independent inR.(ii) For everyi, j ∈ Π such thati and j are incomparable the idealI contains a poly-

nomial of the formxix j −∑λxhxk

with λ ∈ k, h,k∈ Π, h⊑ k, h⊏ i andh⊏ j. The above sum is allowed to run onthe empty-set.

The polynomials in (ii) give a way of rewriting inR the product of two incomparableelements. These relations are called thestraightening relations. Let≺ be a degrevlexmonomial order on a linear extension of⊏. Then the polynomials in (ii) form a Grobnerbasis ofI and in≺(I) = IΠ. Particularly, being square-free, in≺(I) has no embeddedprime ideals. So, we have the following:

Corollary 2.1.12. Let I ⊆ S be a homogeneous ideal defining an ASL. Then, theEisenbud-Goto conjecture2.1.8holds true for I. That is, if S/I is connected in codi-mension 1, then



Proof. Let ≺ be a degrevlex monomial order on a linear extension of the partial ordergiven by the poset underlining the structure of ASL ofS/I . Since in≺(I) is square-free,Theorem2.1.11implies the thesis.

2.1.3 The initial ideal of a Cohen-Macaulay ideal

Combining Theorem2.1.2with PropositionB.1.2 we immediately get the followingnice consequence:

Theorem 2.1.13.Let I be a homogeneous ideal of S, andω ∈ Nn≥1. If depth(S/I) ≥

r + 1, then S/ inω(ω) is r-connected. In particular, if S/I is Cohen-Macaulay, thenS/ inω (I) is connected in codimension 1.

At first blush, the reader might ask whether Theorem2.1.13could be strengthenedsaying that “S/

√inω (I) is Cohen-Macaulay wheneverS/I is Cohen-Macaulay”. How-

ever, this is far to be true, as we are going to show in the following example due toConca.

Example 2.1.14.Consider the graded ideal:

I = (x1x5+ x2x6+ x24, x1x4+ x2

3− x4x5, x21+ x1x2+ x2x5)⊆ C[x1, . . . ,x6] = S.

Using some computer algebra system, for instance Cocoa [23], one can verify that Iis a prime ideal which is a complete intersection of height3. In particular, S/I is aCohen-Macaulay domain of dimension 3. Moreover, S/I is normal. At last, the radicalof the initial ideal of I with respect to the lexicographicalmonomial order is:

√in(I) = (x1,x2,x3)∩ (x1,x3,x6)∩ (x1,x2,x5)∩ (x1,x4,x5).

In accord with Theorem2.1.13, R := S/√

in(I) is connected in codimension 1. How-ever, R is not Cohen-Macaulay. Were it, R℘ would be Cohen-Macaulay, where℘ is thehomomorphic image of the prime ideal(x1,x3,x4,x5,x6). In particular, R℘ would be1-connected by PropsitionB.1.2. The minimal prime ideals of R℘ are the homomorphicimage of those of R which are contained in℘, namely(x1,x3,x6) and(x1,x4,x5). Sincetheir sum is℘, the ring R℘ cannot be1-connected.

This example shows also as the cohomological dimension of anideal, in general,cannot be compared with the one of its initial ideal. In factcd(S, I) = 3 because I isa complete intersection of height3 (see(0.10)). But cd(S, in(I)) = projdim(R) > 3,where the equality follows by a result of Lyubeznik in [73] (it is reported in Theorem0.3.4). On the other hand, examples of ideals J⊆ S such thatcd(S/J)> cd(S/ in(J))can be easily produced following the guideline of Remark2.1.6.

In Example2.1.14, the dimension ofS/I is 3. Moreover, starting from it, we canconstruct other similar example for any dimension greater than or equal to 3. Oppo-sitely, such an example cannot exist in the dimension 2-caseby the following result.

Proposition 2.1.15. Let I ⊆ S be a homogeneous ideal such thatV+(I) ⊆ Pn−1 is apositive dimensional connected projective scheme and≺ a monomial order. Then

depth(S/√

in≺(I))≥ 2.

In particular, depth(S/√

in≺(I))≥ 2 wheneverdepth(S/I)≥ 2.


Proof. Corollary 2.1.5 implies thatV+(√

in≺(I)) ⊆ Pn−1 is connected. As one caneasily check, this is the case if and only if the associated simplicial complex∆ :=∆(√

in≺(I)) is connected. In turn, it is well known that this is the case ifand only ifthe depth ofk[∆] = S/

√in≺(I) is at least 2. The last part of the statement follows at

once by PropositionB.1.2.

As we already said, the higher dimensional analog of the lastpart of the statementof Proposition2.1.15is not true. However, one can notice that the ideal of Example2.1.14is such that its initial ideal is not square-free. Actually,even if Conca attemptedto find such an example with a square-free initial ideal, he could not find it. This factslead him to formulate, in one of our informal discussions, the following question:

Question 2.1.16.Let I ⊆ Sbe a homogeneous ideal and suppose that≺ is a monomialorder onSsuch that in≺(I) is square-free. Is depth(S/I) = depth(S/ in≺(I))?

Actually, we do not know whether to expect an affirmative answer to Question2.1.16or a negative one. In any case, we think it could be interesting to inquire into it.In the rest of this subsection, we are going to show some situations in which Ques-tion 2.1.16has an affirmative answer. First of all, we inform the reader that it iswell known that in general, even without the assumption thatin≺(I) is square-free,depth(S/I) ≥ depth(S/ in≺(I)) (for example see [20, Corollary 3.5]). So, the inter-esting part of Question2.1.16is whether, under the constraining assumption about thesquare-freeness of in≺(I), the inequality depth(S/I)≤ depth(S/ in≺(I)) holds true. Thefirst result we present in this direction is that Question2.1.16has an affirmative answerin low dimension.

Proposition 2.1.17.Let I ⊆ S be a homogeneous ideal and suppose that≺ is a mono-mial order on S such thatin≺(I) is square-free. IfdimS/I ≤ 2, then

depth(S/I) = depth(S/ in≺(I)).

Proof. Since depth(S/ in≺(I)) ≥ 1, being in≺(I) square-free, the only nontrivial caseis when depth(S/I) = 2. In this case Proposition2.1.15supplies the conclusion.

For the next result, we need to recall a definition: Let us suppose thatI ⊆ S is ahomogeneous ideal, contained inm2, such thatS/I is Cohen-Macaulayd-dimensionalring. Theh-vector ofS/I will have the form:

hS/I (z) = 1+(n−d)z+h2z2+ . . .+hsz

s,

where the Cohen-Macaulayness ofS/I forces all thehi ’s to be natural numbers. There-fore, we have that

e(S/I) = hS/I (1)≥ n−d+1.

The ringS/I is said Cohen-Macaulay withminimal multiplicitywhene(S/I) is pre-ciselyn−d+1. We have the following:

Proposition 2.1.18.Let I ⊆ S be a homogeneous ideal contained inm2, and supposethat ≺ is a monomial order on S such thatin≺(I) is square-free. If S/I is Cohen-Macaulay with minimal multiplicity, then S/ in≺(I) is Cohen-Macaulay too.

Proof. Obviously, in≺(I) ⊆ m2. Furthermore,e(S/ in≺(I)) = e(S/I) (for example see[20, Proposition 1.4 (e)]). So, using Theorem2.1.13, we have thatS/ in≺(I) is a d-dimensional Stanley-Reisner ring (whered = dimS/I ) connected in codimension 1such thate(S/ in≺(I)) = n−d+1. Then, a result of our paper with Nam [85, Proposi-tion 4.1] implies thatS/ in≺(I) is Cohen-Macaulay with minimal multiplicity.


A particular case of Question2.1.16is whether depth(S/I) = depth(S/IΠ), whereS/I is a homogeneous ASL on a posetΠ over k. The result below provides an af-firmative answer to this question in a particular case. First, we need to recall that, ifΠ = ([n],⊏) is a poset, then therank of an elementi ∈ Π, is the maximumk such thatthere is a chain

i = ik ⊐ ik−1 ⊐ . . .⊐ i1, i j ∈ Π.

Proposition 2.1.19. Let I ⊆ S be a homogeneous ideal such that S/I is an ASL on aposetΠ overk. Assume that, for each k∈ [n], there are at most two elements ofΠ ofrank k. Then S/I is Cohen-Macaulay if and only if S/IΠ is Cohen-Macaulay.

Proof. The only implication to show is that ifS/I is Cohen-Macaulay, thenS/IΠ isCohen-Macaulay. SinceIΠ is the initial ideal ofI with respect to a degrevlex mono-mial order on a linear extension on[n] of the poset order, Theorem2.1.13yields thatS/IΠ is connected in codimension 1. At this point, the result got by the author andConstantinescu [26, Theorem 2.3] implies that, because the peculiarity ofΠ, S/IΠ isCohen-Macaulay.

2.2 The defining equations of certain varieties

As we already said in the introduction, in this section, making use of results from thesecond part of Chapter1, we will estimate the number of equations needed to definecertain projective schemes.

2.2.1 Notation and first remarks

We want to fix some notation that we will use throughout this section. Byk we denotean algebraically closed field of arbitrary characteristic.We recall that theSegre productof two finitely generated gradedk-algebraA andB is defined as

A♯B :=⊕

k∈NAk⊗kBk.

This is a gradedk-algebra and it is clearly a direct summand of the tensor productA⊗kB. The name “Segre product”, as one can expect, comes from Algebraic Geometry.In the rest of this chapter, we leave the setting of the last section: In fact, instead ofworking with the polynomial ring inn variablesS= k[x1, . . . ,xn], we will work withthe polynomial ring inn+1 variables

R := k[x0, . . . ,xn].

This is due to the fact that in this section we will often have ageometric point of view:Thus, just for the sake of notation, we prefer to work withPn rather thanPn−1. We alsoneed another polynomial ring: Fixed a positive integerm, let

T := k[y0, . . . ,ym]

denote the polynomial ring inm+1 variables overk. Let X ⊆ Pn andY ⊆ Pm be twoprojective schemes defined respectively by the standard graded idealsa⊆ Randb⊆ T.SetA := R/a andB := T/b. Then, we have the isomorphism

X×Y ∼= Proj(A♯B),

2.2 The defining equations of certain varieties 33

whereX×Y is the Segre product ofX andY. Moreover, if

Q := k[xiy j : i = 0, . . . ,n; j = 0, . . . ,m]⊆ k[x0, . . . ,xn,y0, . . . ,ym] = R⊗kT,

thenA♯B∼= Q/I with I ⊆ Q the homogeneous ideal we are going to describe: Ifa=( f1, . . . , fr) andb = (g1, . . . ,gs) with degfi = di and degg j = ej , thenI is generatedby the following polynomials:

(i) M · fi whereM varies among the monomials inTdi for everyi = 1, . . . , r.(ii) g j ·N whereN varies among the monomial inRej for every j = 1, . . . ,s.

We want to presentA♯B as a quotient of a polynomial ring. So, consider the polynomialring in (n+1)(m+1) variables overk:

P := k[zi j : i = 0, . . . ,n : j = 0, . . . ,m].

Moreover, consider thek-algebra homomorphism

φ : Pψ−→ Q

π−→ A♯B,

whereψ(zi j ) := xiy j andπ is just the projection. Therefore, we are interested intodescribeI := Ker(φ). In fact,I is the defining ideal ofX×Y, since we have:

X×Y ∼= Proj(P/I)⊆ PN, N := nm+n+m.

Let us describe a system of generators ofI . For anyi = 1, . . . , r and for all monomialsM ∈ Tdi , let us choose a polynomialfi,M ∈ P such thatψ( fi,M) = M · fi . Analogously,pick a polynomialg j ,N ∈ P for all j = 1, . . . ,sand for each monomialN ∈ Rej . It turnsout that

I = I2(Z)+ J,

where:

(i) I2(Z) denotes the ideal generated by the 2-minors of the matrixZ := (zi j ).(ii) J := ( fi,M ,g j ,N : i ∈ [r], j ∈ [s], M andN are monomials ofTdi andRej ).

Our purpose is to study the defining equations (up to radical)of I in P, and so tocompute the arithmetical rank ofI . In general, this is a very hard problem. It is enoughto think that the case in whicha = b = 0 is nothing but trivial (see [14]). We will givea complete answer to this question in some other special cases. We end this subsectionremarking that, if we are interested in definingX×Y set-theoretically rather than ideal-theoretically, the number of equations immediately lowersa lot.

Remark 2.2.1. It turns out that the number of polynomials generating the kernel ofP→ A♯B is, in general, huge. In fact, for any minimal generatorfi of the ideala⊆ A,we have to consider all the polynomialsfi,M with M varying inTdi : These are

(m+dim

)

polynomials! The same applies for the minimal generators ofb ⊆ B. At the contrary,up to radical, it is enough to choosem+1 monomials for everyfi andn+1 monomialsfor everyg j , in a way we are going to explain.

For everyi = 1, . . . , r andl = 0, . . . ,m, setM := ydil . A possible choice forfi,M is:

fi,l := fi(z0l , . . . ,znl) ∈ P.

In the same way, for everyj = 1, . . . ,s andk= 0, . . . ,n we define:

g j ,k := g j(zk0, . . . ,zkm) ∈ P.


If we call J′ the ideal ofP generated by thefi,l ’s and theg j ,k’s, then we claim that:√

I =√

I2(Z)+ J′.

Sincek is algebraically closed, Nullstellensatz implies that it is enough to prove thatZ (I) = Z (I2(Z) + J′), whereZ (·) denotes the zero locus. Obviously, we haveZ (I)⊆ Z (I2(Z)+ J′). So, pick a point

p := [p00, p10, . . . , pn0, p01, . . . , pn1, . . . , p0m, . . . , pnm] ∈ Z (I2(Z)+ J′).

It is convenient to writep= [p0, . . . , pm], whereph := [p0h, . . . , pnh] is [0,0, . . . ,0] or apoint ofPn. Sincep∈ Z (I2(Z)), it follows that the nonzero points among theph’s areequal as points ofPn. Moreover, if ph is a nonzero point, actually it is a point ofX,becausefi,h(p) = 0 for all i = 1, . . . , r. Then, we get thatfi,M(p) = 0 for everyi,M andany choice offi,M. By the same argument, we can prove that also all theg j ,N’s vanishat p, so we conclude.

The authors of [14] describednm+ n+m− 2 homogeneous equations definingI2(Z) up to radical. So, putting this information together with the discussion above, wegot nm+(s+1)n+(r+1)m+s+ r homogeneous equations defining set-theoreticallyX×Y ⊆ PN.

2.2.2 Lower bounds for the number of defining equations

In this subsection, we provide the necessary lower bounds for the number of (set-theoretically) defining equations of the varieties we are interested in. The results ofthis subsection will be immediate consequences of those of Chapter1. As the readerwill notice, the lower bounds derivable from cohomologicalconsiderations are moregeneral than the gettable upper bounds. However, in general, while to obtain upperbounds one can think up a lot of clever ad hoc arguments, depending on the situation,to get lower bounds, essentially, the only available tools are cohomological consider-ations. In the cases in which they does not work we are, for themoment, completelyhelpless.

Proposition 2.2.2. Let X and Y be smooth projective schemes overk of positive di-mension. Let X×Y ⊆ PN be any embedding of the Segre product X×Y, and let I bethe ideal defining it (in a polynomial ring in N+1 variables overk). Then:

N−2≤ ara(I)≤ arah(I)≤ N.

Proof. The fact that arah(I) ≤ N is a consequence of [36, Theorem 2]. Combining(A.7) with (A.8), we get that

ara(I)≥ ecd(U)−N+1,

whereU := PN \ (X×Y). Eventually, Proposition1.2.11implies ecd(U)≥ 2N−3, sowe conclude.

The lower bound of Proposition2.2.2can be improved by one whenX is a smoothcurve of positive genus.

Proposition 2.2.3. Let X be a smooth projective curve overk of positive genus andY a smooth projective scheme overk. Let X×Y ⊆ PN be any embedding of the Segreproduct X×Y, and let I be the ideal defining it (in a polynomial ring in N+1 variablesoverk). Then:

N−1≤ ara(I)≤ arah(I)≤ N.


Proof. Once again, the fact that arah(I) ≤ N is a consequence of [36, Theorem 2].Setg the genus ofX, and choose an integerℓ prime with char(k). It is well known(see for instance the notes of Milne [83, Proposition 14.2 and Remark 14.4]) thatH1(Xet,Z/ℓZ)∼= (Z/lZ)2g. MoreoverH0(Yet,Z/ℓZ) 6= 0 andH1(PN

et,Z/ℓZ) = 0. Butby Kunneth formula for etale cohomology (for instance seethe book of Milne [82,Chapter VI, Corollary 8.13])H1((X ×Y)et,Z/ℓZ) 6= 0, therefore Theorem1.2.8im-plies that ecd(U) ≥ 2N−2, whereU := PN \ (X×Y). At this point we can concludeas in Proposition2.2.2, since

ara(I)≥ ecd(U)−N+1.

.

2.2.3 Hypersurfaces cross a projective spaces

Proposition2.2.2implies that the number of equations defining set-theoretically X ×Y ⊆ PN described in Remark2.2.1, whereN = nm+n+m, are still too much. In thissubsection we will improve the upper bound given in Proposition 2.2.2in some specialcases. In some situations we will determine the exact numberof equations needed todefineX×Y set-theoretically.

Remark 2.2.4. Assume thatX := V+(F)⊆ Pn is a projective hypersurface defined bya homogeneous polynomialF , m := 1 andY := P1. In this caseX ×Y ⊆ P2n+1 andRemark2.2.1gives us the same upper bound for the arithmetical rank of theideal IdefiningX×Y than Proposition2.2.2, namely

arah(I)≤ 2n+1.

Remark2.2.1also gets an explicit set of homogeneous polynomials generating I up toradical. Using the notation of the book of Bruns and Vetter [15], we denote by[i, j] the2-minorzi0zj1−zj0zi1 of the matrixZ, for everyi and j such that 0≤ i < j ≤ n. In [14]is proven that

I2(Z) =√( ∑i+ j=k

[i, j] : k= 1, . . . ,2n−1)

By Remark2.2.1, to get a set of generator ofI up to radical, we have only to add

F0 := F(z00, . . . ,zn0) and F1 := F(z01, . . . ,zn1).

Notice that the generators up to radical we exhibited have the (unusual) property ofbeing part of a set of minimal generators ofI .

Theorem 2.2.5. Let X= V+(F) ⊆ Pn be a hypersurface such that there exists a lineℓ⊆ Pn that meets X only at a point p. If I is the ideal defining X×P1 ⊆ P2n+1, then:

arah(I)≤ 2n

Proof. By a change of coordinates we can assume thatℓ= V+((x0, . . . ,xn−2)). The set

Ω := [i, j] : 0≤ i < j ≤ n, i + j ≤ 2n−2

is an ideal of the poset of the minors of the matrixZ=(zi j ): That is, for any two positiveintegersh andk such thath< k, if [i, j] ∈ Ω with h≤ i andk ≤ j, then[h,k] ∈ Ω. So,[15, Lemma 5.9] implies:

ara(ΩP)≤ rank(Ω) = 2n−2


(let us remind thatP = k[zi j : i = 0, . . . ,n j = 0,1]). We want to prove thatI =√

K,whereK := ΩP+(F0,F1) (with the notation of Remark2.2.4). To this aim, by Remark2.2.4and Nullstellensatz, it is enough to proveZ (I2(Z)+ (F0,F1)) = Z (K). So, set

q := [q0,q1] = [q00, . . . ,qn0,q01, . . . ,qn1] ∈ Z (K).

If q0 = 0 orq1 = 0 trivially q∈ Z (I2(Z)), so we assume that bothq0 andq1 are pointsof Pn. First let us suppose thatqi j 6= 0 for somei ≤ n−2 and j ∈ 0,1. Let us assumethat j = 0 (the casej = 1 is the same). In such a case, notice that for anyh∈ 0, . . . ,ndifferent fromi we have that[h, i] (or [i,h]) is an element ofΩ. Becauseq∈ Z (K), weget that, settingλ := qi1/qi0:

qh1 = λqh0 ∀ h= 0, . . . ,n.

This means thatq1 andq0 are the same point ofPn, so thatq ∈ Z (I2(Z)). We cantherefore assume thatqi j = 0 for all i ≤ n− 2 and j = 0,1. In this case,q0 andq1

belong toℓ∩X = p, soq0 = q1 = p. Once again, this yieldsq∈ Z (I2(Z)).

Combining Theorem2.2.5and Proposition2.2.2, we get the following corollary.

Corollary 2.2.6. Let X ⊆ Pn be a smooth hypersurface such that there exists a lineℓ⊆ Pn that meets X only at a point p. If I is the ideal defining X×P1 ⊆ P2n+1, then:

2n−1≤ ara(I)≤ arah(I)≤ 2n

Besides, combining Theorem2.2.5 and Proposition2.2.3, we get the followingresult.

Theorem 2.2.7. Let X ⊆ P2 be a smooth projective curve of degree d≥ 3 over k.Assume that there exists a lineℓ ⊆ P2 that meets X only at a point p, and let I be theideal defining the Segre product X×P1 ⊆ P5. Then

ara(I) = arah(I) = 4.

Proof. We recall that the genusg of the curveX is given by the formula:

g= 1/2(d−1)(d−2).

In particular, under our assumptions, it is positive. Thus Propsition2.2.3, together withTheorem2.2.5, lets us conclude.

Remark 2.2.8. In Theorem2.2.7, actually, we prove something more than the arith-metical rank being 4. IfX = V+(F), then the four polynomials generatingI up toradical are the following:

F(z00,z10,z20), F(z10,z11,z12), z00z11− z01z10, z00z12− z02z10.

It turns out that these polynomials are part of a minimal generating set ofI . This is avery uncommon fact.

Theorem2.2.7generalizes [96, Theorem 1.1] and [98, Theorem 2.8]. In fact weget them in the next two corollaries.

Corollary 2.2.9. Let X⊆ P2 be a smooth elliptic curve overk and let I be the idealdefining the Segre product X×P1 ⊆ P5. Then



Proof. It is well known that any plane projective curveX of degree at least three has anordinary flex, or an ordinary inflection point (see the book ofHartshorne [56, ChapterIV, Exercise 2.3 (e)]). That is, a pointp ∈ X such that the tangent lineℓ at p hasintersection multiplicity 3 withX at p. If X has degree exactly 3, which actually is ourcase, such a lineℓ does not intersectX in any other point butp. Therefore we are underthe assumptions of Theorem2.2.7, so we may conclude.

Remark 2.2.10. The equations exhibited in Corollary2.2.9are different from thosefound in [96, Theorem 1.1]. In fact, our result is characteristic free, while the authorsof [96] had to assume char(k) 6= 3.

Corollary 2.2.11. Let X= V+(F)⊆ P2 be a Fermat curve of degree d≥ 2 overk, thatis F = xd

0 + xd1 + xd

2, and let I be the ideal defining the Segre product X×P1 ⊆ P5. Ifchar(k) does not divide d, then:


Proof. Let λ ∈ k be such thatλ d =−1. So, letℓ⊆ P2 be the line defined by the linearform x0−λx1. One can easily check that:

X∩ ℓ= [λ ,1,0].

Eventually, since char(k) does not divided, X is smooth, so Theorem2.2.7 lets usconclude.

Remark 2.2.12. In the situation of Corollary2.2.11, if char(k) dividesd, thenX×P1

is a set-theoretic complete intersection inP5, that is ara(I) = arah(I) = 3. In fact, morein general, this is always the case whenF = ℓd, whereℓ ∈ k[x0,x1,x2]1, independentlyof the characteristic ofk. To see this, up to a change of coordinates we can assume thatℓ= V+(x2). Using the Nullstellensatz as usual, it is easy to see that:

√(z20, z21, z00z11− z01z10) =

√I .

On the other hand, ara(I)≥ 3 by the Hauptidealsatz (0.9).

Curves satisfying the hypothesis of Theorem2.2.7, however, are much more thanthose of Corollaries2.2.9and2.2.11. The next remark will clarify this point.

Remark 2.2.13. In light of Theorem2.2.5, it is natural to introduce the followingset. For every natural numbersn,d ≥ 1 we defineV n−1

d as the set of all the smoothhypersurfacesX ⊆ Pn, modulo PGLn(k), of degreed which have a pointp as in The-orem2.2.5. Notice that any hypersurface ofV n−1

d can be represented, by a change ofcoordinates, byV+(F) with

F = xdn−1+

n−2

∑i=0

xiGi(x0, . . . ,xn),

where theGi ’s are homogeneous polynomials of degreed−1.We start to analyze the casen= 2, and for simplicity we will writeVd instead of

V 1d . So our question is:How many smooth projective plane curves of degree d do

belong toVd? Some plane projective curves belonging toVd are:(i) Obviously, every smooth conic belongs toV2.(ii) Every smooth elliptic curve belongs toV3, as we already noticed in the proof of

Corollary2.2.9.


(iii) Every Fermat’s curve of degreed belongs toVd, as we already noticed in Corol-lary 2.2.11.

In [22, Theorem A], the authors compute the dimension of the lociVd,α , α = 1,2, ofall the smooth plane curves of degreed with exactlyα points as in Theorem2.2.5(ifthese points are nonsingular, as in this case, they are called d-flexes). They showedthatVd,α is an irreducible rational locally closed subvariety of themoduli spaceMg ofcurves of genusg= 1/2(d−1)(d−2). Furthermore the dimension ofVd,α is

dim(Vd,α) =

(d+2−α

2

)−8+3α.

Moreover, it is not difficult to show thatVd,1 is an open Zariski subset ofVd, (see [22,Lemma 2.1.2]), and so

dim(Vd) =

(d+1

2

)−5.

The locusHd of all smooth plane curves of degreed up to isomorphism is a nonempty

open Zariski subset ofP(d+2

2 ) modulo the group PGL2(k), so

dim(Hd) =

(d+2

2

)−9.

Particularly, the codimension ofVd in Hd, providedd ≥ 3, isd−3. So, for example, ifwe pick a quarticX in thehypersurfaceV4 of H4, Theorem2.2.7implies thatX×P1 ⊆P5 can be defined by exactly four equations. However,it remains an open problem tocompute the arithmetical rank of X×P1 ⊆ P5 for all the quartics X⊆ P2.

In the general case (n≥ 2 arbitrary) we can state the following lemma.

Lemma 2.2.14.Let X⊆ Pn be a smooth hypersurface of degree d. If d≤ 2n−3, or ifd ≤ 2n−1 and X is generic, then X∈ V n−1

d .

Proof. First we prove the following:

Claim. If X ⊆ Pn is a smooth hypersurface of degreed ≤ 2n−1 not containinglines, thenX ∈ V n−1

d .We denote byG(2,n+1) the Grassmannian of lines ofPn. Let us introduce the

following incidence variety:

Wn := (p, ℓ) ∈ Pn×G(2,n+1) : p∈ ℓ.

It turns out that this is an irreducible variety of dimension2n−1. Now set

Tn,d := (p, ℓ,F) ∈Wn×Ln,d : i(ℓ,V+(F); p)≥ d,

where byLn,d we denote the projective space of all the polynomials ofk[x0, . . . ,xn]d,and byi(ℓ,V+(F); p) the intersection multiplicity ofℓ andV+(F) at p (if ℓ ⊆ V+(F),then i(ℓ,V+(F); p) := +∞). Assume thatp = [1,0, . . . ,0] and thatℓ is given by theequations

x1 = x2 = . . .= xn = 0.

Then it is easy to see that, for a polynomialF ∈ Ln,d, the condition(p, ℓ,F) ∈ Tn,d

is equivalent to the fact that the coefficients ofxd0, xd−1

0 x1, . . . , x0xd−11 in F are 0.

This implies thatTn,d is a closed subset ofPn×G(2,n+1)×Ln,d: Therefore,Tn,d is a


projective scheme overk. Consider the restriction of the first projectionπ1 : Tn,d −→Wn. Clearlyπ1 is surjective; moreover, it follows by the above discussionthat all thefibers ofπ1 are projective subspaces ofLn,d of dimension dim(Ln,d)− d. Therefore,Tn,d is an irreducible projective variety of dimension

dim(Tn,d) = 2n−1+dim(Ln,d)−d.

Now consider the restriction of the second projectionπ2 : Tn,d −→ Ln,d. Clearly, ifX ∈ π2(Tn,d) is smooth and does not contain any line, thenX ∈ V n−1

d . So, to prove theclaim, we have to check the surjectivity ofπ2 wheneverd ≤ 2n−1. To this aim, sincebothTn,d andLn,d are projective, it is enough to show that for a generalF ∈ π2(Tn,d),the dimension of the fiberπ−1

2 (F) is exactly 2n−1−d. On the other hand it is clearthat the codimension ofπ2(Tn,d) in Tn,d is at leastd−2n+1 whend ≥ 2n. We proceedby induction onn (for n= 2 we already know this).

First consider the case in whichd≤ 2n−3. LetF be a general form ofπ2(Tn,d), andsetr = dim(π−1

2 (F)). By contradiction assume thatr > 2n−1−d. Consider a generalhyperplane section ofV+(F), and letF ′ be the polynomial defining it. Obviously,any element ofπ2(Tn−1,d) comes fromπ2(Tn,d) in this way, soF ′ is a generic form ofπ2(Tn,d). The condition for a line to belong to a hyperplane is of codimension 2, so thedimension of the fiber ofF ′ is at leastr −2. SinceF ′ is a polynomial ofK[x0, . . . ,xn−1]of degreed ≤ 2(n−1)−1, we can apply an induction gettingr −2≤ 2n−3−d, sothatr ≤ 2n−1−d, which is a contradiction.

We end with the case in whichd = 2n−1 (the cased = 2n−2 is easier). LetF andr be as above, and suppose by contradiction thatr ≥ 1. This implies that there exists ahypersurfaceH ⊆G(n,n+1) such that for any generalH ∈ H the polynomial defin-ing V+(F)∩H belongs toπ2(Tn−1,d). This implies that the codimension ofπ2(Tn−1,d)in Tn−1,d is less than or equal to 1, but we know that this is at least 2.

So we proved the claim. Now, we prove the lemma by induction onn. Once again,if n= 2, then it is well known to be true.

If d≤ 2n−3, then we cutX by a generic hyperplaneH. It turns out (using Bertini’stheorem) thatX ∩H ⊆ Pn−1 is the generic smooth hypersurface of degreed ≤ 2(n−1)−1, so by induction there exist a lineℓ⊆H and a pointp∈Pn such that(X∩H)∩ℓ=p. So we conclude thatX ∈ V n−1

d .It is known that the generic hypersurface of degreed ≥ 2n− 2 does not contain

lines. So ifd = 2n−2 ord = 2n−1 the statement follows by the claim.

Combining Lemma2.2.14with Theorem2.2.5we get the following.

Corollary 2.2.15. Let X⊆ Pn be a smooth hypersurface of degree d, and let I be theideal defining the Segre product X×P1 ⊆ P2n+1. If d ≤ 2n−3, or if d ≤ 2n−1 and Xis generic, then

2n−1≤ arah(I)≤ 2n.

Putting some stronger assumptions on the hypersurfaceX, we can even computethe arithmetical rank of the idealI (and not just to give an upper bound as in Theorem2.2.5).

Theorem 2.2.16.Let X = V+(F) ⊆ Pn be such that, F= xdn +∑n−3

i=0 xiGi(x0, . . . ,xn)(Gi homogeneous polynomials of degree d− 1), and let I be the ideal defining theSegre product X×P1 ⊆ P2n+1. Then

arah(I)≤ 2n−1.


Moreover, if X is smooth, then

ara(I) = arah(I) = 2n−1.

Proof. If X is smooth, Proposition2.2.2implies that ara(I) ≥ 2n− 1. Therefore,weneed to prove that the upper bound holds true. Consider the set

Ω := [i, j] : i < j, i + j ≤ 2n−3.

As in the proof of Theorem2.2.5, we have

ara(ΩR)≤ rank(Ω) = 2n−3.

The rest of the proof is completely analog to that of Theorem2.2.5.

Remark 2.2.17. Notice that, ifn≥ 4, the generic hypersurface ofPn defined by theform F = xd

n +∑n−3i=0 xiGi(x0, . . . ,xn) is smooth (whereas ifn ≤ 3 andd ≥ 2 such a

hypersurface is always singular).

The below argument uses ideas from [96]. Unfortunately, to use these kinds oftools, we have to put some assumptions to char(k).

Theorem 2.2.18.Assumechar(k) 6= 2. Let X= V+(F) be a smooth conic ofP2, andlet I be the ideal defining the Segre product X×Pm⊆ P3m+2. Then

ara(I) = arah(I) = 3m.

In particular X×Pm ⊆ P3m+2 is a set-theoretic complete intersection if and only ifm= 1.

Proof. First we want to give 3m homogeneous polynomials of the polynomial ringP = k[zi j : i = 0,1,2, j = 0, . . . ,m] which defineI up to radical. Forj = 0, . . . ,mchoose set, similarly to Remark2.2.1,

Fj := F(z0 j ,z1 j ,z2 j)

Then, for all 0≤ j < i ≤ m, set

Fi j :=2

∑k=0

∂F∂xk

(z0i ,z1i ,z2i)zjk.

Eventually, for allh= 1, . . . ,2m−1, let us put:

Gh := ∑i+ j=h

Fi j .

We claim that

I =√

J, whereJ := (Fi ,G j : i = 0, . . . ,m, j = 1, . . . ,2m−1).

The inclusionJ ⊆ I follows from the Euler’s formula, since char(k) 6= 2. As usual,to proveI ⊆

√J, thanks to Nullstellensatz, we may prove thatZ (J) ⊆ Z (I). Pick

p∈ Z (J), and write it as

p := [p0, p1, . . . , pm] where p j := [p0 j , p1 j , p2 j ].


SinceFi(p) = 0, for everyi = 0, . . . ,m, the nonzeropi ’s are points ofX. So, by Remark2.2.1, it remains to prove that the nonzeropi ’s are equal as points ofP2.

By contradiction, leti be the minimum integer such thatpi 6= 0 and there existsksuch thatpk 6= 0 andpi 6= pk as points ofP2. Moreover letj be the least among thesek (so i < j). Seth := i + j. We claim thatpk = pl provided thatk+ l = h, k< l , k 6= i,pk 6= 0 andpl 6= 0. In fact, if l < j, thenpi = pl by the minimal property ofj. For thesame reason, alsopk = pi , sopk = pl . On the other hand, ifl > j, thenk< i, sopk = pl

by the minimality ofi. SoFlk(p) = 0 for any(k, l) 6= (i, j) such thatk+ l = h, becausepk belongs to the tangent line ofX in pl (beingpl = pk). ThenGh(p) = Fji (p), and so,sincep∈ Z (J), Fji (p) = 0: This means thatpi belongs to the tangent line ofX in p j ,which is possible, beingX a conic, only ifpi = p j , a contradiction.

For the lower bound, we just have to notice that, in this case,Proposition2.2.2yields ara(I)≥ 3m.

Remark 2.2.19. Badescu and Valla, computed recently in [4], independently fromthis work, the arithmetical rank of the ideal defining any rational normal scroll. Sincethe Segre product of a conic withPm is a rational normal scroll, Theorem2.2.18is aparticular case of their result.

We end this subsection with a result that yields a natural question.

Proposition 2.2.20. Let n≥ 2 and m≥ 1 be two integers. Let X= V+(F) ⊆ Pn bea hypersurface smooth overk and let I⊆ P = k[z0, . . . ,zN] be the ideal defining anembedding X×Pm⊆ PN. Then

cd(P, I) =

N−1 if n = 2 and deg(F)≥ 3N−2 otherwise

Proof. By Remark1.2.1we can assumek= C. If Z := X×Pm ⊆ PN, Using equation(1.8) we have

β0(Z) = 1, β1(Z) = β1(X) and β2(Z) = β2(X)+1≥ 2,

where the last inequality follows by (1.11). If n= 2, notice thatβ1(X) 6= 0 if and onlyif deg(F)≥ 3. In fact, equation (1.9) yields

β1(X) = h01(X)+h10(X) = 2h01(X),

where the last equality comes from Serre’s duality (see [56, Chapter III, Corollary7.13]). Buth01(X) is the geometric genus ofX, therefore it is different from 0 if andonly if deg(F) ≥ 3. So if n = 2 we conclude by Theorem1.2.4. If n > 2, then wehaveβ1(X) = 0 by the Lefschetz hyperplane theorem (see the book of Lazarsfeld [71,Theorem 3.1.17], therefore we can conclude once again usingTheorem1.2.4.

In light of the above proposition, it is natural the following question.

Question 2.2.21.With the notation of Proposition2.2.20, if we consider the classicalSegre embedding ofX×Pm (and soN= nm+n+m), do the integers ara(I) and arah(I)depend only onn, m and deg(F)?


2.2.4 The diagonal of the product of two projective spaces

In [99] Speiser, among other things, computed the arithmetical rank of the diagonal

∆ = ∆(Pn)⊆ Pn×Pn,

provided that the characteristic of the base field is 0. In positive characteristic he provedthat the cohomological dimension ofPn×Pn\∆ is the least possible, namelyn−1, buthe did not compute the arithmetical rank of∆. In this short subsection we will givea characteristic free proof of Speiser’s result. Actually Theorem1.2.8easily impliesthat the result of Speiser holds in arbitrary characteristic, since the upper bound foundin [99] is valid in any characteristic. However, since in that paper the author did notdescribe the equations needed to define set-theoretically∆, we provide the upper boundwith a different method, that yields an explicit set of equations for∆.

To this aim, we recall that the coordinate ring ofPn×Pn is

A := k[xiy j : i, j = 0, . . . ,n]

and the idealI ⊆ A defining∆ is

I := (xiy j − x jyi : 0≤ i < j ≤ n).

Proposition 2.2.22. In the situation described aboveara(I) = arah(I) = 2n−1.

Proof. As said above, by [99, Proposition 2.1.1] we already know that arah(I)≤ 2n−1.However, we want to exhibit a new proof of this fact: Let us consider the followingover-ring ofA

R := k[xi ,y j : i, j = 0, . . . ,n].

Notice thatA is a direct summand of theA-moduleR. The extension ofI in R, namelyIR, is the ideal generated by the 2-minors of the following 2× (n+1) matrix:

(x0 . . . xn

y0 . . . yn

).

So, by [15, (5.9) Lemma], a set of generators ofIR⊆ Rup to radical is

gk := ∑0≤i< j≤n

i+ j=k

(xiy j − x jyi), k= 1, . . . ,2n−1.

Since these polynomials belong toA and sinceA is a direct summand ofR, we get√(g1, . . . ,g2n−1)A= I ,

thereforeara(I)≤ arah(I)≤ 2n−1.

For the lower bound chooseℓ coprime with char(k). Kunneth formula for etalecohomology [82, Chapter VI, Corollary 8.13] implies that

H2(Pnet×Pn

et,Z/ℓZ)∼= (Z/ℓZ)2,

while H2(∆et,Z/ℓZ)∼= H2(Pnet,Z/ℓZ)∼= Z/ℓZ. So, Theorem1.2.8yields

ecd(U)≥ 4n−2,

whereU := Pn×Pn\∆. Therefore, combining (A.8) and (A.7), we get

2n−1≤ ara(I)≤ arah(I).

Chapter 3

Relations between Minors

In Commutative Algebra, in Algebraic Geometry and in Representation Theory theminors of a matrix are an interesting object for many reasons. Surprisingly, in general,the minimal relations among thet-minors of am×n generic matrixX are still unknown.In this chapter, which is inspired to our work joint with Bruns and Conca [21], we willinvestigate them.

Let us consider the following matrix:

X :=

(x11 x12 x13 x14

x21 x22 x23 x24

)

where thexi j ’s are indeterminates over a fieldk. We remind that, as in the secondpart of Chapter2, we denote the 2-minor insisting on the columnsi and j of X by [i j ],namely[i j ] := x1ix2 j − x1 jx2i . One can verify the identity:

[12][34]− [13][24]+ [14][23]= 0.

This is one of the celebratedPlucker relations, and in this case it is the only minimalrelation, i.e. it generates the ideal of relations between the 2-minors ofX. Actually,the caset = minm,n is well understood in general, even if anything but trivial:Infact in such a situation the Plucker relations are the only minimal relations among thet-minors ofX. In particular, there are only quadratic minimal relations. This changesalready for 2-minors of a 3×4-matrix. For 2-minors of a matrix with more than 2 rowswe must first modify our notation in order to specify the rows of a minor:

[i j |pq] := xipx jq − xiqx jp.

Of course, we keep the Plucker relations, but they are no more sufficient: Some cubicsappear among the minimal relations, for example the following identity

det

[12|12] [12|13] [12|14][13|12] [13|13] [13|14][23|12] [23|13] [23|14]

= 0

does not come from the Plucker relations, see Bruns [12].One reason for which the case of maximal minors is easier thanthe general case

emerges from a representation-theoretic point of view. LetR be the polynomial ringoverk generated by the variablesxi j , andAt ⊆ R denote thek-subalgebra ofR gen-erated by thet-minors ofX. Whent = minm,n, the ringAt is nothing but than the

44 Relations between Minors

coordinate ring of the GrassmannianG(m,n) of all k-subspaces of dimensionm of ak-vector spaceV of dimensionn (we assumem≤ n). In the general case,At is thecoordinate ring of the Zariski closure of the image of the following homomorphism:

Λt : Homk(W,V)→ Homk(t∧

W,t∧

V), Λt(φ) := ∧tφ ,

whereW is ak-vector space of dimensionm. Notice that the groupG := GL(W)×GL(V) acts on each graded component[At ]d of At . If t = minm,n, each[At ]d is actu-ally an irreducibleG-representation (for the terminology about Representation Theorysee AppendixD). This is far from being true in the general case, complicating thesituation tremendously.

In this paper, under the assumption that the characteristicof k is 0, we will provethat quadric and cubics are the only minimal relations amongthe 2-minors of a 3×nmatrix and of a 4×n matrix (Theorem3.2.8). This confirms the impression of Brunsand Conca in [18]. We will use tools from the representation theory of the generallinear group in order to reduce the problem to a computer calculation. In fact, moregenerally, we will prove in Theorem3.2.6that a minimal relation betweent-minors of am×n matrix must already be in am× (m+ t)-matrix. In general, apart from particularwell known cases, we prove that cubic minimal relations always exist (Corollary3.1.8),and we interpret them in a representation-theoretic fashion. A type of these relationscan be written in a nice determinantal form: A minimal cubic relation is given bythe vanishing of the following determinant (just for a matter of space below we puts= t −1, u= t +1 andv= t +2):

det

[1, . . . ,s, t,u|1, . . . ,s, t] [1, . . . ,s, t,u|1, . . . ,s,u] [1, . . . ,s, t,u|1, . . . ,s,v][1, . . . ,s, t,v|1, . . . ,s, t] [1, . . . ,s, t,v|1, . . . ,s,u] [1, . . . ,s, t,v|1, . . . ,s,v][1, . . . ,s,u,v|1, . . . ,s, t] [1, . . . ,s,u,v|1, . . . ,s,u] [1, . . . ,s,u,v|1, . . . ,s,v]

.

The above polynomial actually corresponds to the highest weight vector of the irre-ducibleG-representationLγW⊗LλV∗, where

γ := (t +1, t+1, t−2) and λ := (t +2, t−1, t−1).

However, fort ≥ 4 there are also other irreducibleG-representations of degree 3 whichcorrespond to minimal relations, as we point out in Theorem3.1.6. We also prove thatthere are no other minimal relations for “reasons of shape”.The only minimal rela-tions of degree more than 3 which might exist would be for “reasons of multiplicity”(Proposition3.2.11).

In Thoerem3.2.1we can write down a formula for the Castelnuovo-Mumford reg-ularity of At in all the cases. This is quite surprising: We do not even knowa set ofminimal generators of the ideal of relations between minors, but we can estimate itsregularity, which is usually computed by its minimal free resolution. Essentially thisis possible thanks to a result in [18] describing the canonical module ofAt , combinedwith the interpretation of the Castelnuovo-Mumford regularity in terms of local co-homology as explained in Subsection0.4.3. The regularity yields an upper bound onthe degree of a minimal relation. Even if such a bound, in general, is quadratic inm(Corollary3.2.7), it is the best general upper bound known to us.

In the last section we also exhibit a finite Sagbi basis ofAt (Theorem3.3.2). Thisproblem was left open by Bruns and Conca in [19], where they proved the existenceof a finite Sagbi basis without describing it. With similar tools, in Theorem3.3.5, we

3.1 Some relations for the defining ideal ofAt 45

give a finite system ofk-algebra generators of the ring of invariantsAUt , whereU :=

U−(W)×U+(V) is the subgroup ofG with U−(W) (respectivelyU+(V)) the subgroupof lower (respectively upper) triangular matrices of GL(W) (respectively of GL(V))with 1’s on the diagonals. This is part of a classical sort of problems in invarianttheory, namely the “first main problem” of invariant theory.

3.1 Some relations for the defining ideal ofAt

Let k be a field of characteristic 0,m andn two positive integers such thatm≤ n and

X :=

x11 x12 · · · · · · x1n

x21 x22 · · · · · · x2n...

.... . .

. . ....

xm1 xm2 · · · · · · xmn

a m×n matrix of indeterminates overk. Moreover let

R(m,n) := k[xi j : i = 1, . . . ,m, j = 1, . . . ,n]

be the polynomial ring inm· n variables overk. As said in the introduction, we areinterested in understanding the relations between thet-minors ofX. In other words, wehave to consider the kernel of the following homomorphism ofk-algebras:

π : St(m,n)−→ At(m,n),

whereAt(m,n) is thealgebra of minors, i.e. thek-subalgebra ofR(m,n) generatedby thet-minors ofX, andSt(m,n) is the polynomial ring overk whose variables areindexed on thet-minors ofX. Since all the generators of thek-algebraAt(m,n) havethe same degree, namelyt, we can “renormalize” them: That is, at-minor will havedegree 1. This wayπ becomes a homogeneous homomorphism. LetW andV be twok-vector spaces of dimensionm andn. Of course we have the identification

St(m,n)∼= Sym(t∧

W⊗t∧

V∗) =⊕

i∈NSymd(

t∧W⊗

t∧V∗),

thusG := GL(W)×GL(V) acts onSt(m,n). The algebra of minorsAt(m,n) is also aG-representation, as explained in AppendixD, SectionD.3. Actually, for any naturalnumberd, the graded componentsSt(m,n)d andAt(m,n)d are finite dimensionalG-representations, and the mapπ is G-equivariant. Let us denote

Jt(m,n) := Ker(π).

When it does not raise confusion we just writeR, St , Jt andAt in place ofR(m,n),St(m,n), Jt(m,n) andAt(m,n).

Remark 3.1.1. Consider the following numerical situations:

t = 1 or n≤ t +1 (3.1)

t = m (3.2)

In the cases (3.1) the algebraAt is a polynomial ring, so thatJt = 0 (for the casem= n= t +1 look at the book of Bruns and Vetter, [15, Remark 10.17]). In the case


(3.2) At is the coordinate ring of the GrassmannianG(m,n) of k-subspaces ofV ofdimensionm. In this case, if 2≤ m≤ n− 2, the idealJt is generated by the Pluckerrelations. In particular it is generated in degree two. Clearly the Plucker relations occurin all the remaining cases too. So the idealJt is generated in degree at least 2 in thecases different from (3.1).

Because Remark3.1.1, throughout the paper we will assume to being in cases dif-ferent from (3.1) and (3.2). So, from now on, we feel free to assume

1< t < m and n> t +1.

Our purpose for this section is to show that in the above rangeminimal generators ofdegree 3 always appear inJt . Notice that, sinceπ is G-equivariant, thenJt is a G-subrepresentation ofSt . Moreover, ifLγW⊗LλV∗ is an irreducible representation ofSt

then Schur’s LemmaD.1.1implies that either it collapses to zero or it is mapped 1-1 toitself. In other wordsπ is a “shape selection”. Therefore, TheoremD.3.2implies thatLγW⊗LλV∗ ⊆ Jt wheneverLγW⊗LλV∗ ⊆ St , andγ andλ are different partitions ofa natural numberd. However, it is difficult to say something more at this step. In fact,in contrast withAt , a decomposition ofSt as direct sum of irreducible representationsis unknown. For instance a decomposition of Sym(

∧mV) as a GL(V)-representationis known just form≤ 2, see Weyman [107, Proposition 2.3.8]. Whenm> 2, thisis an open problem in representation theory, which is numbered among the plethysm’sproblems. In order to avoid such a difficulty, we go “one step more to the left”, in a waythat we are going to outline. For any natural numberd we have the natural projection

pd :d⊗(

t∧W⊗

t∧V∗)−→ Symd(

t∧W⊗

t∧V∗)∼= St(m,n)d.

The projectionpd is a G-equivariant surjective map. We have also the followingG-equivariant isomorphism

fd : (d⊗ t∧

W)⊗ (d⊗ t∧

V∗)−→d⊗(

t∧W⊗

t∧V∗).

Therefore, for any natural numberd, we have theG-equivariant surjective map

φd := pd fd : (d⊗ t∧

W)⊗ (d⊗ t∧

V∗)−→ St(m,n)d.

Putting together theφd’s we get the following (G-equivariant and surjective) homoge-neousk-algebra homomorphism

φ : Tt(m,n) :=⊕

d∈N((

d⊗ t∧W)⊗ (

d⊗ t∧V∗))−→ St(m,n).

When it does not raise confusion we will writeTt for Tt(m,n).

Remark 3.1.2. Notice thatTt is not commutative. The idealsI ⊆ Tt we consider willalways be two-sided, i.e. they arek-vector spaces such thatst and ts belong toIwhenevert ∈ I ands∈ Tt . Moreover, if we say that an idealI ⊆ Tt is generated byt1, . . . , tq we mean that

I = q

∑i=1

fitigi : fi ,gi ∈ Tt(m,n).

In this senseTt is Noetherian. However,Tt is neither left-Noetherian nor right-Noetherian.


By definition and by meaning of thefd’s, the kernel ofφ is generated in degree 2,namely

Ker(φ) = (( f ⊗ f ′)⊗ (e⊗e′))− ( f ′⊗ f )⊗ (e′⊗e)) : f , f ′ ∈t∧

W, e,e′ ∈t∧

V∗).

Finally, we have aG-equivariant surjective graded homomorphism

ψ := π φ : Tt(m,n)→ At(m,n).

We callKt(m,n) (Kt when it does not raise any confusion) the kernel of the above map.Since Ker(φ) is generated in degree two andJt is generated in degree at least two, inorder to understand which is the maximum degree of a minimal generator ofJt , we canstudy which is the maximum degree of a minimal generator ofKt .

Lemma 3.1.3. Let d be an integer bigger than2. There exists a minimal generator ofdegree d in Kt if and only if there exists a minimal generator of degree d in Jt .

The advantages of passing toTt are that it “separates rows and columns” and thatit is available a decomposition of it in irreducibleG-representations, see Proposition3.1.4. The disadvantage is that we have to work in a noncommutativesetting.

Proposition 3.1.4. As a G-representation Tt decomposes as

Tt(m,n)d∼=

⊕

γ andλ d-admissibleht(γ)≤ m,ht(λ)≤ n

(LγW⊗LλV∗)n(γ,λ ),

where the multiplicities n(γ,λ ) are the nonzero natural numbers described recursivelyas follows:

1. If d = 1, i.e. if γ = λ = (t), then n(γ,λ ) = 1;2. If d> 1, then n(γ,λ )=∑n(γ ′,λ ′)where the sum runs over the(d−1)-admissible

diagramsγ ′ andλ ′ such thatγ ′ ⊆ γ ⊆ γ ′(t) andλ ′ ⊆ λ ⊆ λ ′(t) (for the notationsee TheoremD.2.11).

Proof. SinceTt(m,n)1 =∧t W⊗∧t V, we can prove the statement by induction. So

suppose that

Tt(m,n)d−1∼=

⊕

γ andλ (d−1)-admissibleht(γ)≤ m, ht(λ)≤ n

(LγW⊗LλV∗)n(γ,λ )

for d ≥ 2. Therefore, sinceTt(m,n)d∼= Tt(m,n)d−1⊗ (

∧t W⊗∧t V∗), we have

Tt(m,n)d∼=

⊕

γ andλ (d−1)-admissibleht(γ)≤ m, ht(λ)≤ n

((LγW⊗t∧

W)⊗ (LλV∗⊗t∧

V∗))n(γ,λ ).

At this point we get the desired decomposition from Pieri’s TheoremD.2.11.

As the reader will realize during this chapter, the fact thatthe n(γ,λ )’s may be,and in fact usually are, bigger than 1, is one of the biggest troubles to say somethingabout the relations between minors. We will say that a pair ofd-admissible diagrams,a d-admissiblebi-diagramfor short,(γ ′|λ ′) is a predecessorof a (d+ 1)-admissible


bi-diagram(γ|λ ) if γ ′ ⊆ γ ⊆ γ ′(t) andλ ′ ⊆ λ ⊆ λ ′(t). In order to simplify the notation,from now on a sentence like “(γ|λ ) is an irreducible representation ofTt (or St or Jt

etc.)” will mean thatLγW⊗LλV∗ occurs with multiplicity at least 1 in the decomposi-tion of Tt (or St or Jt etc.). Furthermore, since bothJt andKt are generated, as ideals, byirreducible representations, we say that “(γ|λ ) is a minimal irreducible representationof Jt (or of Kt )” if the elements inLγW⊗LλV∗ are minimal generators ofJt (or Kt ).

We are going to list some equations of degree 3 which are minimal generators ofJt (in situations different from (3.1) and (3.2)). To this purpose we define some special

bi-diagrams(γ i |λ i) for any i = 1, . . . ,⌊ t

2

⌋, for which bothγ i andλ i have exactly 3t

boxes. In Theorem3.1.6, we will prove that some of these diagrams will be minimalirreducible representations of degree 3 inJt . It is convenient to consider separately thecases in whicht is even or odd.

Suppose thatt is even. For eachi = 1, . . . ,t2

we setai :=3t2− i+1, bi := 2(i−1),

ci := 2(t − i +1) anddi :=t2+ i −1. Then

γ i := (ai ,ai ,bi), (3.3)

λ i := (ci ,di ,di).

It turns out thatγ i andλ i are both partitions of 3t.

If t is odd, then for eachi = 1, . . . ,t −1

2seta′i :=

3t −12

− i+1, b′i := 2(i−1)+1,

c′i := 2(t − i +1)−1 andd′i :=

t +12

+ i −1. Then

γ i := (a′i ,a′i ,b

′i), (3.4)

λ i := (c′i ,d′i ,d

′i ).

Once again it turns out thatγ i andλ i are both partitions of 3t.

Example 3.1.5.For t = 2 there is only one(γ i |λ i), namely(γ1|λ 1). The picture belowfeatures this bi-diagram.

(γ1|λ 1) =

Notice that the above bi-diagram has only one predecessor, namely

(α1|α1) =

It turns out that(α1|α1) has multiplicity one and is symmetric. As we are going to seein the proof of Theorem3.1.6this facts holds true in general, and it is the key to findminimal relations in Jt .


Theorem 3.1.6. The bi-diagram(γ i |λ i) is a minimal irreducible representation of Jt

of degree3 in the following cases:

1. if t is even, whent2≥ i ≥ max

0,

3t2−m+1, t− n

2+1

.

2. if t is odd, whent −1

2≥ i ≥ max

0,

3t −12

−m+1, t− n+12

.

Proof. By Lemma3.1.3it is enough to show that the bi-diagram(γ i |λ i) is a minimalirreducible representation of degree 3 ofKt . Bothγ i andλ i are 3-admissible partitions.Furthermore, the assumptions oni imply thatγ i

1 ≤mandλ i1 ≤ n. Therefore(γ i |λ i) is an

irreducible representation of[Tt ]3 by Proposition3.1.4. Notice that[Tt ]3 decomposes as[Kt ]3⊕ [At ]3. But (γ i |λ i) is anasymmetric bi-diagram, i.e. γ i 6= λ i , thus it cannot be anirreducible representation ofAt by TheoremD.3.2. This implies that it is an irreduciblerepresentation of[Kt ]3.

It remains to prove that(γ i |λ i) is minimal. First of all we show that it has multiplic-ity 1 in Tt . If t is even the unique predecessor of(γ i |λ i) is the symmetric bi-diagram(α i |α i), where:

α i1 =

3t2− i +1

α i2 =

t2+ i −1.

Also if t is odd the unique predecessor of(γ i |λ i), which we denote again by(α i |α i),is symmetric. TheG-representation[Tt ]2 decomposes as[Kt ]2 ⊕ [At ]2. Now, the ir-reducible representations ofTt of degree 2 have obviously multiplicity 1, since theirunique predecessor is((t)|(t)) that has multiplicity 1. So(α i |α i) is not an irreduciblerepresentation of[Kt ]2, because it is an irreducible representation of[At ]2 by TheoremD.3.2and it has multiplicity 1 in[Tt ]2. Therefore we conclude that(γ i |λ i) is a minimalirreducible representation ofKt of degree 3.

Definition 3.1.7. The bi-diagram(γ i |λ i) such thati satisfies the condition of Theorem3.1.6are calledshape relations.

Corollary 3.1.8. In our situation, i.e. if1< t < m and n> t +1, the ideal Jt has someminimal generators of degree3.

Proof. It is enough to verify that there is at least onei satysfying the conditions ofTheorem3.1.6.

3.1.1 Make explicit the shape relations

The reader might recriminate that we have not yet described explicitly the degree 3minimal relations we found in Theorem3.1.6. Actually we think that the best way topresent them is by meaning of shape, as we did. But of course itis legitimate to pretendto know the polynomials corresponding to them, therefore weare going to explain howto pass from bi-diagrams to polynomials. Consider one of thebi-diagrams(γ i |λ i) ofTheorem3.1.6. Let us call it (γ|λ ). Then choose two tableuxΓ and Λ of shape,respectively,γ andλ and such thatc(Γ) = c(Λ) = 3t. We can consider the followingmap

e(Γ)⊗e(Λ)∗ :3t⊗

W⊗3t⊗

V∗ −→3t⊗

W⊗3t⊗

V∗,


wheree(·) are the Young symmetrizers (seeD.2.1). Actually the image ofe(Γ)⊗e(Λ)∗is in

⊗3∧t W⊗⊗3∧t V∗, therefore we can compose it with the map

3⊗ t∧W⊗

3⊗ t∧V∗ −→ [St ]3.

The polynomials in the image do not depend from the chosen tableu, sinceLγW⊗LλV∗

has multiplicity one inTt .Actually, among the bi-diagrams(γ i |λ i) of Theorem3.1.6, there is one such that the

relativeU-invariant (see SubsectionD.3) can be written in a very plain way. Namely,the bi-diagram under discussion is(γ|λ ) := (γ⌊t/2⌋|λ ⌊t/2⌋), namely:

(γ|λ ) = ((t +1, t+1, t−2)|(t+2, t−1, t−1)).

The corresponding equation can be written in a determinantal form: More precisely, itis given by the vanishing of the following determinant (justfor a matter of space belowwe puts= t −1, u= t +1 andv= t +2):

det

([1, . . . ,s, t,u|1, . . . ,s, t] [1, . . . ,s, t,u|1, . . . ,s,u] [1, . . . ,s, t,u|1, . . . ,s,v][1, . . . ,s, t,v|1, . . . ,s, t] [1, . . . ,s, t,v|1, . . . ,s,u] [1, . . . ,s, t,v|1, . . . ,s,v][1, . . . ,s,u,v|1, . . . ,s, t] [1, . . . ,s,u,v|1, . . . ,s,u] [1, . . . ,s,u,v|1, . . . ,s,v]

)= 0. (3.5)

We are going to prove that (3.5) gives actually theU-invariant of the irreducible rep-resentation(γ|λ ) described above. In particular, this means that the determinant (3.5)generatesLγW⊗LλV∗ as aG-module. Furthermore, notice that in the casest = 2,3 theunique minimal irreducible representation ofJt of degree 3 we described in Theorem3.1.6is (γ|λ ). So, in these two cases we have a very good description of the (guessed)minimal equations betweent-minors.

Proposition 3.1.9. If 1 < t < m and n> t + 1, the equation(3.5) supplies the U-invariant of the minimal irreducible representation(γ|λ ) of Jt , whereγ = (t +1, t +1, t−2) andλ = (t +2, t−1, t−1). Particularly, it is a minimal cubic generator of Jt .

Proof. First of all notice that the polynomial of the determinant of(3.5) is nonzeroin St . This is obvious, since it is the determinant of a 3×3 matrix whose entries arevariables all different between them.

Let us call f the polynomial associated to the determinant of (3.5). To see thatfis anU-invariant, we have to show that(A,B) · f = α f for someα ∈ k \ 0, whereA∈U−(W)⊆ GL(W) is a lower triangular matrices andB∈U+(V)⊆ GL(V) is an up-per triangular matrices. To prove this, it suffices to show that f vanishes formally everytime that we substitute an index, of the columns or of the rowsof the variables appear-ing in f , with a smaller one. This is easily checkable: For example, if we substitute the(t +2)th row with thetth one, that is, using the notation of (3.5), if we substitute thevth row with thetth one, thenf becomes:

det

[1, . . . ,s, t,u|1, . . . ,s, t] [1, . . . ,s, t,u|1, . . . ,s,u] [1, . . . ,s, t,u|1, . . . ,s,v]0 0 0

−[1, . . . ,s,u, t|1, . . . ,s, t] −[1, . . . ,s,u, t|1, . . . ,s,u] −[1, . . . ,s,u, t|1, . . . ,s,v]

,

which is obviously formally 0. Or, if we substitute thevth column with thetth one, fbecomes:

3.2 Upper bounds on the degrees of minimal relations among minors 51

det

[1, . . . ,s, t,u|1, . . . ,s, t] [1, . . . ,s, t,u|1, . . . ,s,u] [1, . . . ,s, t,u|1, . . . ,s, t][1, . . . ,s, t,v|1, . . . ,s, t] [1, . . . ,s, t,v|1, . . . ,s,u] [1, . . . ,s, t,v|1, . . . ,s, t][1, . . . ,s,u,v|1, . . . ,s, t] [1, . . . ,s,u,v|1, . . . ,s,u] [1, . . . ,s,u,v|1, . . . ,s, t]

,

which is formally 0 since the first and third columns are the same.To see that (3.5) becomes 0 inAt , underπ , we have to observe thatf becomes the

expansion int-minors of the determinant of the following 3t×3t matrix:

x11 x12 . . . x1s x1t x11 x12 . . . x1s x1u x11 x12 . . . x1s x1vx21 x22 . . . x2s x2t x21 x22 . . . x2s x2u x21 x22 . . . x2s x2v...

......

......

......

......

......

......

......

xs1 xs2 . . . xss xst xs1 xs2 . . . xss xsu xs1 xs2 . . . xss xsv

xt1 xt2 . . . xts xtt xt1 xt2 . . . xts xtu xt1 xt2 . . . xts xtv

xu1 xu2 . . . xus xut xu1 xu2 . . . xus xuu xu1 xu2 . . . xus xuv


......

......

......

......

......

......

......


xt1 xt2 . . . xts xtt xt1 xt2 . . . xts xtu xt1 xt2 . . . xts xtv

xv1 xv2 . . . xvs xvt xv1 xv2 . . . xvs xvu xv1 xv2 . . . xvs xvv


......

......

......

......

......

......

......


xu1 xu2 . . . xus xut xu1 xu2 . . . xus xuu xu1 xu2 . . . xus xuv

xv1 xv2 . . . xvs xvt xv1 xv2 . . . xvs xvu xv1 xv2 . . . xvs xvv

.

Such a determinant is zero (for instance because the 1st row is equal to the(t +1)th).So far, we have shown that the polynomialf ∈ St associated to (3.5) is a nonzero

U-invariant ofJt . Now, it clearly has bi-weigth(tγ|tλ ), so it is theU-invariant of theirreducible representation(γ|λ ). Particularly, it is a minimal cubic generator ofJt byTheorem3.1.6.

Remark 3.1.10. Actually, in the proof of Proposition3.1.9it is not necessary to showthat the determinant of (3.5) vanishes inAt . In fact, once proved that it is theU-invariantof (γ|λ ), that it goes to zero follows becauseγ 6= λ . However we wanted to show that itvanishes because the huge matrix of Proposition3.1.9has been the first mental imagethat suggested us the equation3.5.

3.2 Upper bounds on the degrees of minimal relationsamong minors

During the previous section we have found out some cubics among the minimal gener-ators ofJt . The authors of [18] said that there are indications that quadrics and cubicsare enough for generatingJt , at least fort = 2. In this section we are going to prove thattheir guess is right for 3×n and 4×n matrices. The way to get these results will be to


notice that the “minimal”U-invariants of aJ2(3,n) must be already inJ2(3,5), and asimilar fact for a 4×n-matrix. These cases are then doable with a computer calculation.Furthermore, we will give some reasons to believe to the guess of [18] in general, evenfor t ≥ 3. First of all, anyway, we will give a formula for the Castelnuovo-Mumfordregularity ofAt(m,n) in all the cases. SinceAt(m,n) = St(m,n)/Jt(m,n), we will alsoget a general upper bound for the degree of a minimal relationbetween minors (see0.4.3).

3.2.1 Tha Castelnuovo-Mumford regularity of the algebra ofmi-nors

The authors of [18] noticed thatAt is a Cohen-Macaulayk-algebra with negativea-invariant. This implies that the degrees of the minimal generators ofJt are less than orequal to dimAt = mn (for the last equality see [15, Proposition 10.16]). It would bedesirable to know the exact value ofa(At), equivalently of reg(At). To this aim we willpass through a toric deformation ofAt . If ≺ is a diagonal term order onR=R(m,n), i.e.such that in([i1 . . . ip| j1 . . . jp]) = xi1 j1 · · ·xip jp, then the initial algebra in(At) is a finitelygenerated normal Cohen-Macaulayk-algebra, see [19, Theorem 7.10]. Furthermore in[18, Lemma 3.3], the authors described the canonical moduleωin(At ) of in(At). By(0.12) we have thata(in(At)) = −mind : [ωin(At )]d 6= 0. Thus it is natural to expectto get the Castelnuovo-Mumford regularity of in(At) from ωin(At). Actually this is true,albeit anything but trivial, and we are going to prove it in Theorem3.2.1. Eventually,it is easy to prove that reg(At) = reg(in(At)).

Theorem 3.2.1.Suppose to be in cases different from(3.1) and(3.2), that is1< t < mand n> t + 1. Then At and in(At) are finitely generated graded Cohen-Macaulayalgebras such that:

(i) If m+n−1< ⌊mn/t⌋, then

a(At) = a(in(At)) =−⌈mn/t⌉,reg(At) = mn−⌈mn/t⌉.

(ii) Otherwise, i.e. if m+n−1≥ ⌊mn/t⌋, we have

a(At) = a(in(At)) =−⌊m(n+ k0)/t⌋,reg(At) = mn−⌊m(n+ k0)/t⌋.

where k0 = ⌈(tm+ tn−mn)/(m− t)⌉.

Proof. Thek-algebrasAt and in(At) are finitely generated and Cohen-Macaulay by [19,Theorem 7.10]. By [18, Lemma 3.3], it turns out that the canonical moduleω = ωin(At)

of the initial algebra ofAt with respect to a diagonal term order≺ is the ideal of in(At)generated by in(∆), where∆ is a product of minors ofX of shapeγ = (γ1, . . . ,γh) where|γ|= td, h< d and such thatx := ∏xi j divides in(∆). To prove the theorem we need tofind the leastd for which such a∆ exists. Of course such ad must be such thattd≥mn,but in general this is not sufficient. First we prefer to illustrate the strategy we will useto located with an example:


Example 3.2.2.Set t= 3 and m= n= 5. So our matrix looks like

X =

x11 x12 x13 x14 x15

x21 x22 x23 x24 x25

x31 x32 x33 x34 x35

x41 x42 x43 x44 x45

x51 x52 x53 x54 x55

.

Notice that we are in the case (ii) of the theorem. We are interested in finding the leastd ∈ N such that there exists a product of minors∆ ∈ A3(5,5) of shapeγ = (γ1, . . . ,γh)such that|γ|= 3d, h< d andx= ∏xi j dividesin(∆). The first product of minors whichcomes in mind is

∆2 := [12345|12345][1234|2345][2345|1234][123|345][345|123][12|45][45|12][1|5][5|1].

Obviouslyin(∆2) is a multiple ofx, but its shape isγ2 = (5,4,4,3,3,2,2,1,1). This isa partition of25, which is not divisible by3. This means that∆2 does not even belongto A3(5,5). Moreover the parts ofγ2 are 9, whereas⌊25/3⌋= 8 (it should be biggerthan9). To fix this last problem, it is natural to multiply∆2 for 5-minors till the desiredresult is gotten. For instance, setting∆1 := ∆2 · [12345|12345]3, we have that∆1 is aproduct of minors of shapeγ1 = (5,5,5,5,4,4,3,3,2,2,1,1). This is a partition of40with 12parts, and⌊40/3⌋= 13> 12. However∆1 is still not good, because40 is not amultiple of3. In some sense∆1 is too big, in fact we can replace in its shape a5-minorby a4-minor, to get a partition of39. For instance, set

∆ := ∆2 · [12345|12345]2[1234|1234].

The shape of∆ is γ = (5,5,5,4,4,4,3,3,2,2,1,1), which is a partition of39 with 12parts. Since39= 3 ·13, 12< 13 and x dividesin(∆), the natural number d we werelooking for is at most13. Actually it is exactly13, and we will prove that the strategyused here to find it works in general. Before coming back to thegeneral case, noticethat in this case k0 = 3, and13= ⌊40/3⌋= ⌊m(n+ k0)/t⌋.

First let us suppose to be in the case (i). Let us define the product of minors

Π := π1 · · ·πm+n−1

where

πi :=

[m− i +1,m− i +2, . . .,m|1,2, . . . , i] if 1 ≤ i < m

[1,2, . . . ,m|i −m+1, i −m+2, . . ., i] if m≤ i ≤ n

[1,2, . . . ,m+n− i|i −m+1, i −m+2, . . .,n] if n< i ≤ m+n−1

The shape ofΠ is λ = (mn−m+1,(m− 1)2,(m− 2)2, . . . ,12), which is a partition ofmn. Moreoverx divides in(Π). Let r0 be the unique integer such that 0≤ r0 < t andmn+ r0 = d0t. If r0 = 0, then we put∆ := Π andγ := λ . Thenγ hasm+n−1 partsandm+n−1< ⌊mn/t⌋= mn/t = d0. Since in(∆) is a multiple ofx, we deduce thatωd0 6= 0, i.e.

a(At)≥−d0 =−mn/t =−⌈mn/t⌉. (3.6)

Now suppose thatr0 > 0, i.e. mn is not a multiple oft. So, in this case,⌈mn/t⌉ =⌊mn/t⌋+1. We multiplyΠ by anr0-minor, for instance set

∆ := Π · [1,2, . . . , r0|1,2, . . . , r0].


The shape of∆ is γ = (mn−m+1,(m− 1)2,(r0 + 1)2, r30,(r0 − 1)2, . . . ,12). This is a

partition ofd0t with m+n parts. Sincem+n−1< ⌊mn/t⌋, we getm+n< ⌈mn/t⌉=d0. Sincex divides in(∆) (because it divides in(Π)), we getωd0 6= 0, i.e.

a(At)≥−d0 =−⌈mn/t⌉. (3.7)

Clearly equality must hold true both in (3.6) and in (3.7), since ifx divides a monomialin(Γ) for someΓ ∈ At , then deg(Γ)≥ ⌈mn/t⌉.

Now let us assume to be in the case (ii). Notice that the integer k0 of the assumptionis bigger than 0. Letp0 be the unique integer such that 0≤ p0 < t andm(n+ k0) =d0t + p0. Let us define the product of minors

∆ := Π · [1,2, . . . ,m|1,2, . . . ,m]k0−1 · [1,2, . . . ,m− p0|1,2, . . . ,m− p0].

The shape of∆ is

γ = (mk0+n−m,(m−1)2,(m− p0+1)2,(m− p0)3,(m− p0−1)2, . . . ,12).

This is a partition ofd0t with k0+n+m−1 parts. By the choice ofk0, one can verifythat k0+ n+m− 1< d0. Furthermore, being∆ a multiple ofΠ, x divides in(∆). Soin(∆) ∈ ω , which impliesωd0 6= 0 and

a(At)≥−d0 =−⌊m(n+ k0)/t⌋. (3.8)

To see that thea-invariant of in(At) is actually−d0 in (3.8), we need the following easylemma.

Lemma 3.2.3.With a little abuse of notation set X:= xi j : i = 1, . . . ,m, j = 1, . . . ,n.Define a poset structure on X in the following way:

xi j ≤ xhk if i = h and j= k or i < h and j< k.

Suppose that X= X1∪ . . .∪Xh where each Xi is a chain, i.e. any two elements of Xi arecomparable, and set N:= ∑h

i=1 |Xi |. Then

h≥ Nm+m−1.

Proof. For anyℓ= 1, . . . ,m−1 set

X(ℓ) := xi j ∈ X : or m+ j − i = ℓ or n+(i − j) = ℓ.

It is easy to see that for anyℓ, |X(ℓ)|= 2ℓ. Moreover, ifY is a chain such thatX(ℓ)∩Y 6=/0 for someℓ, then|Y| ≤ ℓ. Notice thatxmk∈ X(k) for all k= 1, . . . ,m−1. So, choosinganik such thatxmk∈ Xik, we have that|Xik| ≤ k. Note thatik 6= ih wheneverk 6= h sincexmk andxmh are not comparable. In the same way we choose aXjk containingx1,n+1−k

for anyk= 1, . . . ,m−1. Once again|Xjk| ≤ k sincex1,n+1−k ∈ X(k). Furthermore thejk’s are distinct because differentx1,n+1−k’s are incomparable. Actually, for the samereason, all theih’s and jk’s are distinct. In general, for anyi = 1, . . . ,h we have|Xi | ≤m.Thus, settingA := ik, jk : k= 1, . . . ,m−1, we get

N =h

∑i=1

|Xi |= ∑i∈A

|Xi |+ ∑i∈1,...,h\A

|Xi | ≤ (m−1)m+m(h−2m+2),

which supplies the desired inequalityh≥ Nm+m−1.


Now take a product of minors∆ = δ1 · · ·δh such that in(∆) ∈ ω . Let λ be the shapeof ∆ and suppose that|λ |= td with d < d0. For anyi = 1, . . . ,h set

Xi := xpr : xpr| in(δi).

Sincex divides in(∆), with the notation of Lemma3.2.3we have thatX =∪hi=1Xi where

eachXi is a chain with respect to the order defined onX. So by Lemma3.2.3we havethat

h≥ dtm

+m−1.

We recall thatd0t = mn+mk0− p0, where 0≤ p0 < t. Of course we can write in aunique waydt = mn+ms−q, where 0≤ q< m. Before going on, notice thatk0 is thesmallest natural numberk satisfying the inequality

m+n+ k−1<

⌊m(n+ k)

t

⌋.

By what saids≤ k0. There are two cases:1. If s= k0, consider the inequalities

m+n+(s−1)−1=dt+q

m+m−2<

dtm

+m−1≤ h≤ d−1.

Notice that, sinced < d0, we have thatq≥ p0+ t. Moreoverm< 2t, otherwisewe would be in case (i) of the theorem. Thus

d−1=m(n+ s)−q− t

t≤⌊

m(n+(s−1))t

⌋.

The inequalities above contradicts the minimality ofk0.2. If s< k0, then

n+ s+m−1=dt+q

m+m−1≤ h< d =

m(n+ s)−qt

≤⌊

m(n+ s)t

⌋.

Once again, this yields a contradiction to the minimality ofk0.Finally, it turns out that the Hilbert function of a gradedk-algebra and the one of its

initial algebra (with respect to any term order) coincide, see Conca, Herzog and Valla[24, Proposition 2.4]. In particular we have HFAt = HFin(At) and HPAt = HPin(At ). Soby the characterization of thea-invariant given in (0.14), we have

a(At) = a(in(At)).

Furthermore reg(At) = dimAt +a(At) from (0.11), and dimAt =mnby [15, Proposition10.16].

Remark 3.2.4. Let us look at the cases in Theorem3.2.1.(i) If X is a square matrix, that ism= n, one can easily check that we are in case

(i) of Theorem3.2.1if and only if m is at least twice the size of the minors, i.e.m≥ 2t.

(ii) The natural numberk0 of Theorem3.2.1may be very large: For instance, let usconsider the caset = m−1 andn = m+ 1. Since in the interesting situations


m≥ 3, one can easily check that we are in the case (ii) of Theorem3.2.1. In thiscase we havek0 = m2−2m−1. Therefore Theorem3.2.1yields

a(Am−1(m,m+1)) =−m2

andreg(Am−1(m,m+1)) = m.

Since reg(Jt) = reg(At)+ 1 and reg(Jt) bounds from above the degree of a mini-mal generator ofJt by Theorem0.4.7, as a consequence of Theorem3.2.1we get thefollowing:

Corollary 3.2.5. Let us call d the maximum degree of a minimal generator of Jt .(i) If m+n−1< ⌊mn/t⌋, then

d ≤ mn−⌈mn/t⌉+1.

(ii) Otherwise, i.e. if m+n−1≥ ⌊mn/t⌋, we have

d ≤ mn−⌊m(n+ k0)/t⌋+1,

where k0 = ⌈(tm+ tn−mn)/(m− t)⌉.

For instance, the degree of a minimal generator ofJ2 can be at most⌊mn/2⌋+1.Instead, a minimal generator ofJm−1(m,m+1) has degree at mostm+1.

3.2.2 The independence onn for the minimal relations

Given a degreed minimal relation betweent-minors of them×n matrix X, clearly itinsists at most ontd columns ofX. Therefore it must be a minimal relation already ina m× td matrix. This fact can be useful whend is small: For instance, to see if thereare minimal relations of degree 4 between 2-minors of a 3× n-matrix, it is enoughto check if they are in a 3×8 matrix. However, even once checked that there are nominimal relations of degree 4, there might be anyway of degree 5, 6 or so on; andwith d growing up this observation is useless. In fact, unfortunately, so far the bestupper bound we have for the degree of a minimal relation is given by the Castelnuovo-Mumford regularity ofAt(m,n), see Theorem3.2.1. For example, the degree of aminimal generator ofJ2(3,n) might bed = ⌊3n/2⌋+1, providedn≥ 6. So we shouldcontrol if such a minimal relation is in am× td matrix; but td = 2⌊3n/2⌋+ 2 > n,so we would be in a worse situation than the initial one. In this subsection we willmake an observation somehow similar, but finer, to the one discussed above. The sizeof the “reduction-matrix”, besides being independent onn, will not even depend ond.Precisely, in Theorem3.2.6we will show that a a degreed minimal relation betweent-minors of anm×n matrix must be already in am×(m+ t)matrix! Thus, in principle,once fixedm we could check by hand the maximum degree of a minimal generator ofJt(m,n). In practice, however, a computer can actually supply an answer just for smallvalues ofm. Actually the proof of Theorem3.2.6is not very difficult: Essentially wehave just to exploit the structure asG-representations of our objects.

Theorem 3.2.6. Let d(t,m,n) denote the highest degree of a minimal generator ofJt(m,n). Then

d(t,m,n)≤ d(t,m,m+ t)


Proof. We can assume to be in a numerical case different from (3.1). Sod(t,m,n) isthe highest degree of a minimal generator ofKt(m,n), too (Lemma3.1.3). Thereforesuppose that(γ|λ ) is a minimal irreducible representation ofKt(m,n). Then we claimthatλ1 ≤ m+ t: If not, for any predecessor(γ ′|λ ′) of (γ|λ ) we have thatλ ′

1 > m. Onthe other sideγ ′1 ≤ m. This implies that any predecessor of(γ|λ ) is asymmetric, and sofor any(γ ′|λ ′) predecessor of(γ|λ )

Lγ ′W⊗Lλ ′V∗ ⊆ Kt(m,n).

Since there must exist some predecessor(γ ′|λ ′) such that

LγW⊗LλV∗ ⊆ (Lγ ′W⊗Lλ ′V∗)⊗ (t∧

W⊗t∧

V∗),

it turns out that(γ|λ ) cannot be minimal. So if(γ|λ ) is minimal in Kt(m,n), thenλ1 ≤m+t. Let us consider the canonical bi-tableu of(γ|λ ) , namely(cγ |cλ ). Such a bi-tableu corresponds to a minimal generator ofKt(m,n). By the definition of the Youngsymmetrizers (seeD.2.1), actually(cγ |cλ ) can be seen as an element ofTt(m,m+ t),becauseλ1 ≤ m+ t. So it belongs toKt(m,m+ t). Furthermore, if(cγ |cλ ) were notminimal in Kt(m,m+ t), all the more reason it would not be minimal inKt(m,n). Thisthereby implies the thesis.

So, putting together Corollary3.2.5and Theorem3.2.6, we get:

Corollary 3.2.7. Let d(t,n,m) be as in Theorem3.2.6.(i) If m+ t−1< ⌊m2/t⌋, then

d(t,m,n)≤ m2+m(t −1)−⌈m2/t⌉+1.

(ii) Otherwise, we have

d(t,m,n)≤ m2+m(t−1)−⌊m(m+ k0)/t⌋+1,

where k0 = ⌈(t2+ tm−m2)/(m− t)⌉.

For instance, Corollary3.2.7implies that a minimal relation between 2-minors of a3×n matrix has degree at most 7. Actually we will see in the next subsection that sucha relation is at most a cubic.

3.2.3 Relations between2-minors of a 3×n and a4×n matrix

In this paragraph we want to explain the strategy to prove thefollowing result:

Theorem 3.2.8.For all n≥ 4, the ideals J2(3,n) and J2(4,n) are generated by quadricsand cubics.

Of course we have to use Theorem3.2.6. It implies thatd(2,3,n) ≤ d(2,3,5)and d(2,4,n) ≤ d(2,4,6). Since we already know that there are minimal relationsof degree 2 and 3 by Corollary3.1.8, we have just to show thatd(2,3,5) ≤ 3 andd(2,4,6) ≤ 3. The strategy to prove these inequalities is the same; however from acomputational point of view we must care more attentions to show the second one,since its verification might require some days. For this reason we will present thestrategy to prove the second inequality:


(1) SetJ := J2(4,6), S:= S2(4,6) and, for anyd ∈ N, let J≤d ⊆ J denote the idealgenerated by the polynomials of degree less than or equal tod of J. By elimina-tion (for instance see Eisenbud [34, 15.10.4]) one can compute a set of generatorsof J≤3.

(2) Fixed some term order, for instance degrevlex, we compute a Grobner basis ofJ≤3 up to degree 13. So we getB := in(J≤3)≤13.

(3) Let us compute the Hilbert function ofS/B. Clearly we have

HFS/J≤3(d)≤ HFS/B(d),

where equality holds true provided thatd ≤ 13.(4) We recall that a decomposition ofAt(m,n) in irreducibleG-representations is

known by TheoremD.3.2. Moreover any irreducibleG-module appearing init is of the formLλW⊗ LλV∗ for some partitionλ . It turns out that there is aformula to compute the dimension of such vector spaces, namely the hook lengthformula (for instance see the book of Fulton [42, p. 55]). So we can quickly getthe Hilbert function ofA2(4,6) = S/J. Let us compute it up to degree 13.

(5) SinceJ≤3 ⊆ J, we will have that HFS/B(d)≥ HFS/J(d). However, comparing thetwo Hilbert functions, one can check that HFS/B(d) = HFS/J(d) for anyd ≤ 13.This implies thatJ≤3 = J≤13.

(6) Corollary3.2.5yields that a minimal generator ofJ has at most degree 13. There-foreJ≤13= J. So we are done, sinceJ≤3 = J≤13.

We used the computer algebra system Singular, [30]. The employed machine tookabout 60 hours to compute the generators ofJ≤3, and about 15 hours to compute aGrobner basis of it up to degree 13. This means that probablyis necessary to boundthe degrees, if not the computation might not finish. At the contrary, the computationof J2(3,5) can be done without any restrictions: It is just a matter of seconds.

3.2.4 The uniqueness of the shape relations

Theorem3.1.6implies that some cubics lie inJt . But are we sure that we cannot findsome other higher degrees minimal relations with similar arguments? We recall thatthose cubics correspond to some bi-diagrams, namely the shape relations, which areirreducible representations ofTt . The crucial properties we needed to prove that thesebi-diagrams actually correspond to minimal generators ofJt have been that they areasymmetric diagrams of multiplicity one, whose only predecessor is symmetric. Inthis subsection we are going to prove that, but the shape relations and those of degreeless than 3, there are no other bi-diagrams ofTt satisfying these properties. In a certainsense, this implies that the bi-diagrams that are minimal irreducible representations ofJt for reasons of shapeare just in degree 2 and 3. So the question is: Are there anyother minimal irreducible representations ofJt for reasons of multiplicity?

To our purpose we introduce some notation and some easy facts. For the nextlemma let us work just withV. Letλ =(λ1, . . . ,λk) be a partition. Pieri’s formula (The-oremD.2.11) implies that a diagramLλV is an irreducible GL(V)-subrepresentationof⊗d∧t V if and only if |λ | = td and k ≤ d. As in the case of bi-diagrams we

say that a partitionλ ′ ⊢ t(d− 1) with at mostd− 1 parts is a predecessor ofλ ifλ ′ ⊆ λ ⊆ λ ′(d). Moreover we define thedifference sequence∆λ := (∆λ1, . . . ,∆λk)where∆λi := λi − λi+1, settingλk+1 = 0. Notice that in order to get a predecessorof λ we can removeq boxes from itsith row only if ∆λi ≥ q. Moreover notice that∆λ1+ . . .+∆λk = λ1 ≥ t.


Lemma 3.2.9. Letλ = (λ1, . . . ,λk) ⊢ td be a partition such that LλV ⊆⊗d∧t V, withd > 1. Thenλ has a unique predecessor if and only if eitherλ1 = . . . = λk (λ is arectangle) or there exist i such thatλ1 = . . . = λi > λi+1 = . . . = λk and k= d (λ is afat hook).

Proof. If λ is a rectangle its difference sequence is

∆λ = (0,0, . . . ,0︸︷︷︸k−1 times

,λ1).

This means that the only way to removet boxes from different columns ofλ is to do itfrom its last row. Thereforeλ has a unique predecessor, namely

λ ′ = (λ1,λ1, . . . ,λ1︸︷︷︸k−1 times

,λ1− t).

If λ is a fat hook, instead, its difference sequence is

∆λ = (0,0, . . . ,0︸︷︷︸i−1 times

,λ1−λk, 0,0, . . . ,0︸︷︷︸k− i−1 times

,λk).

Sincek= d, to get a predecessor ofλ we must remove completely the last row. So theonly nonzero entry which remains is∆λi. Therefore the only predecessor ofλ is

λ ′ = (λ1,λ1, . . . ,λ1︸︷︷︸i−1 times

,λ1+λk− t,λk,λk, . . . ,λk︸︷︷︸k− i−1 times

).

For the converse, first suppose thatk< d. Then in order to get a predecessor ofλ wehave not to care about removing the last row. Moreover, in this case,∆λ1+ . . .+∆λk =λ1 > t. This implies that if two of the∆λi ’s were bigger than 0, more than one choicewould be possible: Soλ would have more than one predecessor. So there must be justonei such that∆λi > 0. In other words,λ has to be a rectangle.

Now supposek= d. In this case we must remove thekth row. If λk = t, thenλ willbe a rectangle. So we can assume thatλk < t (consequentlyλ1 > t). This means thatafter removing the last row we can remove freelyt −λk boxes from the other. Because∆λ1 + . . .+ ∆λk−1 = λ1 − λk > t − λk, the choice of removing some boxes to get apredecessor will be unique only if just one among the∆λi is bigger than 0. This meansthatλ has to be a fat hook.

Corollary 3.2.10. Given a partitionλ = (λ1, . . . ,λk) ⊢ td (with d≥ 2), the irreducibleGL(V)-representation LλV appears with multiplicity one in

⊗d∧t V if and only ifλhas only one predecessor which either is(t), or, in turn, has only one predecessor.These facts are equivalent to the fact thatλ is a diagram of the following list:

1. a rectangle with one row (k= 1);2. a rectangle with d−1 rows;3. a rectangle with d rows;4. a fat hook of the typeλ1 > λ2 = . . .= λk with k= d;5. a fat hook of the typeλ1 = . . .= λk−1 > λk with k= d;

Proof. By Pieri’s formula (TheoremD.2.11), LλV has multiplicity one in⊗d∧t V

if and only if λ has only a predecessorλ ′ such thatLλ ′V has multiplicity one in


⊗d−1∧t V. Thus we can argue by induction ond. For d = 2, the only predecessorof λ is λ ′ = (t), andL(t)V ∼=

∧t V has obviously multiplicity one in∧t V. For d > 2,

theλ ’s appearing in the list of the statement have just one predecessor by Lemma3.2.9.Furthermore, it is easy to check that ifλ is a partition in the list, its unique predecessorλ ′ is in the list as well asλ (of course we mean withd replaced byd−1). SoLλ ′Vhas multiplicity one in

⊗d−1∧t V by induction. To see that there are no otherλ butthose in the list, we have to check that: If a partitionλ is not in the list and has onlyone predecessorλ ′, thenλ ′ has more than one predecessor. This is easily checkableusing Lemma3.2.9.

Now we are ready to prove the announced fact that the only minimal relations forreasons of shape are in degree 2 and 3.

Proposition 3.2.11. Let (γ|λ ) be an asymmetric bi-diagram such that LγW⊗ LλV∗

appears in Td with multiplicity one and such that its only predecessor is symmetric.Then d= 2 or d = 3 and(γ|λ ) is a shape relation.

Proof. Since(γ|λ ) is an irreducibleG-subrepresentation ofTd, we have|γ|= |λ |= td.For the proof we have to reason by cases. Set:

A1 := α d-admissible :α is a rectangle with 1 row

A2 := α d-admissible :α is a rectangle withd−1 rows

A3 := α d-admissible :α is a rectangle withd rows

A4 := α d-admissible :α is a fat hook of the typeλ1 > λ2 = . . .= λk with k= dA5 := α d-admissible :α is a fat hook of the typeλ1 = . . .= λk−1 > λk with k= d

Since(γ|λ ) has multiplicity one, by Corollary3.2.10there existi, j such thatγ ∈Ai

andλ ∈A j . Denote by(γ ′|λ ′) the only predecessor of(γ|λ ). We can assume thatd≥ 3.

1. γ ∈ A1.

(a) If λ ∈ A1 then(γ|λ ) would not be asymmetric;(b) If λ ∈ A2 then(γ ′|λ ′) would not be symmetric;(c) If λ ∈ A3 then(γ ′|λ ′) would not be symmetric;(d) If λ ∈ A4 then(γ ′|λ ′) would not be symmetric;(e) If λ ∈ A5 then(γ ′|λ ′) would not be symmetric;

2. γ ∈ A2.

(a) If λ ∈ A2 then(γ|λ ) would not be asymmetric;(b) If λ ∈ A3 then(γ ′|λ ′) would not be symmetric;(c) If λ ∈ A4 and if d ≥ 4 then(γ ′|λ ′) would not be symmetric. Ifd = 3 then

actually(γ|λ ) is a shape relation;(d) If λ ∈ A5 then(γ ′|λ ′) would not be symmetric (γ ′1 > λ ′

1);

3. γ ∈ A3.

(a) If λ ∈ A3 then(γ|λ ) would not be asymmetric;(b) If λ ∈ A4 then(γ ′|λ ′) would not be symmetric (γ ′2 = t > λ ′

2);(c) If λ ∈ A5 then(γ ′|λ ′) would not be symmetric (γ ′1 = t < λ ′

1);

4. γ ∈ A4.

3.3 The initial algebra and the algebra ofU-invariants of At 61

(a) If λ ∈ A4 then, since(γ ′|λ ′) has to be symmetric,(γ|λ ) would have to besymmetric as well;

(b) If λ ∈ A5 andd ≥ 4, then(γ ′|λ ′) would not be symmetric; ifd = 3, then(γ|λ ) is a shape relation.

5. γ ∈ A5.

(a) If λ ∈ A5 then, since(γ ′|λ ′) has to be symmetric,(γ|λ ) would have to besymmetric as well.

3.3 The initial algebra and the algebra ofU-invariantsof At

In this section we will exhibit a system of generators for: (i). The initial algebra ofAt

with respect to a diagonal term order. (ii). The subalgebra of U-invariants ofAt . In gen-eral both the initial algebra and the ring of invariants of a finitely generatedk-algebramight be not finitely generated. We will prove that the twok-algebras considered aboveare finitely generated, but especially we will give a finite system of generators of them.

3.3.1 A finite Sagbi basis ofAt

In [17, Theorem 3.10], Bruns and Conca described a Grobner basis,with respect to anydiagonal term order, for every powers of the determinantal idealsIt . This allows us todescribe when a monomial ofR= k[X] belongs to in(Ad). In fact, we have

in(At) =⊕

d∈Nin(Id

t ∩Rtd).

Lemma 3.3.1. A monomial M∈ R belongs toin(At)d if and only if M= M1 · · ·Mk

where the Mq’s are monomials of R such that1. For any q= 1, . . . ,k the monomial Mq is the initial term of an rq-minor.2. The partition(r1, r2, . . . , rk) is d-admissible.

Although in [17, Theorem 3.11] the authors showed that in(At) is a finitely gener-atedk-algebra, they could not specify a finite Sagbi basis ofAt (see [19, Remark 7.11(b)]). Actually they could not even give an upper bound for the degree of a minimalgenerator of in(At). We will be able to do it. To this aim, first, we need some notionsabout partitions of integers (to delve more into such an argument see the sixth chapterof the book of Sturmfels [101]).

A partition identityis any identity of the form

a1+a2+ . . .+ak = b1+b2+ . . .+bl (3.9)

whereai ,b j ≥ 1 are integers. Fixed a positive integerq we will say that (3.9) is apartition identity with entries in[q] if furthermoreai ,b j ≤ q. The partition identity(3.9) is calledprimitive if there is no proper subidentity

ai1 +ai2 + . . .+air = b j1 +b j2 + . . .+b js


with r + s< k+ l . We say that the partition identity3.9 is homogeneousif k= l . It ishomogeneous primitiveif there is no proper subidentity

ai1 +ai2 + . . .+air = b j1 +b j2 + . . .+b jr

with r < k.

Theorem 3.3.2. Let d denote the maximum degree of a minimal generator lying inin(At). Then d≤ m−1. Furthermore d≤ m−2 if and only ifGCD(m−1, t−1) = 1.

Proof. Let M := M1 · · ·Mk be a product of initial terms of minors, sayMi := in(δi)whereδi is anmi-minor ofX. Let td be the degree ofM, so that∑k

i=1mi = td.If k< d the monomialM cannot be a minimal generator ofAt : In fact, sincem1 > t,

M1 = in(δ ′1) · in(δ ′′

1 ),

whereδ ′1 is anm1− t-minor andδ ′′

1 is at-minor. Sincek≤ d−1, the monomial

M′ := in(δ ′1) ·M2 · · ·Mk

belongs to in(At)d−1 by Lemma3.3.1. Moreover in(δ ′′1 ) obviously belongs to in(At)1.

Thus, asM = M′ · in(δ ′′1 ), it is not a minimal generator.

So we can assume thatk = d. This means that we have a homogeneous partitionidentity with entries in[m] of the kind

m1+m2+ . . .+md = t + t+ . . .+ t︸︷︷︸d times

.

If d ≥ m then the above is not a homogeneous primitive partition identity by [101,Theorem 6.4]. Therefore there is a subsetS( [d] such that

∑i∈S

mi = |S| · t,

and as a consequence

∑i∈[d]\S

mi = (d−|S|) · t.

Therefore by Lemma3.3.1M′ :=∏i∈SMi ∈ in(At)|S| andM′′ :=∏i∈[d]\SMi ∈ in(At)d−|S|,so thatM = M′ ·M′′ is not a minimal generator of in(A).

For the second part of the statement, arguing as in the proof of [101, Theorem 6.4]one can deduce that the only possible homogeneous primitivepartition identity withentries in[m] of the kind

λ1+λ2+ . . .+λm−1 = t + t+ . . .+ t︸︷︷︸m−1 times

is the following one:

1+1+ . . .+1︸︷︷︸m−t times

+m+m+ . . .+m︸︷︷︸t−1 times

= t + t+ . . .+ t︸︷︷︸m−1 times

.

The above is not a homogeneous primitive partition identiy if and only if there existtwo natural numbers,p≤m− t andq≤ t−1, such thatq(m− t) = p(t−1) andp+q<m−1. This is possible if and only if GCD(m− t, t−1) = GCD(m−1, t−1)> 1.

3.3 The initial algebra and the algebra ofU-invariants of At 63

Corollary 3.3.3. A finite Sagbi basis of At is formed by the product of minors belongingto it and of degree at most m−1.

Remark 3.3.4. An upper bound for the degree of the minimal generators ofJt mightbe gotten by studying the defining ideal of the initial algebra of At . The minimalgenerators of such an ideal are binomials, so it could be not impossible to find them.Although there is no hope that this ideal is generated in degree at most 3, there is ahope to obtain, in this way, an upper bound linear inm (the one we got in Corollary3.2.7is quadratic inm). Furthermore passing through the initial algebra would saveus from using representation theory, so the results would hold in any not exceptionalcharacteristic (char(k) = 0 or char(k)> mint,m− t). In this direction we can provethat the ideal defining the initial algebra ofA2(3,n) is generated in degree 2, 3 and4. However we will not include this result in the thesis, since its proof is boring andit would extend too weakly (with respect to the effort we would do proving it) ourknowledge about the relations.

3.3.2 A finite system of generators forAUt

Techniques similar to those used in the previous subsectioncan be also used to exhibita finite system of generators for an important subalgebra ofAt . LetU−(W) be the sub-group of the lower triangular matrices of GL(W) with 1’s on the diagonal andU+(V)be the subgroup of the upper triangular matrices of GL(V) with 1’s on the diagonal.Then, set

U :=U−(W)×U+(V)⊆ G.

The groupU plays an important role: In fact one can show that a polynomial f ∈ Ris U-invariant if and only if it belongs to thek-vector space generated by the highestbi-weight vectors with respect toB (for the definition of highest bi-weight vector andof B seeD.3, for the proof of the above fact see [15, Proposition 11.22]). Therefore,we have a description for the ring ofU-invariants ofAt , namely

AUt = k⊕< [cλ |cλ ] : λ is an admissible partition> .

In this subsection we will find a finite set of “basic invariants” of At , that is a finitesystem ofk-algebra generators ofAU

t . In other words, we will solve what is known asthefirst main problemof invariant theory in the case ofAU

t .

Theorem 3.3.5.The ring of invariants AUt is generated by the product of initial minors[cλ |cλ ] whereλ is an admissible partition with at most m− 1 parts. Furthermore isgenerated up to degree m.

Proof. Let A be the subalgebra ofAUt generated by[cλ |cλ ] whereλ is an admissible

partition with at mostm−1 parts. We have to prove that actuallyA= AUt .

Thus letλ = (λ1, . . . ,λk) be an admissible partition withk≥msuch that, by contra-diction, [cλ |cλ ] does not belong toA. Furthermore assume that among such partitionsλ is minimal with respect tok. First, let us suppose that|λ |= kt. Then, as in the proofof Theorem3.3.2, the following

λ1+λ2+ . . .+λk = t + t+ . . .+ t︸︷︷︸k times

is not a homogeneous primitive partition identity by [101, Theorem 6.4]. Thereforethere is a subsetS( [k] such that∑i∈Sλi = |S| · t and∑i∈[k]\Sλi = (k− |S|) · t. We set


λS := (λi : i ∈ S) andλ[k]\S := (λi : i ∈ [k]\S). These are both admissible partitions so[cλS

|cλS] and[cλ[k]\S]

|cλ[k]\S] are elements ofA. We claim that

[cλS|cλS

] · [cλ[k]\S|cλ[k]\S

] = [cλ |cλ ].

In fact [cλS|cλS

] · [cλ[k]\S|cλ[k]\S

] is obviouslyU-invariant. Moreover it has bi-weight

((tλ1, . . . ,t λm)|(−tλn, . . . ,−tλ1)), so it is actually[cλ |cλ ].

It remains the case in which|λ | = dt with d > k. Notice that we can assumethat λi 6= t for all i = 1, . . . ,k: Otherwise, puttingλ ′ := (λ j : j 6= i), we would have[cλ |cλ ] = [cλ ′ |cλ ′ ] · [12|12]. We have the partition identity with entries in[m]

λ1+λ2+ . . .λk = t + t+ . . .+ t︸︷︷︸d times

.

Sets := maxi : λi > t. We can consider the partition identity with entries in[m]

δ1+ δ2+ . . .+ δs= δs+1+ δs+2+ . . .+ δd (3.10)

whereδi := |λi − t| for i ≤ k andδi = t for k< i ≤ d. We claim that the above partitionidentity is primitive. By the contrary, suppose that there exist i1, . . . , ip ⊆ [s] and j1, . . . , jq ⊆ s+1, . . . ,d with p+q< d such that

δi1 + δi2 + . . .+ δip = δ j1 + δ j2 + . . .+ δ jq.

Therefore ifu1, . . . ,us−p = [s] \ i1, . . . , ip andv1, . . . ,vd−s−q = s+ 1, . . . ,d \ j1, . . . , jq we also have

δu1 + δu2 + . . .+ δus−p = δv1 + δv2 + . . .+ δvd−s−q.

Settingr := maxa : ja ≤ k andl := maxb : vb ≤ k we get

λi1 +λi2 + . . .+λip +λ j1 + . . .+λ jr = t + t+ . . .+ t︸︷︷︸p+q times

andλu1 +λu2 + . . .+λus−p +λv1 + . . .+λvl = t + t+ . . .+ t︸︷︷︸

d−p−q times

.

Let α := (λi1, . . . ,λip,λ j1, . . . ,λ jr ) andβ := (λu1, . . . ,λus−p,λv1, . . . ,λvl ) be the respec-tive two partitions. Sincep+ r ≤ p+ q ands− p+ l ≤ d− p− q both α andβ areadmissible partitions. So, as above[cλ |cλ ] = [cα |cα ] · [cβ |cβ ] belongs toA, which is acontradiction. Therefore the partition (3.10) must be primitive. So, using [101, Corol-lary 6.2] we get,d ≤ m− t+ t = m, and thereforek≤ m−1.

As a consequence of Theorem3.3.5we have an upper bound on the maximumdegree of a generator of any prime ideal ofAt which is aG-subrepresentation, for shortaG-stable prime ideal.

Corollary 3.3.6. A G-stable prime ideal of At(m,n) is generated in degree less than orequal to m.

Proof. Suppose to have a minimal generator of degree greater thanm in a G-stableprime ideal℘⊆ At . Since℘ is aG-subrepresentation ofAt , than we can suppose thatsuch an element is of the type[cλ |cλ ] with |λ | > mt. By Theorem3.3.5there existtwo admissible partitionsγ andδ such that|γ|, |δ |< |λ | and[cγ |cγ ] · [cδ |cδ ] = [cλ |cλ ].Since[cλ |cλ ] is a minimal generator, this contradicts the primeness of℘.

Chapter 4

Symbolic powers and Matroids

This chapter is shaped on our paper [105]. It is easy to show that, ifS:= k[x1, . . . ,xn]is a polynomial ring inn variables over a fieldk andI is a complete intersection ideal,thenS/Ik is Cohen-Macaulay for all positive integerk. A result of Cowsik and Nori,see [27], implies that the converse holds true, provided the ideal is homogeneous andradical. Therefore, somehow the above result says that there are no ideals with thisproperty but the trivial ones. On the other hand, ifS/Ik is Cohen-Macaulay, thenIk hasnot any embedded prime ideals, so by definition it is equal to thekth symbolic powerof I , namelyIk = I (k) (seeE.1). So, it is natural to ask:

For which I⊆ S is the ring S/I (k) Cohen-Macaulay for any positive integer k?

We will give a complete answer to the above question in the case in which I isa square-free monomial ideal. Notice that whenn = 4 this problem was studied byFrancisco in [41]. These kind of ideals, whose basic properties are summarized inE,supply a bridge between combinatorics and commutative algebra, given by attachingto any simplicial complex∆ on n vertices the so-calledStanley-Reisner ideal I∆ andStanley-Reisner ringk[∆] = S/I∆. One of the most interesting parts of this theory isfinding relationships between combinatorial or topological properties of∆ and ring-theoretic ones ofk[∆]. For instance, Reisner gave a topological characterization ofthose simplicial compexes∆ for whichk[∆] is Cohen-Macaulay (for example see Millerand Sturmfels [81, Theorem 5.53]). Such a characterization depends on certain singularhomology groups with coefficients ink of topological spaces related to∆. In fact,the Cohen-Macaulayness of∆ may depend on char(k). At the contrary, a wide openproblem is, once fixed the characteristic, to characterize in a purely graph-theoreticfashion those graphsG for whichk[∆(G)] is Cohen-Macaulay, where∆(G) denotes theindependence complex ofG. For partial results around this problem see, for instance,Herzog and Hibi [59], Herzog, Hibi and Zheng [62], Kummini [70] and our paper jointwith Constantinescu [25]. Thus, in general, even if a topological characterizationof the“Cohen-Macaulay simplicial complex” is known, a purely combinatorial one is still amystery. In this chapter we are going to give a combinatorialcharacterization of those

simplicial complexes∆ such thatS/I (k)∆ is Cohen-Macaulay for all positive integerk.The characterization is quite amazing, in fact the combinatorial counter-party of the

algebraic one “S/I (k)∆ Cohen-Macaulay” is a well studied class of simplicial complexes,whose interest comes besides this result. Precisely we willshow in Theorem4.1.1:

The ring S/I (k)∆ is Cohen-Macaulay for all k> 0 ⇐⇒ ∆ is a matroid.

66 Symbolic powers and Matroids

Matroids have been introduced as an abstraction of the concept of linear indepen-dence. We briefly discuss some basic properties of them in Section E.2, however thereare entire books treating this subject, such as that of Oxley[88] or the one of Welsh[106]. It must be said that Theorem4.1.1has been proved independently and withdifferent methods by Minh and Trung in [84, Theorem 3.5].

Actually, we will show a more general version of Theorem4.1.1described above,namely Theorem4.1.6, regarding a class of monomial ideals more general than thesquare-free ones. Let us say that this generalization appears for the first time, since itwas not present in our original paper [105]. As a first tool for our proof, we need aduality for matroids(see TheoremE.2.2), which will allow us to switch the problemfrom I∆ to the cover idealJ(∆). The if-part of the proof, at this point, is based on theinvestigation of the symbolic fiber cone ofJ(∆), the so-called algebra of basic coversA(∆) (more generallyA(∆,ω), whereω is a weighted function on the facets of∆).Such an algebra was introduced by Herzog during the summer schoolPragmatic 2008,in Catania. Among other things, he asked for its dimension. Since then, even if acomplete answer is not yet known, some progresses have been done: In our paper [25],a combinatorial characterization of the dimension ofA(∆) is given for one-dimensionalsimplicial complexes∆. Here we will show that dim(A(∆)) is minimal whenever∆is a matroid. To this purpose anexchange property for matroids, namely (E.4), isfundamental. Eventually, Proposition4.1.7implies the if-part of Theorem4.1.1.

In [105], we actually showed that dim(∆) is minimalexactlywhen∆ is a matroid,getting also theonly-if partof Theorem4.1.1. Here, however, we decided to prove this

part in an other way. Namely, we study the associated primes of the polarization ofI (k)∆(Corollary4.1.11), showing that if∆ is not a matroid, then there is a lack of connect-edness of certain ideals related to∆, obstructing their Cohen-Macaulayness (Theorem

4.1.12). In particular, this will imply that ifS/In−dim(∆)+2∆ is Cohen-Macaulay, then∆

has to be a matroid, a stronger statement than the only-if part of Theorem4.1.1.It turns out that Theorem4.1.1has some interesting consequences. For instance,

in Corollary4.2.1we deduce that dim(A(∆)) = k[∆] if and only if ∆ is a matroid, andthat, in this situation, the “multiplicity ofA(∆)” is bounded above from the multiplic-ity of k[∆] (here the commas are due to the fact that, in general,A(∆) is not standardgraded). An other consequence regards the problem of set-theoretic complete inter-sections (Corollary4.2.4): After localizing at the maximal irrelevant ideal, I∆ is aset-theoretic complete intersection whenever∆ is a matroid.

4.1 Towards the proof of the main result

In this section we prove the main theorem of the chapter. For the notation and thedefinitions of the basic objects we remind to AppendixE. In particular,k will denotea field,S := k[x1, . . . ,xn] will be the polynomial ring inn variables overk, andm :=(x1, . . . ,xn)⊆ Swill denote the maximal irrelevant ideal ofS.

Theorem 4.1.1.Let ∆ be a simplicial complex on[n]. The following are equivalent:

(i) S/I (k)∆ is Cohen-Macaulay for any k∈ N≥1.(ii) S/J(∆)(k) is Cohen-Macaulay for any k∈ N≥1.(iii) ∆ is a matroid.

Remark 4.1.2. Notice that condition (iii) of Theorem4.1.1does not depend on thecharacteristic ofk. Thus, as a consequence of Theorem4.1.1, conditions (i) and (ii) donot depend on char(k) as well as (iii). This fact was not clear a priori.

4.1 Towards the proof of the main result 67

Remark 4.1.3. If ∆ is them-skeleton of the(n−1)-simplex,−1≤ m≤ n−1, then∆is a matroid. So Theorem4.1.1implies that all the symbolic powers ofI∆ are Cohen-Macaulay.

Actually, we will prove a slightly more general version of Theorem4.1.1, since thisdoes not require much more effort. More precisely, we will show Theorem4.1.1for alarger class of pure monomial ideals than the pure square-free ones. We are going tointroduce them below: For a simplicial complex∆ on [n], a function

ω : F (∆) → N\ 0F 7→ ωF

is calledweighted function(this concept has been introduced by Herzog, Hibi andTrung in [61]). Moreover, the pair(∆,ω) is called aweighted simplicial complex. Theauthors of [61] studied the properties of theweighted monomial ideal

J(∆,ω) :=⋃

F∈F (∆)℘ωF

F .

If ω is thecanonical weighted function, that isωF = 1 for anyF ∈ F (∆), it turns outthanJ(∆,ω) = J(∆) is nothing but than the cover ideal of∆. In particular the classof all the weighted monomial ideals contains the square-free ones. The class we wantto define stays between the pure square-free monomial idealsand the weighted puremonomial ideals. We say that a weighted functionω is agood-weighted functionif itis induced by aweight on the variables, namely if there exists a functionλ : [n]→R>0

such thatω = ωλ , where

ωλF := ∑

i∈F

λ (i) ∀ F ∈ F (∆).

In this case the pair(∆,ω) will be called agood-weighted simplicial complexand theidealJ(∆,ω) a good-weighted monomial ideal.

Remark 4.1.4. If I ⊆Sis a pure square-free monomial ideal, then it is a good-weightedmonomial ideal. In fact, we have thatI = J(∆) for some pure simplicial complex∆. Let d− 1 be the dimension of∆. Then, because∆ is pure,|F | = d for all facetsF ∈ F (∆). Defining the functionλ : [n] → R>0 asλ (i) := 1/d for any i ∈ N, wehave that the canonical weighted function is induced byλ , and soJ(∆) = J(∆,ωλ ) isa good-weighted monomial ideal.

The assumption thatI is pure, in Remark4.1.4, is necessary, as we are going toshow in the next example.

Example 4.1.5.Let ∆ be the simplicial complex on1, . . . ,5 such that

F (∆) = 1,2,1,3,1,4,3,4,2,3,5.

If the canonical weighted function on∆ were induced by some weightλ on the vari-ables, then we should haveλ (1) = λ (2) = λ (3) = λ (4) = 1/2. This, since2,3,5 ∈F (∆), would implyλ (5) = 0, a contradiction.

We will prove the following theorem which, as we are going to show just below,implies Theorem4.1.1.

Theorem 4.1.6.Let J= J(∆,ω)⊆ S be a good-weighted monomial ideal. Then S/J(k)

is Cohen-Macaulay for any k∈ N≥1 if and only if∆ is a matroid.


Proof. (of Theorem4.1.1from Theorem4.1.6). (ii) =⇒ (iii). SinceS/J(∆) is Cohen-Macaulay,∆ has to be pure. ThereforeJ(∆) is good-weighted by Remark4.1.4, andTheorem4.1.6implies that∆ is a matroid.

(iii) =⇒ (ii). ∆ is pure because it is a matroid (seeE.2). ThusJ(∆) is good-weighted by Remark4.1.4, and Theorem4.1.6yields thatS/J(∆)(k) is Cohen-Macaulayfor all k∈ N≥1.

(i) ⇐⇒ (ii). By TheoremE.2.2a simplicial complex∆ is a matroid if and only if∆c is a matroid. SinceI∆ = J(∆c) andI∆c = J(∆), the equivalence between (ii) and (iii)yields the equivalence between (i) and (ii).

4.1.1 The algebra of basick-covers and its dimension

In AppendixE we introduced the concept of vertex cover of a simplicial complex,emphasizing how it lends the name to the cover ideal. The concept of vertex covercan be extended in the right way to a weighted simplicial complex as follows: For allnatural numberk, a nonzero functionα : [n]→N is ak-coverof a weighted simplicialcomplex(∆,ω) on [n] if

∑i∈F

α(i) ≥ kωF ∀ F ∈ F (∆).

If ω is the canonical weighted function, i.e. if(∆,ω) is an ordinary simplicial com-plex, then vertex covers are in one-to-one correspondence with 1-covers with entries in0,1. Moreover, it is easy to show

J(∆,ω) = (xα(1)1 · · ·xα(n)

n : α is a 1-cover).

Actually k-covers come in handy to describe a set of generators of all thekth symbolicpowers ofJ(∆,ω). In fact, by (E.3), we get

J(∆,ω)(k) =⋂

F∈F (∆)℘kωF

F .

Therefore, as before, one can show

J(∆,ω)(k) = (xα(1)1 · · ·xα(n)

n : α is ak-cover).

A k-coverα of (∆,ω) is said to bebasicif for any k-coverβ of (∆,ω) with β (i)≤α(i)for any i ∈ [n], we haveβ = α. Of course to the basick-covers of(∆,ω) correspondsa minimal system of generators ofJ(∆,ω)(k).

Now let us consider the multiplicative filtrationS ymb(∆,ω) := J(∆,ω)(k)k∈N≥0

(for any idealI in a ringR, we setI (0) := I0 = R). We can form the Rees algebra ofSwith respect to the filtrationS ymb(∆,ω), namely

A(∆,ω) :=⊕

k∈NJ(∆,ω)(k).

Actually A(∆,ω) is nothing but than the symbolic Rees algebra ofJ(∆,ω). In [61,Theorem 3.2], Herzog, Hibi and Trung proved thatA(∆,ω) is noetherian. In particular,the associated graded ring ofSwith respect toS ymb(∆,ω)

G(∆,ω) :=⊕

k∈NJ(∆,ω)(k)/J(∆,ω)(k+1)


and the special fiber

A(∆,ω) := A(∆,ω)/mA(∆,ω) = G(∆,ω)/mG(∆,ω)

are noetherian too. The algebraA(∆,ω) is known as thevertex cover algebraof (∆,ω),and its properties have been intensively studied in [61]. The name comes from the factthat, writing

A(∆,ω) =⊕

k∈NJ(∆,ω)(k) · tk ⊆ S[t]

and denoting byA(∆,ω)k = J(∆,ω)(k) · tk, it turns out that, fork≥ 1, a (infinite) basisfor A(∆,ω)k as ak-vector space is

xα(1)1 · · ·xα(n)

n · tk : α is ak-cover of(∆,ω).

The algebraA(∆,ω) is more subtle to study with respect to the vertex cover algebra:For instance it is not evan clear which is its dimension. It iscalled thealgebra of basiccoversof (∆,ω), and its properties have been studied for the first time by theauthorwith Benedetti and Constantinescu in [6] when∆ is a bipartite graph. More generally,in [25], we studied it for any 1-dimensional simplicial complex∆. Clearly, the gradingdefined above onA(∆,ω) induces a grading onA(∆,ω), and it turns out that a basis forA(∆,ω)k, for k≥ 1, as ak-vector space is

xα(1)1 · · ·xα(n)

n · tk : α is a basick-cover of(∆,ω).

Notice that ifα is a basick-cover of (∆,ω), thenα(i) ≤ λk for any i ∈ [n], whereλ := maxωF : F ∈ F (∆). This implies thatA(∆,ω)k is a finitek-vector space foranyk∈N. So we can speak about the Hilbert function ofA(∆,ω), denoted by HFA(∆,ω),and from what said above we have, fork≥ 1,

HFA(∆,ω)(k) = |basick-covers of(∆,ω)|.

The key to prove Theorem4.1.1is to compute the dimension ofA(∆,ω). So we need acombinatorial description of dim(A(∆,ω)). Being in general non-standard graded, thealgebraA(∆,ω) could not have a Hilbert polynomial. However, by [61, Corollary 2.2]we know that there existsh∈ N such that(J(∆,ω)(h))k = J(∆,ω)(hk) for all k ≥ 1. Itfollows that thehth Veronese of the algebra of basic covers, namely

A(∆,ω)(h) :=⊕

k∈NA(∆,ω)hk,

is a standard gradedk-algebra. Notice that if a set f1, . . . , fq generatesA(∆,ω) as

a k-algebra, then the set f i11 · · · f

iqq : 0 ≤ i1, . . . , iq ≤ h− 1 generatesA(∆,ω) as

a A(∆,ω)(h)-module. Thus dim(A(∆,ω)) = dim(A(∆,ω)(h)). SinceA(∆,ω)(h) has aHilbert polynomial, we get a useful criterion to compute thedimension ofA(∆,ω).First let us remind that, for two functionsf ,g : N→ R, the writing f (k) = O(g(k))means that there exists a positive real numberλ such thatf (k) ≤ λ ·g(k) for k ≫ 0.Similarly, f (k) = Ω(g(k)) if there is a positive real numberλ such thatf (k)≥ λ ·g(k)for k≫ 0

Criterion for detecting the dimension ofA(∆,ω). If HFA(∆,ω)(k) = O(kd−1), then

dim(A(∆,ω))≤ d. If HFA(∆,ω)(k) = Ω(kd−1), then dim(A(∆,ω))≥ d.

The following proposition justifies the introduction ofA(∆). It is an alternativeversion of a result got by Eisenbud and Huneke in [38].


Proposition 4.1.7. For any simplicial complex∆ on [n] we have

dim(A(∆,ω)) = n−mindepth(S/J(∆,ω)(k)) : k∈ N≥1

Proof. ConsiderG(∆,ω), the associated graded ring ofSwith respect toS ymb(∆,ω).SinceG(∆,ω) is noetherian, it follows by Bruns and Vetter [15, Proposition 9.23] that

mindepth(S/J(∆,ω)(k)) : k∈ N≥1= grade(mG(∆,ω)).

We claim thatG(∆,ω) is Cohen-Macaulay.In fact the Rees ring ofS with respect tothe filtrationS ymb(∆,ω), namelyA(∆,ω), is Cohen-Macaulay by [61, Theorem 4.2].Let us denote byM := m⊕A(∆,ω)+ the unique bi-graded maximal ideal ofA(∆,ω).The following short exact sequence

0−→ A(∆,ω)+ −→ A(∆,ω)−→ S−→ 0

yields the long exact sequence on local cohomology

. . .−→ H i−1M

(S)−→ H iM(A(∆,ω)+)−→ H i

M(A(∆,ω))−→ . . . .

By the independence of the base in computing local cohomology modules (see Lemma0.2.2(ii)) we haveH i

M(S) =H i

m(S) = 0 for anyi < n by (0.8). Furthermore, using onceagain (0.8), H i

M(A(∆,ω)) = 0 for anyi ≤ n sinceA(∆,ω) is a Cohen-Macaulay(n+1)-

dimensional (see Bruns and Herzog [13, Theorem 4.5.6]) ring. ThusH iM(A(∆,ω)+) =

0 for any i ≤ n by the above long exact sequence. Now let us look at the other shortexact sequence

0−→ A(∆,ω)+(1)−→ A(∆,ω)−→ G(∆,ω)−→ 0,

whereA(∆,ω)+(1) meansA(∆,ω)+ with the degrees shifted by 1, and the correspond-ing long exact sequence on local cohomology

. . .−→ H iM(A(∆,ω))−→ H i

M(G(∆,ω))−→ H i+1M

(A(∆,ω)+(1))−→ . . . .

BecauseA(∆,ω)+ and A(∆,ω)+(1) are isomorphicA(∆,ω)-module, we have thatH iM(A(∆,ω)+(1)) = 0 for any i ≤ n. ThusH i

M(G(∆,ω)) = 0 for any i < n. Since

G(∆,ω) is ann-dimensional ring (see [13, Theorem 4.5.6]) this implies, using onceagain Lemma0.2.2(ii) and (0.8), thatG(∆,ω) is Cohen-Macaulay.

SinceG(∆,ω) is Cohen-Macaulay, grade(mG(∆,ω)) = ht(mG(∆,ω)) (for instancesee Matsumura [80, Theorem 17.4]). So, becauseA(∆,ω) = G(∆,ω)/mG(∆,ω), weget

dim(A(∆,ω)) = dim(G(∆,ω))−ht(mG(∆,ω)) = n−grade(mG(∆,ω)),

and the statement follows at once.

4.1.2 If-part of Theorem 4.1.6

In this subsection we show theif-part of Theorem4.1.6, namely:If ∆ is a matroid, thenS/J(∆,ω)(k) is Cohen-Macaulay for each good-weighted functionω on∆. To this aimwe need the following lemma, which can be interpreted as a sort of rigidity property ofthe basic covers of a good-weighted matroid.


Lemma 4.1.8. Let ∆ be a matroid on[n], ω a good-weighted function on it and k apositive integer. Let us fix a facet F∈F (∆), and letα : F →N be a function such that∑i∈F α(i) = kωF . Then there exists an unique way to extendα to a basic k-cover of(∆,ω).

Proof. That such an extension exists is easy to prove, and to get it wedo not even needthat∆ is a matroid nor thatω is a good-weighted function. Setα ′ : [n]→N the nonzerofunction such thatα ′ |F= α andα ′(i) := Mk if i ∈ [n]\F, whereM := maxωF : F ∈F (∆). Certainlyα ′ will be a k-cover of(∆,ω). If it is not basic, then there exists avertexi ∈ [n] such that∑ j∈G α ′( j) > kωG for anyG containingi. Thus we can lowerthe valueα ′(i) of one, getting a newk-cover. Such a newk-cover is still an extensionof α, since obviouslyi /∈ F. Going on in such a way we will eventually find a basick-cover extendingα.

The uniqueness, instead, is a peculiarity of matroids. Leti0 ∈ [n] be a vertex whichdoes not belong toF , and let us denote byα ′ a basick-cover of(∆,ω) extendingα.Sinceα ′ is basic, there must exist a facetG of ∆ such thati0 ∈ G and∑i∈G α ′(i) = kωG.By the exchange property for matroids (E.4), there exists a vertexj0 ∈ F such that

G′ := (G\ i0)∪ j0 ∈ F (∆) and F ′ := (F \ j0)∪i0 ∈ F (∆).

So, denoting byλ the weight on the variables inducingα ′, we have

∑i∈G′

α ′(i)≥ kωG′ = ∑i∈G′

kλ (i),

which yieldsα ′( j0)− kλ ( j0)≥ α(i0)− kλ (i0), and

∑j∈F ′

α ′( j)≥ kωF ′ = ∑j∈F ′

kλ ( j),

which yieldsα ′( j0)− kλ ( j0) ≤ α(i0)− kλ (i0). Therefore there is only one possiblevalue to assign toj0, namely:

α ′( j0) = α(i0)− kλ (i0)+ kλ ( j0).

Now we are ready to prove the if-part of Theorem4.1.6.

Proof. If-part of Theorem4.1.6. Let us consider a basick-cover α of our good-weighted matroid(∆,ω). Sinceα is basic there is a facetF of ∆ such that∑ j∈F α( j) =kωF . Setd := |F|. By Lemma4.1.8we deduce that the values assumed byα on [n]are completely determined by those onF . Furthermore, the ways to give values on thevertices ofF in such a manner that∑i∈F α(i) = kωF , are exactly:

(kωF +d−1

d−1

),

which is a polynomial ink of degreed−1. This implies that, fork≥ 1,

HFA(∆,ω)(k) = |basick-covers of(∆,ω)| ≤ |F (∆)| ·(

kM+d−1d−1

),

whereM := maxωF : F ∈ F (∆). Since|F (∆)| does not depend onh, we get

HFA(∆,ω)(k) = O(kd−1).


So dim(A(∆,ω)) ≤ d = dim(∆)+ 1 (the last equality is because∆ is pure, seeE.2).Since dim(S/J(∆)) = n−d, by Proposition4.1.7we get

d ≥ dim(A(∆,ω)) = n−mindepth(S/J(∆,ω)(k)) : k∈N≥1 ≥ d,

from whichS/J(∆,ω)(k) is Cohen-Macaulay for anyk∈N≥1.

Remark 4.1.9. If ∆ is a matroid on[n], may exist a weighted functionω of ∆ such thatS/J(∆,ω)(k) is Cohen-Macaulay for allk∈ N≥1 andω is not good. For instance, con-sider the complete graph∆ = K4 on 4 vertices, which clearly is a matroid. Furthermoreconsider the following ideal:

J(K4,ω) := (x1,x2)2∩ (x1,x3)∩ (x1,x4)∩ (x2,x3)∩ (x2,x4)∩ (x3,x4).

The corresponding weighted function is clearly not good, however one can show thatS/J(K4,ω)(k) is Cohen-Macaulay for allk ∈ N≥1 using the result of Francisco [41,Theorem 5.3 (iii)].

4.1.3 Only if-part of Theorem 4.1.6

To this aim we need to describe the associated prime ideals ofthe polarization (seeE.3)of a weighted monomial idealJ(∆,ω), namely

J(∆,ω)⊆ S.

Since polarization commutes with intersections (see (E.6)), we get:

J(∆,ω) =⋂

F∈F (∆)℘ωF

F .

Thus, to understand the associated prime ideals ofJ(∆,ω) we can focus on the de-

scription of Ass(℘ωFF ). So let us fix a subsetF ⊆ [n] and a positive integerk. We

have:℘k

F ⊆ S= k[xi, j : i = 1, . . . ,n, j = 1, . . . ,k].

Being a monomial ideal, the associated prime ideals of℘kF are ideals of variables.

For this reason is convenient to introduce the following notation: For a subsetG :=i1, . . . , id⊆ [n] and a vectora :=(a1, . . . ,ad)⊆Nd with 1≤ ai ≤ k for anyi = 1, . . . ,d,we set

℘G,a := (xi1,a1,xi2,a2, . . . ,xid,ad).

Lemma 4.1.10.Let F := i1, . . . , id ⊆ [n] and k be a positive integer. Then, a prime

ideal℘⊆ S is associated to℘kF if and only if℘=℘F,a with |a| := a1 + . . .+ad ≤

k+d−1.

Proof. A minimal generator of℘kF is of the form

xF,b :=b1

∏p=1

xi1,p ·b2

∏p=1

xi2,p · · ·bd

∏p=1

xid,p,

whereb := (b1, . . . ,bd) ∈ Nd is such that|b| = k. Let us callBk ⊆ Nd the set of suchvectors. An associated prime of℘F is forced to be generated byd variables: In fact,


℘kF is Cohen-Macaulay of heightd like ℘k

F from TheoremE.3.4. Moreover, it is easyto check that a prime ideal of variables

℘ := (x j1,c1, x j2,c2, . . . , x jd,cd)⊆ S,

wherec := (c1, . . . ,cd) ∈ [k]d, is associated to℘kF if and only if

∀ b ∈ Bk ∃ p∈ [d] such thatx jp,cp|xF,b. (4.1)

So, if we chooseb := (0,0, . . . ,0,k,0, . . . ,0), where the nonzero entry is at the placep,we get thatip is in j1, . . . , jd. Letting varyp∈ [d], we eventually get j1, . . . , jd=F ,i.e.℘=℘F,c. Moreover notice that

xip,cp|xF,b ⇐⇒ cp ≤ bp. (4.2)

Suppose by contradiction that|c| ≥ k+d and setp0 := minp : ∑pi=1ci ≥ k+ p ≤ d.

Then chooseb := (b1, . . . ,bd) ∈ Bk in this way:

bp :=

cp−1 if p< p0

k−∑p0−1i=1 (ci −1) if p= p0

0 if p> p0

Notice that the property ofp0 implies thatbp0 < cp0. Moreover its minimality impliesbp0 >0. Sincebp< cp for all p= 1, . . . ,d, it cannot exist anyp∈ [d] such thatxip,cp|xF,b

by (4.2). Therefore℘F,c /∈ Ass(℘kF) by (4.1).

On the other hand, ifc∈ [k]d is such that|c| ≤ k+d−1, then for allb ∈ Bk thereexistsp∈ [d] such thatcp ≤ bp, otherwise|b| ≤ |c|−d ≤ k−1. Thereforexip,cp|xF,b

by (4.2) and℘F,c ∈ Ass(℘kF) by (4.1).

Corollary 4.1.11. Let ∆ be a simplicial complex on[n] andω a weighted function on

it. A prime ideal℘⊆ S is associated toJ(∆,ω) if and only if℘=℘F,a with F ∈ F (∆)and|a| ≤ ωF + |F|−1.

Proof. Recall thatJ(∆,ω) =⋂

F∈F (∆)℘ωF . By (E.6), we have

J(∆,ω) =⋂

F∈F (∆)℘ωF

F .

Therefore, by Lemma4.1.10, we get

J(∆,ω) =⋂

F∈F (∆)

⋂

|a|≤ωF+|F|−1

℘F,a

.

Thus we conclude.

Theorem 4.1.12.Let ∆ be a simplicial complex on[n] andω a weighted function onit. Furthermore assume thatωF ≥ dim(∆)+2 for all facet F of∆. If S/J(∆,ω) isCohen-Macaulay, then∆ is a matroid.


Proof. Suppose, by contradiction, that∆ is not a matroid. Then there exist two facetsF,G∈ F (∆) and a vertexi ∈ F such that

(F \ i)∪ j /∈ F (∆) ∀ j ∈ G. (4.3)

BeingS/J(∆,ω) Cohen-Macaulay, then∆ has to be pure. So, setting dim(∆) := d−1,we can assume:

F = i1, . . . , id, G= j1, . . . , jd and i = i1.

Let us define:

a := (d+1,1,1, . . . ,1) ∈Nd and b := (2,2, . . . ,2) ∈Nd.

Notice that|a| = |b| = 2d. So, since by assumptionωF andωG are at leastd+ 1,Corollary4.1.11implies:

℘F,a, ℘G,b ∈ Ass(J(∆,ω)).

SinceS/J(∆,ω) is Cohen-Macaulay,S/J(∆,ω) has to be Cohen-Macaulay as well byTheoremE.3.4. So, the localization

(S/J(∆,ω))℘F,a+℘G,b

is Cohen-Macaulay. Particularly it is connected in codimension 1 (see PropositionB.1.2). This translates into the existence of a sequence of prime ideals

℘F,a =℘0, ℘1, . . . , ℘s=℘G,b

such that℘i ∈ Ass(J(∆,ω)), ℘i ⊆℘F,a+℘G,b and ht(℘i +℘i−1) ≤ d+ 1 for eachindex i making sense (see LemmaB.0.7). In particular, ht(℘F,a+℘1) ≤ d+1. Since

℘1 ∈ Ass(J(∆,ω)) and it is contained in℘F,a+℘G,b, there must existp,q∈ [d] suchthatc= (a1,a2, . . . ,ap−1,bq,ap+1, . . . ,ad) ∈ Nd and

℘1 = (xi1,a1, . . . , xip−1,ap−1, x jq,bq, xip+1,ap+1, . . . , xid,ad) =℘(F\ip)∪ jq,c.

But this is eventually a contradiction: In fact, ifp= 1, then(F \i)∪ jq would be afacet of∆, a contradiction to (4.3). If p 6= 1, then|c|= 2d+1. This contradicts Lemma

4.1.10, since℘1 =℘(F\ip)∪ jq,c is associated toJ(∆,ω).

Eventually, Theorem4.1.12implies the only-if part of Theorem4.1.6.

Proof. Only if-part of Theorem4.1.6. SinceωF is a positive integer for any facetF ∈ F (∆), then(dim(∆) + 2)ωF ≥ dim(∆) + 2 for all facetsF ∈ F (∆). So, sinceS/J(∆,ω)(dim(∆)+2) is Cohen-Macaulay,∆ has to be a matroid by Theorem4.1.12.

Remark 4.1.13. Actually Theorem4.1.12is much stronger than the only-if part ofTheorem4.1.6: In fact there are not any assumptions on the weighted function ω .Moreover, Theorem4.1.12implies that it is enough that an opportune symbolic powerof J(∆,ω) is Cohen-Macaulay to force∆ to be a matroid!

4.2 Two consequences 75

4.2 Two consequences

We end the the chapter stating some applications of Theorem4.1.6. We recall thatwe already introduced in Chapter2 the concept of multiplicity of a standard gradedk-algebraR, namelye(R), in terms of the Hilbert series ofR. It is well known thate(R) can be also defined as the leading coefficient of the Hilbert polynomial times(dim(R)− 1)!, see [13, Proposition 4.1.9]. Geometrically, let ProjR⊆ PN, i.e. R=K[X0, . . . ,XN]/J for a homogeneous idealJ. The multiplicity e(R) counts the numberof distinct points of ProjR∩H, whereH is a generic linear subspace ofPN of dimensionN−dim(ProjR). Before stating the next result, let us say that for a simplicial complex∆, if ω is the canonical weighted function, then we denoteA(∆,ω) simply byA(∆).

Corollary 4.2.1. Let ∆ be a simplicial complex andω a good-weighted function on it.Then∆ is a (d−1)-dimensional matroid if and only if:

dim(A(∆,ω)) = dim(k[∆]) = d.

Moreover, if∆ is a matroid, then

HFA(∆)(k)≤e(k[∆])

(dim(A(∆))−1)!kdim(A(∆))−1+O(kdim(A(∆))−2).

Proof. The first fact follows putting together Theorem4.1.6and Proposition4.1.7. Forthe second fact, we have to recall that, during the proof of the if-part of Theorem4.1.6,we showed that for a(d−1)-dimensional matroid∆ we have the inequality

HFA(∆)(k)≤ |F (∆)| ·(

k+d−1d−1

).

It is well known that if ∆ is a pure simplicial complex then|F (∆)| = e(K[∆]) (forinstance see [13, Corollary 5.1.9]), so we get the conclusion.

Example 4.2.2. If ∆ is not a matroid the inequality of Corollary4.2.1may not be true.For instance, take∆ :=C10 the decagon (thus it is a1-dimensional simplicial complex).Since C10 is a bipartite graphA(C10) is a standard gradedk-algebra by [61, Theorem5.1]. In particular it admits a Hilbert polynomial, and for k≫ 0 we have

HFA(C10)(k) =

e(A(C10))

(dim(A(C10))−1)!kdim(A(C10))−1+O(kdim(A(C10))−2).

In [25] it is proved that for any bipartite graph G the algebraA(G) is a homogeneousASL on a poset described in terms of the minimal vertex coversof G. So the multiplicityof A(G) can be easily read from the above poset. In our case it is easy to check thate(A(C10)) = 20, whereas e(k[C10]) = 10.

For the next result we need the concepts of “arithmetical rank” of an ideal and “set-thoeretic complete intersections”, introduced in0.4.2. By 0.10, if an ideala of a ringR is a set-theoretical complete intersection, then cd(R,a) = ht(a). In the case in whichR= Sanda = I∆ where∆ is some simplicial complex on[n], by a result of Lyubeznikgot in [73] (see Theorem0.3.4), it turns out that cd(S, I∆) = n−depth(k[∆]). So, if I∆is a set-theoretic complete intersection, thenk[∆] will be Cohen-Macaulay.


Remark 4.2.3. In general, even ifk[∆] is Cohen-Maculay, thenI∆ might not be aset-theoretic complete intersection. For instance, if∆ is the triangulation of the realprojective plane with 6 vertices described in [13, p. 236], thenk[∆] is Cohen-Macaulaywhenever char(k) 6= 2. However, for any characteristic ofk, I∆ need at least (actuallyexactly) 4 polynomials ofk[x1, . . . ,x6] to be defined up to radical (see the paper ofYan [108, p. 317, Example 2]). Since ht(I∆) = 3, this means thatI∆ is not a completeintersection for any field.

Corollary 4.2.4. Let k be an infinite field. For any matroid∆, the ideal I∆Sm is aset-theoretic complete intersection in Sm.

Proof. By the duality on matroids it is enough to prove thatJ(∆)Sm is a set-theoreticcomplete intersection. Forh≫ 0 it follows by [61, Corollary 2.2] that thehth Veroneseof A(∆),

A(∆)(h) =⊕

m≥0

A(∆)hm,

is standard graded. ThereforeA(∆)(h) is the ordinary fiber cone ofJ(∆)(h). MoreoverA(∆) is finite as aA(∆)(h)-module. So the dimensions ofA(∆) and ofA(∆)(h) are thesame. Therefore, using Corollary4.2.1, we get

ht(J(∆)Sm) = ht(J(∆)) = dimA(∆)(h) = ℓ(J(∆)(h)) = ℓ((J(∆)Sm)(h)),

whereℓ(·) is the analytic spread of an ideal, i.e. the Krull dimension of its ordinary fibercone. From a result by Northcott and Rees in [86, p. 151], sincek is infinite, it followsthat the analytic spread of(J(∆)Sm)(h) is the cardinality of a set of minimal generatorsof a minimal reduction of(J(∆)Sm)(h). Clearly the radical of such a reduction is thesame as the radical of(J(∆)Sm)(h), i.e. J(∆)Sm, so we get the statement.

Remark 4.2.5. Notice that a reduction ofISm, whereI is a homogeneous ideal ofS,might not provide a reduction ofI . So localizing at the maximal irrelevant ideal is acrucial assumption of Corollary4.2.4. It would be interesting to know whetherI∆ is aset-theoretic complete intersection inSwhenever∆ is a matroid.

4.3 Comments

One of the keys to get our proof of Theorem4.1.1is to characterize the simplicial com-plexes∆ on [n] for which the Krull dimension ofA(∆) is the least possible. We suc-ceeded in this task showing that this is the case if and only if∆ is a matroid (Corollary4.2.1). In view of this, it would be interesting to characterize ingeneral the dimen-sion of A(∆) in terms of the combinatorics of∆. In [6, Theorem 3.7], we located thefollowing range for the dimension of the algebra of basic covers:

dim∆+1≤ dimA(∆)≤ n−⌊

n−1dim∆+1

⌋.

In [25], we characterized in a combinatorial fashion the dimension of A(G) for a graphG, that is a 1-dimensional simplicial complex. However, already in that case, mattershave been not easy at all. We will not say here which is the graph-theoretical invariantallowing to read off the dimension ofA(G), since it is a bit technical. It suffices tosay that, for reasonable graphs, one can immediately compute such an invariant, andtherefore the dimension of the algebra of basic covers. The following is an interestingexample.

4.3 Comments 77

Example 4.3.1.Let G=C6 the hexagon, namely:

C6 : s

s

s

s

s

s

6

2

4

1

5

3

By [25, Theorem 3.8], it follows thatdimA(C6) = 3. So, Proposition4.1.7yields:

mindepth(S/J(C6)(k)) : k∈ N≥1= 6−3= 3,

where J(C6) is the cover ideal:

J(C6) = (x1,x2)∩ (x2,x3)∩ (x3,x4)∩ (x4,x5)∩ (x5,x6)∩ (x6,x1).

Notice that S/J(C6) is a4-dimensionalk-algebra which is not Cohen-Macaulay. To seethis it suffices to consider the localization of S/J(C6) at the residue class of the primeideal (x1,x2,x4,x5) ⊆ S= k[x1, . . . ,x6]. The resultingk-algebra is a2-dimensionallocal ring not1-connected. Therefore, PropositionB.1.2implies that it is not Cohen-Macaulay. A fortiori, the original ring S/J(C6) is not Cohen-Macaulay too. So we getdepth(S/J(C6))< dimS/J(C6) = 4, which impliesdepth(S/J(C6)) = 3 (from what saidjust below the picture). Moreover, it follows from a generalfact (see Herzog, Takayamaand Terai [60, Theorem 2.6]) thatdepth(S/J(C6))≥ depth(S/J(C6)

(k)) for all integersk≥ 1. Thus, eventually, we have:

depth(S/J(C6)(k)) = 3< dim(S/J(C6)) ∀ k∈ N≥1

Example4.3.1suggests us the following question:Which are the simplicial com-

plexes∆ on [n] such thatdepth(S/I (k)∆ ) is constant with k varying among the positive in-tegers?Equivalently, which are the simplicial complexes∆ such that depth(S/J(∆)) =dimA(∆)? Theorem4.1.1shows that matroids are among these simplicial complexes,however Example4.3.1guarantees that there are others.

Appendix A

Other cohomology theories

Besides local cohomology, in mathematics many other cohomologies are available. Inthis section we want to introduce some cohomology theories we will use in Chapter1,underlying the relationships between them.

A.1 Sheaf cohomology

In algebraic geometry the most used cohomology issheaf cohomology. It can be de-fined once given a topological spaceX and a sheafF of abelian groups onX. WhenXis an open subset of an affine or of a projective scheme andF is a quasi-coherent sheaf,then sheaf cohomology and local cohomology are strictly related, as we are going toshow.

Actually to define sheaf cohomology it is not necessary to have a topological space,but just to have a special categoryC: For a topological spaceX such a category issimply Op(X) (see ExampleA.1.1below). The definition in this more general contextdoes not involve much more effort and is useful since this thesis makes use of etalecohomology, so we decided to present sheaf cohomology in such a generality.Etalecohomology was introduced by Grothendieck in [49].

Let C be a category. For each objectU of C suppose to have a distinguished setof family of morphisms(Ui → U)i∈I , calledcoveringsof U , satisfying the followingconditions:

(i) For any covering(Ui →U)i∈I and any morphismV →U the fiber productsUi ×U

V exist, and(Ui ×U V)i∈I is a covering ofV.(ii) If (Ui → U)i∈I is a covering ofU and(Ui j → Ui) j∈Ji is a covering ofUi, then

(Ui j →U)(i, j)∈I×Jiis a covering ofU .

(iii) For each objectU of C the family(UidU−−→U) consisting of one morphism is a

covering ofU .The system of coverings is calledGrothendieck topology, and the categoryC togetherwith it is calledsite.

Example A.1.1. Given a topological space X, we can consider the categoryOp(X)in which the objects are the open subsets of X and whose morphisms are the inclusionmaps: SoHom(V,U) is non-empty if and only if V⊆ U. Moreover, in such a caseit consists in only one element. There is a natural Grothendieck topology onOp(X),

80 Other cohomology theories

namely the coverings of an object U are its open coverings. Clearly, if V and W areopen subsets of an open subset U, we have V×U W =V ∩W.

A presheaf (of Abelian groups)on a citeC is a contravariant functorF : C → Ab,whereAb denotes the category of the Abelian groups. IfF is a presheaf of Abelian

groups andVφ−→ U is a morphism inC, for anys∈ F (U) the elementF (φ)(s) ∈ V

is denoted bys|V , although this can be confusing because there may be more than onemorphism fromV to U . A presheafF of Abelian groups is asheafif for any objectUof C and any covering(Ui →U)i∈I , we have:

(i) For anys∈ F (U), if s|Ui = 0 for anyi ∈ I , thens= 0.(ii) If si ∈F (Ui) for anyi ∈ I are such thatsi |Ui×UU j = sj |Ui×UU j for all (i, j) ∈ I × I ,

then there existss∈ F (U) such thats|Ui = si for everyi ∈ I .A morphism of presheavesis a natural transformation between functors. Amorphismof sheavesis a morphism of presheaves.

The definition of Grothendieck topology gives rise to the study of various cohomol-ogy theories: (Zariski) cohomology, etale cohomology, flat cohomology etc. Actuallyall of these cohomologies are constructed in the same way: What changes is the site.

Example A.1.2. In this thesis we are interested just in two sites, both related to ascheme X.

(i) The Zariski site on X, denoted by XZar, is the site on X regarded as a topologicalspace as in ExampleA.1.1. Since this is the most natural structure of site on X,we will not denote it by XZar, but simply by X.

(ii) The etale site on X, denoted by Xet, is constructed as follows: The objects arethe etale morphisms U→ X. The arrows are the X-morphisms U→ V, and

the coverings are the families ofetale X-morphisms(Uiφi−→U)i∈I such that U=

∪φi(Ui).

The functor from the category of sheaves on a siteC to Ab, defined byΓ(U, ·) :F 7→ Γ(U,F ) := F (U), is left exact for every objectU of C, and the category ofsheaves onC has enough injective objects (see Artin [1, p. 33]). Therefore it is possibleto define the right derived functors ofΓ(U, ·), which are denoted byH i(U, ·). For aschemeX and a sheafF on X, the ith cohomology ofF is H i(X,F ). If F is asheaf onXet, the ith etale cohomology ofF is H i(X → X,F ), and it is denoted byH i(Xet,F ).

As we already anticipated, the cohomology of a quasi-coherent sheaf on an opensubset of a noetherian affine schemeX is related to local cohomology: LetX :=Spec(R), a ⊆ R an ideal andU := X \ V (a). Recall that ifF is a quasi-coherentsheaf onX thenF is the sheafication of theR-moduleM := Γ(X,F ). By combining[10, Theorem 20.3.11] and [10, Theorem 2.2.4], shown in the book of Brodmann andSharp, we have that there is an exact sequence

0→ Γa(M)→ M → H0(U,F )→ H1a(M)→ 0, (A.1)

and, for alli ∈ N≥1, isomorphisms

H i(U,F )∼= H i+1a (M). (A.2)

There is also a similar correspondence in the graded case: Let Rbe a graded ring andabe a graded ideal ofR. SetX := Proj(R) andU := X \V+(a). Let M be aZ-gradedR-module. In this case the homomorphism appearing in [10, Theorem 2.2.4] are graded.

A.1 Sheaf cohomology 81

So by [10, Theorem 20.3.15] we have an exact sequence of gradedR-modules

0→ Γa(M)→ M →⊕

d∈ZH0(U,M(d))→ H1

a(M)→ 0, (A.3)

and, for alli ∈N≥1, graded isomorphisms

⊕

d∈ZH i(U,M(d))∼= H i+1

a (M). (A.4)

(The notation · above means the graded sheafication, see Hartshorne’s book [56, p.116]. We used the same notation also for the “ordinary” sheafication, however it shouldalways be clear from the context which sheafication is meant). In particular, ifm :=R+,

⊕

d∈ZH i(X,M(d))∼= H i+1

m (M) ∀ i ∈N≥1.

Given a schemeX, thecohomological dimensionof X is

cd(X) := mini : H j(X,F ) = 0 for all quasi-coherent sheavesF and j > i.

Suppose thata is an ideal of a ringR which is not nilpotent. Then (A.1) and (A.2)imply

cd(R,a)−1= cd(Spec(R)\V (a)). (A.5)

Moreover, ifR is standard graded over a fieldk anda is a homogeneous ideal which isnot nilpotent, using (A.3) and (A.4) we also have

cd(R,a)−1= cd(Spec(R)\V (a)) = cd(Proj(R)\V+(a)). (A.6)

For a schemeX it is also available theetale cohomological dimension. To give itsdefinition we recall that a sheafF on Xet is torsion if for any etale morphismU → Xsuch thatU is quasi-compact,F (U → X) is a torsion Abelian group.

Example A.1.3. We can associate to any abelian group G the constant sheaf on atopological space X . It is denoted by G, and for any open subset U⊆ X it is definedas

G(U) := Gπ0(U),

whereπ0(U) is the number of connected components of U. If X is a scheme, wecanassociate to G the constant sheaf on Xet in the same way:

G(U → X) := Gπ0(U).

Clearly, if G is a finite group, then Gis a torsion sheaf on Xet.

We can define the etale cohomological dimension of a schemeX as:

ecd(X) := mini : H j(Xet,F ) = 0 for all torsion sheavesF and j > i.

If X is an-dimensional scheme of finite type over a separably closed field, then ecd(X)is bounded above by 2n (see Milne’s book [82, Chapter VI, Theorem 1.1]). If moreoverX is affine, then ecd(X)≤ n ([82, Chapter VI, Theorem 7.2]).

We have seen in (0.10) that the cohomological dimension provides a lower boundfor the arithmetical rank. An analog fact holds true also forthe etale cohomological


dimension. Assume thatX is a scheme and pick a closed subschemeY ⊆ X. SupposethatU := X \Y can be cover byk affine subsets ofX. The etale cohomological dimen-sion of these affine subsets is less than or equal ton for what said above. So, usingrepetitively the Mayer-Vietoris sequence ([82, Chapter III, Exercise 2.24]), it is easy toprove that

ecd(U)≤ n+ k−1 (A.7)

Notice that ifX := Spec(R) andY := V (a), wherea is an ideal of the ringR, thenX \Ycan be covered by ara(a) affine subsets ofR. The same thing happens in the gradedsetting, with ara(a) replaced by arah(a). Therefore (0.10) actually provides a lowerbound for the arithmetical rank!

We want to finish the section noticing the formula for etale cohomoligical dimen-sion analog to (A.6). Let R be standard graded over a separably closed fieldk anda ahomogeneous ideal ofR. Then, using a result of Lyubeznik [76, Proposition 10.1], onecan prove that:

ecd(Spec(R)\V (a)) = ecd(Proj(R)\V+(a))+1. (A.8)

A.2 GAGA

To compare the cohomology theories we are going to define we need to introducesome notions from a foundamental paper by Serre [95]. When we consider a quasi-projective scheme over the field of complex numbersC, two topologies are available:The euclideian one and the Zariski one. Serre’s work allows us to use results fromcomplex analysis for the algebraic study of such varieties.The name GAGA comesfrom the title of the paper, “Geometrie Algebrique et Geometrie Analytique”.

If X is a quasi-projective scheme overC, we can associate to it an analytic spaceXh. Roughly speaking, we have to consider an affine coveringUii∈I of X. Let usembed everyUi as a closed subspace of a suitableAN

C. The ideal defining eachUi isgenerated by a set of polynomials ofC[x1, . . . ,xN]. Because a polynomial is a holomor-phic function, such a set defines a closed analytic subspace of CN, which we denote byUh

i . We obtainXh gluing theUhi ’s together. Below follow some expected properties of

this construction we will use during the thesis:(i) (Pn)h ∼= P(Cn+1).

(ii) If Y is another quasi-projective scheme overC, thenXh×Yh ∼= (X×Y)h.Furthermore, to any sheafF of OX-modules we can associate functorially a sheafF h

of OXh-modules ([95, Definition 2]). The sheafF h is, in many cases, the one weexpect: For instance, denoting byΩX/C the sheaf of Kahler differentials ofX overC,the sheafΩh

X/C is nothing but than the sheaf of holomorphic 1-forms onXh, namely

ΩXh. More generally, settingΩpX/C := ΛpΩX/C, the sheaf(Ωp

X/C)h is Ωp

Xh, the sheaf of

holomorphicp-forms onXh. Another nice property is thatF h is coherent wheneverF is coherent ([95, Proposition 10 c)]). One of the main results of [95] is the following([95, Theoreme 1]):

Theorem A.2.1. Let X be a projective scheme overC andF be a coherent sheaf onX. For any i∈N there are functorial isomorphisms

H i(X,F )∼= H i(Xh,F h).

A.3 Singular homology and cohomology 83

Thanks to GAGA, we can borrow techniques from complex analysis when we studyalgebraic varieties over a field of characteristic 0. Often this is useful because eu-clideian topology is much finer than the Zariski one: For instance, ifX is irreducibleoverC, the cohomology groupsH i(X,C) vanish for anyi > 0. Instead,H i(Xh,C) ∼=H i

Sing(Xh,C) (see SubsectionA.3), which in general are nonzero. When the character-

istic of the base field is positive these methods are not available. However, a valuableanalog of euclideian topology is the etale site. The following comparison theorem ofGrothendieck (see Milne’s notes [83, Theorem 21.1]) confirms the effectiveness of theetale site.

Theorem A.2.2. Let X be an affine or a projective scheme smooth overC and let G bea finite abelian group. For any i∈ N there are isomorphisms

H i(Xet,G)∼= H i(Xh,G).

A.3 Singular homology and cohomology

If X is a topological space andG an abelian group we will denote byHSingi (X,G)

thei-th singular homology group of X with coefficients in G(for instance see Hatcher’sbook [58, p. 153]). WhenG=Z, we just writeHSing

i (X) for HSingi (X,Z). We recall that

HSingi (X,G) andHSing

i (X) are related by the universal coefficient theorem for homology([58, Theorem 3A.3]):

Theorem A.3.1. If X is a topological space and G an abelian group, for any i∈ Nthere is a split exact sequence

0−→ HSingi (X)⊗G−→ HSing

i (X,G)−→ Tor1(HSingi−1 (X),G)−→ 0

The i-th singular cohomology group of X with coefficients in Gwill be denotedby H i

Sing(X,G) (see [58, p. 197]). Singular cohomology and singular homology arerelated by the universal coefficient theorem for cohomology([58, Theorem 3.2]):

Theorem A.3.2. If X is a topological space and G an abelian group, for any i∈ Nthere is a split exact sequence

0−→ Ext1(HSingi−1 (X),G)−→ H i

Sing(X,G)−→ Hom(HSingi (X),G)−→ 0

Example A.3.3. Let us consider the n-dimensional projective space over thecomplexnumbers,P(Cn+1), supplied with the euclideian topology. Then it is possibleto com-pute its singular homology groups:

HSingi (P(Cn+1))∼=

Z if i = 0,2,4, . . . ,2n

0 otherwise

Using TheoremsA.3.1andA.3.2, sinceC is a free abelian group, we get

HSingi (P(Cn+1),C)∼= HSing

i (P(Cn+1))⊗C∼=C if i = 0,2,4, . . . ,2n

0 otherwise

and

H iSing(P(C

n+1),C)∼= Hom(H iSing(P(C

n+1)),C)∼=C if i = 0,2,4, . . . ,2n

0 otherwise


Suppose thatX is an analytic space and thatG= C. Then, for anyi ∈ N, we havea functorial isomorphism

H iSing(X,C)∼= H i(X,C), (A.9)

for instance see the notes of Deligne [31, Proposition 1.1].

A.4 Algebraic De Rham cohomology

For any regular projective schemeX over a fieldk of characteristic 0, and actuallyfor much more generals schemes, it is possible to define itsalgebraic De Rham coho-mology groups, which we will denote byH i

DR(X) as shown by Grothendieck in [48].Consider the complex of sheaves

Ω•X/k : OX −→ ΩX/k −→ Ω2

X/k −→ . . . .

The De Rham cohomology is defined to be the hypercohomologyofthe complexΩ•X/k.

In symbolsH iDR(X) := Hi(X,Ω•

X/k). The De Rham cohomology theory can also bedeveloped in the singular case, see Hartshorne [55], but we do not need it in this thesis.

If k= C, we can consider the complex

Ω•Xh : OXh −→ ΩXh −→ Ω2

Xh −→ . . . .

A theorem of Grothendieck [48, Theorem 1’] tells us that, under the above hypothesis,

H iDR(X)∼=Hi(Xh,ΩXh). (A.10)

The complex form of Poincare’s lemma shows thatΩ•Xh is a resolution of the constant

sheafC. ThereforeHi(Xh,ΩXh) is nothing but thanH i(Xh,C). So (A.10) and (A.9)yield functorial isomorphisms

H iDR(X)∼= H i

Sing(Xh,C) (A.11)

for all i ∈N.

Appendix B

Connectedness in noetheriantopological spaces

In this appendix, we discuss a notion which generalizes thatof connectedness onNoetherian topological spaces. As example of Noetherian topological spaces, we sug-gest to keep in mind schemes.

Remark B.0.1. Let X be an affine or a projective scheme smooth over the complexnumbers. ThenXh is endowed with a finer topology than the Zariski one. Therefore,if Xh is connected, thenX is connected as well. What about the converse implication?Quite surprisingly,Xh is connected if and only ifX is connected. This can be shownusing TheoremA.2.2. In fact, if G is a finite abelian group, then

Gπ0(X) ∼= H0(Xet,G)∼= H0(Xh,G)∼= Gπ0(Xh),

whereπ0 counts the connected components of a topological space. Therefore, to figurea connected projective scheme, one can trust the Euclideianperception. However, werecommend carefulness about the fact that this occurrence fails as soon asX is not anaffine or a projective scheme. For instance, ifX is the affine line without a point, thenit is connected, butXh is obviously not.

For simplicity, from now on it will always be implied that ournoetherian topologi-cal spaces have finite dimension.

Definition B.0.2. A noetherian topological spaceT is said to ber-connectedif thefollowing holds: if Z is a closed subset ofT such thatT \ Z is disconnected, thendimZ≥ r. (We use the convention that the emptyset is disconnected ofdimension−1.)

Remark B.0.3. Let us list some easy facts about DefinitionB.0.2:(i) If T is r-connected, thenr ≤ dimT. MoreoverT is always(−1)-connected.(ii) If T is r-connected, then it iss-connected for anys≤ r.(iii) T is connected if and only if it is 0-connected.(iv) T is irreducible if and only if it is(dimT)-connected.

If, for a positive integerd, T is (dim(T)−d)-connected we say thatT is connectedin codimension d. If R is a ring, we say thatR is r-connected if Spec(R) is.

Example B.0.4. BesidesSpec(R) we will consider other two noetherian topologicalspaces related to special noetherian rings R.

86 Connectedness in noetherian topological spaces

(i) If (R,m) is a local ring the punctured spectrum of R isSpec(R) \ m, wherethe topology is induced by the Zariski topology onSpec(R). Notice that thepunctured spectrum of R is r-connected if and only if R is(r +1)-connected (alocal ring is obviously always0-connected).

(ii) If R is graded, we can consider the noetherian topological spaceProj(R). Onecan easily show that if R+ is a prime ideal, thenProj(R) is r-connected if andonly if the punctured spectrum of RR+ is r-connected if and only if R is(r +1)-connected.

We list three useful lemmas which allow us to interpret in different ways the con-cept introduced in DefinitionB.0.2.

Lemma B.0.5. Let T be a nonempty noetherian r-connected topological space. LetT1, . . . ,Tm be the irreducible components of T :

(i) If A and B are disjoint nonempty subsets of1, . . . ,m such that A∪B= 1, . . . ,m,then

dim((⋃

i∈A

Ti)∩ (⋃

j∈B

Tj))≥ r. (B.1)

Moreover, if T is not(r +1)-connected, then it is possible to choose A and B insuch a way that equality holds in(B.1).

(ii) If T is not (r +1)-connected, then the dimension of each Ti is at least r. More-over, if T has is reducible, thendimTi > r for any i = 1, . . . ,m (we recall ourassumptiondim(T)< ∞).

Proof. For (i) see Brodmann and Sharp [10, Lemma 19.1.15], for (ii) look at [10,Lemma 19.2.2]).

Lemma B.0.6. A connected ring R with more than one minimal prime ideals is notr-connected if and only if there exist two idealsa andb such that:

(i)√a and

√b are incomparable.

(ii) a∩b is nilpotent.(iii) dimR/(a+b)< r.

Proof. SetT := Spec(R).If a andb are ideals ofR satisfying (i), (ii) and (iii), then considerZ := V (a+b).

It is clear thatV (a)∩ (T \Z) andV (b)∩ (T \Z) provide a disconnection forT \Z.Since dimZ < r we have thatR is notr-connected.

If R is not r-connected, then there exists a closed subsetZ ⊆ T such thatT \Z isdisconnected and dimZ = s< r. We can assume thatR is s-connected. BecauseT \Zis disconnected, there are two closed subsetsA andB of T \Z such that:

(i) A andB are nonempty.(ii) A∪B= T \Z.(iii) A∩B= /0.

Let us choose two idealsa andb of R such thatA= V (a)\Z andB= V (b)\Z. Thefact thatA∩B= /0 impliesV (a+b)⊆ Z. Therefore dimR/(a+b)≤ s< r. Moreover,sinceA∪B= T \Z, we have thatV (a∩b)⊇ T \Z. If ℘ is a minimal prime ideal ofR, by LemmaB.0.5 (ii) dim R/℘> s. Then℘ /∈ Z, so it belongs toV (a∩ b). Thisimplies thata∩b is nilpotent. Finally, sinceA andB are nonempty, the radicals ofaandb cannot be comparable: For instance, if

√a were contained in

√b, we would get

V (a)⊇ V (b). SinceA∩B= /0, we would deduceV (b)⊆ Z, but this would contradictthe fact thatB is non-empty.

87

Lemma B.0.7. For a noetherian topological space T, the following are equivalent:(i) T is r-connected;(ii) For each T′ and T′′ irreducible components of T , there exists a sequence T′ =

Ti0,Ti1, . . . ,Tis = T ′′ such that Ti is an irreducible component of T for all i=0, . . . ,s anddim(Ti j ∩Ti j−1)≥ r for all j = 1, . . . ,s.

A sequence like in (ii) will be referred as an r-connected sequence.

Proof. Let T1, . . . ,Tm be the irreducible components ofT.(ii) =⇒ (i). Suppose thatT is notr-connected. Pick two disjoint nonempty subsets

A,B ⊆ 1, . . . ,m such thatA∪B = 1, . . . ,m, and choosep ∈ A and q ∈ B. SetT ′ := Tp andT ′′ := Tq. By the hypothesis there is anr-connected sequence betweenT ′

andT ′′, namely:T ′ = Ti0,Ti1, . . . ,Tis = T ′′.

Of course there existsk∈ 1, . . . ,s such thatTik−1 ∈ A andTik ∈ B. Therefore,

dim((⋃

i∈A

Ti)∩ (⋃

j∈B

Tj))≥ dim(Tik−1 ∩Tik)≥ r.

This contradicts LemmaB.0.5(i).(i) =⇒ (ii). SetT ′ = Tp for somep∈ 1, . . . ,m. By LemmaB.0.5(i),

dim(T ′∩ (⋃

i 6=p

Ti))≥ r.

Therefore there exists an irreducible closed subsetS1 ⊆ T ′ ∩ (⋃

i 6=pTi) of dimensionbigger than or equal tor. Being S1 irreducible, there existsj1 6= p such thatS1 ⊆T ′∩Tj1. So

dim(T ′∩Tj1)≥ r.

Set A1 := p, j1 and B1 := 1, . . . ,m \ p, j1. Arguing as above, LemmaB.0.5implies that there isj2 ∈ B1 such that:

dim(T ′∩Tj2)≥ r or dim(Tj1 ∩Tj2)≥ r.

Going on this way, we can show that the graph whose vertices are theTi ’s and such thatTi ,Tj is an edge if and only if dim(Ti ∩Tj) ≥ r is a connected graph. This is in turnequivalent to (ii).

As we saw in Equation (0.4), the cohomological dimension does not change undercompletion. This is not the case for the connectedness. Since in Chapter1 we comparedthe cohomological dimension cd(R,a) with the connectedness ofR/a, the followinglemma is necessary.

Lemma B.0.8. Let R be a local ring. The following hold:(i) If R is r-connected, then R is r-connected as well.(ii) if ℘R∈ Spec(R) for all minimal prime ideals℘ of R, thenR is r-connected if

and only if R is r-connected.

Proof. The reader can find the proof in [10, Lemma 19.3.1].

Because LemmaB.0.8 (ii), it is interesting to know local rings whose minimalprime ideals extend to prime ideals of the completion. The following Lemma providesa good class of such rings. Even if the proof is standard, we include it here for theconvenience of the reader.


Lemma B.0.9. Let R be a graded ring andm := R+ denote the irrelevant ideal of R.If ℘ is a graded prime of R, then℘Rm ∈ Spec(Rm). In particular, if R is a domain,Rm

is a domain as well.

Proof. We prove first that ifR is a domain thenRm is a domain as well. Considerthe multiplicative filtrationF := (Im)m∈N of ideals ofR, whereIm are defined asIm =( f ∈Rj : j ≥ m). Let us consider the associated graded ring associated toF , namelygrF (R) = ⊕∞

m=0Im/Im+1. Obviously there is a graded isomorphism ofR0-algebras

betweenR and grF (R). Now, let RF denote the completion ofR with respect to the

filtration F , and letG be the filtration(ImRF )m∈N. It is well known that

grF (R)∼= grG (RF ).

By these considerations we can assert that grG (RF ) is a domain; since∩m∈NImRF = 0,

RF is a domain as well. Since the inverse families of ideals(I j) j∈N and(m j) j∈N are

cofinal,Rm is a domain. For the more general claim of the lemma we have only to notethat, if℘ is a graded prime ofR, thenR/℘ is an graded domain and use the previouspart of the proof.

Remark B.0.10. Thanks to LemmaB.0.9, we can apply LemmaB.0.8(ii) to (localiza-tions of) graded ringsR. In fact, every minimal prime ideals ofR is graded (see Brunsand Herzog [13, Lemma 1.5.6]).

B.1 Depth and connectedness

A result of Hartshorne in [50] (see also Eisenbud’s book [34, Theorem 18.12]), assertsthat a Cohen-Macaulay ring is connected in codimension 1. Infact there is a relation-ship between depth and connectedness. To explain it we need the following lemma.

Lemma B.1.1. Let (R,m) be local. Then

dimR/a≥ depth(R)−grade(a,R).

Proof. Setk := depth(R) andg := grade(a,R). Let f1, . . . , fg ∈ a be anR-sequence. IfJ := ( f1, . . . , fg) we must have

a⊆⋃

℘∈Ass(R/J)

℘,

so there exists℘∈ Ass(R/J) such thata ⊆℘. Since depth(R/J) = k−g, moreover,we have dimR/℘≥ k−g (for instance see Matsumura [80, Theorem 17.2]).

The relation between connectedness and depth is expressed by the following tworesults.

Proposition B.1.2. If R is a local ring such thatdepth(R)= r+1, then it is r-connected.

Proof. If R has only one minimal prime, then the proposition is obvious,therefore wecan suppose thatR has at least two minimal prime ideals. IfR were notr-connected,by LemmaB.0.6there would exist two idealsa andb whose radicals are incomparable,such thata∩b is nilpotent and such that dimR/(a+ b) < r. The first two conditionstogether with ([34, Theorem 18.12]) imply grade(a+ b,R) ≤ 1. Then, from LemmaB.1.1, we would have dimR/(a+b)≥ r, which is a contradiction.

B.1 Depth and connectedness 89

Corollary B.1.3. Let R be a catenary local ring satisfying S2 Serre’s condition. ThenR is connected in codimension 1.

Proof. If Rhas only one minimal prime, then it is clearly connected in codimension 1,therefore we can assume thatR has at least two minimal prime ideals. IfR were notconnected in codimension 1, by LemmaB.0.6there would exist idealsa andb whoseradicals are incomparable, such thata∩b is nilpotent and such that dimR/(a+ b) <dimR−1. Let us localize at a minimal prime℘of a+b: SinceR is catenary ht(℘)≥ 2.It follows by the assumption that depth(R℘)≥ 2. ButV (aR℘) andV (bR℘) provide adisconnection for the punctured spectrum ofR℘. ThereforeR℘ is not 1-connected, andthis contradicts PropositionB.1.2.

Appendix C

Grobner deformations

There are a lot of references concerning Grobner deformations. We decided to follow,for our treatment, especially the lecture notes by Bruns andConca [20].

C.1 Initial objects with respect to monomial orders

Let S:= k[x1, . . . ,xn] be the polynomial ring inn variables over a fieldk. A monomialof S is an element ofS of the formxα := xα1

1 · · ·xαnn with α ∈ Nn. We will denote by

M (S) the set of all monomials ofS. A total order≺ on M (S) is called amonomialorder if:

(i) For all m∈ M (S)\ 1, 1≺ m.(ii) For all m1,m2,n∈ M (S), if m1 ≺ m2, thenm1n≺ m2n.

Given a monomial order≺, a nonzero polynomialf ∈ Shas a unique representation:

f = λ1m1+ . . .+λkmk,

whereλi ∈ k \ 0 for all i = 1, . . . ,k andm1 ≻ m2 ≻ . . . ≻ mk. The initial monomialwith respect to≺ of f , denoted by in≺( f ), is, by definition,m1. Furthermore, ifV ⊆ Sis a nonzerok-vector space, then we will call thespace of initial monomialsof V thefollowing k-vector space:

in≺(V) :=< in≺( f ) : f ∈V \ 0>⊆ S.

Remark C.1.1. Actually, we will usually deal with initial objects of more sophisticatedstructures than vector spaces, namely algebras and ideals.The justifications of thefollowing statements can be found in [20, Remark/Definition 1.5].

(i) If A is ak-subalgebra ofS, then in≺(A) is ak-subalgebra ofSas well, and it iscalled theinitial algebra of A with respect to≺. However, even ifA is finitelygenerated, in≺(A) might be not (see [20, Example 1.7]).

(ii) If A is a k-subalgebra ofS and I is an ideal ofA, then in≺(I) is an ideal ofin≺(A), and it is called theinitial ideal of I with respect to≺. In particular, sincein≺(S) = S, we have that in≺(I) is an ideal ofSwheneverI is an ideal ofS.

A subsetG of ak-subalgebraA⊆ S is calledSagbi baseswith respect to≺ if

in≺(A) = k[in≺(g) : g∈ G ]⊆ S.

92 Grobner deformations

As it is easy to see, a Sagbi basis ofA must generateA as ak-algebra. Of course, aSagbi basis ofA always exists, but, unfortunately, it might do not exist finite: This isthe case if and only if in≺(A) is not finitely generated. Analogously, a subsetG of anidealI of ak-subalgebraA⊆ S is said aGrobner basiswith respect to≺ of I if

in≺(I) = (in≺(g) : g∈ G )⊆ in≺(A).

Also in this case, one can show that a Grobner basis ofI must generateI as an idealof A. Moreover, one can show that if in≺(A) is finitely generated, then Noetherianityimplies the existence of a finite Grobner basis of any idealI ⊆ A. In particular, anyideal of the polynomial ringS admits a finite Grobner basis. From now on, we willomit the subindex≺ when it is clear which is the monomial order, writing in(·) inplace of in≺(·).

C.2 Initial objects with respect to weights

In this section we introduce the notion of initial objects with respect to weights. Avectorω := (ω1, . . . ,ωn) ∈ Nn

≥1 supplies an alternative graded structure onS, namelythe one induced by putting degω(xi) := ωi . Therefore, the degree with respect toω of amonomialxα = xα1

1 · · ·xαnn ∈M (S)\1 will be ω1α1+ . . .+ωnαn ≥ 1. For a nonzero

polynomial f ∈ S, we call theinitial form with respect toω of f the part of maximumdegree off . I.e., if f = λ1m1 + . . .+ λkmk with λi ∈ k \ 0 andmi ∈ M (S) for alli = 1, . . . ,k, then

inω( f ) = λ j1mj1 + . . .+λ jhmjh,

where degω (mj1) = . . .= degω (mjh) = degω ( f ) := maxdegω (mi) : i = 1, . . . ,k. Theω-homogenization off is the polynomial homω( f ) of the polynomial ringS[t] withone more variable, defined as

homω ( f ) :=k

∑i=1

λimitdegω ( f )−degω (mi) ∈ S[t].

Notice that homω ( f ) is homogeneous with respect to theω-graduationon S[t], givenby setting degω (xi) := ωi and deg(t) := 1. Moreover, note that we have that

inω( f )(x1, . . . ,xn) = homω ( f )(x1, . . . ,xn,0).

Analogously to SectionC.1, for a nonzerok-vector spaceV ⊆S, we define thek-vectorspace

inω (V) :=< inω( f ) : f ∈V \ 0>⊆ S

and thek[t]-module

homω(V) := k[t]< homω( f ) : f ∈V \ 0>⊆ S[t].

Remark C.2.1. The present remark is parallel toC.1.1.(i) If A is ak-subalgebra ofS, then inω (A) is ak-subalgebra ofS, called theinitial

algebra of A with respect toω , and homω(A) is ak-subalgebra ofS[t]. However,even ifA is finitely generated, both inω(A) and homω (A) might be not.

(ii) If A is a k-subalgebra ofS and I is an ideal ofA, then inω(I) is an ideal ofinω(A), called theinitial ideal of I with respect toω , and homω(I) is an idealof homω(A). In particular, since inω (S) = S and homω(S) = S[t], we have thatinω(I) is an ideal ofSand homω (I) is an ideal ofS[t] wheneverI is an ideal ofS.

C.3 Some properties of the homogenization 93

The two application inω and homω are related, roughly speaking, by the fact thathomω supplies a flat family whose fiber at 0 is inω . More precisely:

Proposition C.2.2.Let I⊆S be an ideal andω a vector inNn≥1. The ring S[t]/homω (I)

is a freek[t]-module. In particular, t−λ is a nonzero divisor of S[t]/homω(I) for anyλ ∈ k. Furthermore, we have:

S[t]/(homω (I)+ (t−λ ))∼=

S/I if λ 6= 0

S/ inω(I) if λ = 0.

PropositionC.2.2, whose proof can be found in [20, Proposition 2.4], is the maintool to pass to the initial ideal retaining properties from the original ideal, or viceversa.Moreover, the next result says that initial objects with respect to monomial orders area particular case of those with respect to weights. Thus, PropositionC.2.2is availablealso in the context of SectionC.1.

Theorem C.2.3. Let A be ak-subalgebra of S and Ii ideals of A for i= 1, . . . ,k. If ≺is a monomial order such thatin≺(A) is finitely generated, then there exists a vectorω ∈Nn

≥1 such thatin≺(A) = inω(A) and in≺(Ii) = inω (Ii) for all i = 1, . . . ,k.

Proof. See [20, Proposition 3.8].

If ≺ andω are like in TheoremC.2.3, then we say thatω represents≺ for A andIi .

C.3 Some properties of the homogenization

In this section we discuss some properties which will be useful in Chapter2. First ofall we need to introduce the following operation of dehomogenization:

π : S[t] −→ SF(x1, . . . ,xn, t) 7→ F(x1, . . . ,xn,1)

Remark C.3.1. Let ω ∈ Nn≥1. Then the following properties are easy to verify:

(i) For all f ∈ Swe haveπ(homω( f )) = f .(ii) Let F ∈ S[t] be an homogeneous polynomial (with respect to theω-graduation)

such thatF /∈ (t). Then homω(π(F)) = F; moreover, for allk ∈ N, if G= tkFwe have homω(π(G))tk = G.

(iii) If F ∈ homω(I), thenπ(F) ∈ I .

In the next lemma we collect some easy and well known facts:

Lemma C.3.2. Let ω ∈ Nn≥1 and I and J two ideals of S. Then:

(i) homω(I ∩J) = homω (I)∩homω (J).(ii) I is prime if and only ifhomω(I) is prime.(iii) homω(

√I) =

√homω(I).

(iv) I ⊆ J if and only ifhomω (I)⊆ homω(J).(v) ℘1, . . . ,℘s are the minimal primes of I if and only ifhomω(℘1), . . . ,homω(℘s)

are the minimal primes ofhomω (I);(vi) dimS/I +1= dimS[t]/homω(I).

Proof. For (i), (ii) and (iii) see the book of Kreuzer and Robbiano [69, Proposition4.3.10] (for (ii) see also the lecture notes of Huneke and Taylor [65, Lemma 7.3, (1)]).

(iv). This follows easily from RemarkC.3.1.

94 Grobner deformations

(v). If ℘1, . . . ,℘s are the minimal primes ofI , then⋂s

i=1℘i =√

I . So (i), (iii) and(iv) imply

s⋂

i=1

homω(℘i) =√

homω(I).

Then (ii) implies that all minimal primes of homω(I) are contained in the set

homω (℘1), . . . ,homω(℘s).

Moreover, by (iv), all the primes in this set are minimal for homω (I). Conversely, ifhomω (℘1), . . . ,homω(℘s) are the minimal primes of homω(I), then

⋂si=1homω(℘i) =√

homω(I). So, from (i), (iii) and (iv), it follows that⋂s

i=1℘i =√

I . Therefore, (ii)yields that all the minimal primes ofI are contained in the set℘1, . . . ,℘s. Againusing (iv), the primes in this set are actually all minimal for I .

(vi). If ℘0 ℘1 . . . ℘d is a strictly increasing chain of prime ideals suchthat I ⊆℘0, then (ii) and (iv) get that homω(℘0) homω(℘1) . . . homω(℘d) (x1, . . . ,xn, t) is a strictly increasing chain of prime ideals containing homω (I). The lastinclusion holds true because(x1, . . . ,xn, t) is the unique maximal ideal ofS[t] which isω-homogeneous. Furthermore it is a strict inclusion becauseobviouslyt /∈ homω(H)for any idealH ⊆ S. So, dimS[t]/homω(I) ≥ dimS/I +1. Similarly, ht(homω (I)) ≥ht(I), thus we conclude.

Appendix D

Some facts of representationtheory

In this appendix we want to recall some facts of Representation Theory we used inChapter3.

D.1 General facts on representations of group

Let k be a field,E a k-vector space (possibly not finite dimensional) andG a group.By GL(E) we denote the group of allk-automorphisms ofE where the multiplicationis φ ·ψ = ψ φ . A representationof G onV is a homomorphism of groups

ρ : G−→ GL(E).

We will say that(E,ρ) is a G-representation of dimension dimkE. When it is clearwhat isρ we will omit it, writing just “E is aG-representation”. If it is clear also whatis G, we will call E simply a representation. Moreover, we will often write justgv forρ(g)(v) (g∈ G andv∈ E). A map between twoG-representations(E,ρ) and(E′,ρ ′)is a homomorphism ofk-vector spaces, sayφ : E → E′, such that

φ ρ(g) = ρ ′(g)φ ∀ g∈ G.

We will often call φ a G-equivariant map. It is straightforward to check that ifφ isbijective thanφ−1 is G-equivariant. In such a case, therefore, we will say thatV andW are isomorphicG-representations. Asubrepresentationof a representationV is ak-subspaceW ⊆V which is invariant under the action ofG. A representationV is calledirreducible if its subrepresentations are just itself and< 0 >. Equivariant maps andirreducible representations are the ingredients of the, soeasy to prove as fundamental,Schur’s Lemma.

Lemma D.1.1. (Schur’s Lemma) Let V be an irreducible G-representation and V′ bea G-representation. If

φ : V −→V ′

is a G-equivariant map, then it is either zero or injective. If also V′ is irreducible, andφ is not the zero map, then V and V′ are isomorphic G-representations.

96 Some facts of representation theory

Proof. We get the statement because both Ker(φ) and Im(φ) areG-representations.

Given twoG-representationsV andW, starting from them we can construct manyotherG-representations:

(i) Thek-vector spaceV⊕W inherits a natural structure ofG-representation setting

g(v+w) := gv+gw ∀ g∈ G, ∀ v∈V and∀ w∈W.

(ii) The tensor productV ⊗W (⊗ stands for⊗k) becomes aG-representation via

g(v⊗w) := gv⊗gw ∀ g∈ G, ∀ v∈V and∀ w∈W.

In particular, thedth tensor product

d⊗V :=V ⊗V⊗ . . .⊗V︸︷︷︸

d times

becomes aG-representation in a natural way.(iv) If char(k) = 0, the exterior power

∧dV and the symmetric power SymdV canboth be realized ask-subspaces of

⊗dV. As it is easy to check, it turns outthat actually they areG-subrepresentations of

⊗dV. Particularly⊗dV is not

irreducible provided thatd ≥ 2.(iv) The dual spaceV∗ = Hom(V,k) has a privileged structure ofG-representation

too:

(gv∗)(v) := v∗(g−1v) ∀ g∈ G, ∀ v∈V and∀ v∗ ∈V∗.

This definition comes from the fact that, this way, we have(gv∗)(gv) = v∗(v).A representationV is saiddecomposableif there exist two nonzero subrepresentationsW andU of V such thatW⊕U = V. It is indecomposableif it is not decomposable.Obviously, an irreducible representation is indecomposable. Mashcke showed that thereverse implication holds true, provided that the groupG is finite and that char(k)does not divide the order ofG (for instance see the book of Fulton and Harris [43,Proposition 1.5]). Actually, it is true the following more general fact:

Theorem D.1.2. The following are equivalent:(i) Every indecomposable G-representations is irreducible.

(ii) G is finite andchar(k) does not divide|G|.

Proof. See the notes of Del Padrone [32, Theorem 3.1].

Thus, ifG is a finite group whose order is not a multiple of char(k), then any finitedimensionalG-representationV admits a decomposition

V =Va11 ⊕ . . .⊕Vak

k ,

where theVi ’s are distinct irreducible subrepresentations ofV. Moreover Schur’slemma [43, Lemma 1.7] ensures us that such a decomposition is unique.

D.2 Representation theory of the general linear group 97

D.2 Representation theory of the general linear group

From now on, in this appendix we assume char(k) = 0. LetV be ann-dimensionalk-vector space. We want to investigate on the representationsof thegeneral linear groupG= GL(V).

Remark D.2.1. After choosing ak-basis ofV, we can identify GL(V) with the groupof invertiblen×n invertible matrices with coefficient ink. We will often speak of thesetwo groups without distinctions. However, we will punctually remark those situationswhich depend on the choice of a basis.

Since GL(V) is infinite, TheoremD.1.2 implies that there are indecomposableGL(V)-representations which are not irreducible. Below is an example of such a rep-resentation.

Example D.2.2. Let k = R be the field of real numbers, G:= GL(R) and E be a2-dimensionalR-vector space. So, to supply E with a structure of G-representation weneed to give a homomorphismR∗ →GL(E), whereR∗ denotes the multiplicative groupof nonzero real numbers. Set

ρ :R∗ → GL(E)

a 7→(

1 log|a|0 1

).

Sinceρ(a)(x,y) = (x+ log|a|y,y), it is clear that the subspace

F = (x,0) : x∈ R ⊆ E

is a subrepresentation of E. So E is not irreducible. However, one can easily checkthat F is the only G-invariant1-dimensional subspace of E, so E is indecomposable.

To avoid an inconvenient like that in ExampleD.2.2, we introduce a particularkind of GL(V)-representations. AnN-dimensional GL(V)-representationE is calledrational if in the homomorphism

kn2 ⊇ GL(V)→ GL(E)⊆ kN2

each of theN2 coordinate function is a rational function in then2 variables. Analo-gously we define apolynomial representation. Notice that the representation of Exam-ple D.2.2was not rational. In fact it turns out that the analog of Mashcke’s theoremholds true for rational representations of GL(V) (for instance see Weyman [107, Theo-rem 2.2.10]).

Theorem D.2.3. Any indecomposable rational representation ofGL(V) is irreducible(recall thatchar(k) = 0).

Remark D.2.4. TheoremD.2.3does not hold in positive characteristic. For instance,let k be a field of characteristic 2,V a 2-dimensional vector space andE = Sym2(V)with the natural action. Letx,y a basis ofV. With respect to the basisx2,xy,y2 ofE, the action is

ρ : GL(V) → GL(E)(

a bc d

)7→

a2 0 b2

ac ad+bc bdc2 0 d2

.


So ρ is a polynomial representation. Particularly it is rational. Moreover, since thesubspaceF =< x2,y2 >⊆ E is invariant,E is not irreducible. However, it is easy tocheck that there are no invariant subspaces ofE but 0, F andE. ThereforeE isindecomposable.

A crucial concept in representation theory is that of weightvectors. Let us callH ⊆GL(V) the subgroup of diagonal matrix, and, for elementsx= x1, . . . ,xn ∈ k\0,let us denote by diag(x) ∈ H the diagonal matrix

diag(x) :=

x1 0 · · · 00 x2 · · · 0...

..... .

...0 0 · · · xn

.

If E is a rational GL(V)-representation, a nonzero elementv ∈ E is called aweightvectorof weightα = (α1, . . . ,αn) ∈ Zn if

diag(x)v= xαv, xα := xα11 · · ·xαn

n

for all diagonal matrices diag(x) ∈ H.

Remark D.2.5. Let E be a rational GL(V)-representation,B = e1, . . . ,en ak-basis ofV, σ a permutation of the symmetric groupSn andBσ thek-basiseσ(1), . . . ,eσ(n). Wecan think toE as a representation with respect toB or toBσ . However a weight vectorv∈ E of weight(α1, . . . ,αn) with respect toB will have weight(ασ(1), . . . ,ασ(n)) withrespect toBσ . This is not a big deal, but it is better to be aware that such situations canhappen. Sometimes it will also happen that we consider weights β with β ∈ Zp andp< n. In such cases we always meanβ with zeroe-entries added up ton. For instance,if β = (4,1,−1,−2,−3) andn= 7, for weightβ we mean(4,1,0,0,−1,−2,−3).

As one can show, it turns out thatE is the direct sum of itsweight spaces:

E =⊕

α∈Zn

Eα , Eα := v∈ E : diag(x)v= xαv ∀ diag(x) ∈ H.

Let B−(V) ⊆ GL(V) be the subgroup of the lower triangular matrices. A nonzeroelementv∈ E is called ahighest weight vectorif B−(V)v⊆< v>. It is straightforwardto check that a highest weight vector actually is a weight vector. (One can also definethe highest weight vector with respect to the subgroupB+(V)⊆GL(V) of all the uppertriangular matrices: the theory does not change). Highest weight vectors supply the keyto classify the irreducible rational representations of GL(V). In fact:

(i) A rational representation is irreducible⇐⇒ it has only one highest weightvector (up to multiplication by scalars).By what said in RemarkD.2.5, we can, and from now on we do, suppose that theonly highest weight of an irreducible rational representation is α = (α1, . . . ,αn) withα1 ≥ . . .≥ αn. After this observation, we can state a second crucial property of highestweight:

(ii) Two rational irreducible representations are isomorphic ⇐⇒ their highestweight vectors have the same weight.The third and last fact we want to let the reader know is:

(iii) Given α = (α1, . . . ,αn) ∈ Zn with α1 ≥ . . . ≥ αn, there exists an irreduciblerational representation whose highest weight vector has weight α.


D.2.1 Schur modules

In this subsection we are going to give an explicit way to construct all the irreduciblerational representations of GL(V). The plan is first to describe such representationswhenα ∈ Nn. In this case the corresponding representation is actuallypolynomial.To get the general case it will be enough to tensorize the polynomial representationsby a suitable negative power of the determinant representation. We will soon be moreprecise. Let us start introducing the notion of partition: Avectorλ = (λ1, . . . ,λk) ∈Nk

is a partition of a natural numberm, written λ ⊢ m, if λ1 ≥ λ2 ≥ . . . ≥ λk ≥ 1 andλ1+λ2+ . . .+λk = m. We will say that the partitionλ hask partsandheightht(λ ) =λ1. Sometimes will be convenient to group together the equal terms of a partition:For instance we will write(kd) for the partition(k,k, . . . ,k) ∈ Nd, or (33,22,14) for(3,3,3,2,2,1,1,1,1). From a representationE, we can build new representations forany partitionλ , namelyLλ E. If E =V is the representation with the obvious action ofGL(V) the obtained representationsLλV, calledSchur module, will be polynomial andirreducible. Thek-vector spaceLλV is obtained as a suitable quotient of

λ1∧V ⊗

λ2∧V ⊗ . . .⊗

λk∧V,

and the action of GL(V) is the one induced by the natural action onV. To see theprecise definition look at [107, Chapter 2]. We can immediately notice thatLλV = 0if ht(λ ) > n= dimkV. However in characteristic 0, that is our case, the above Schurmodule can be defined as a subrepresentation of

⊗mV: Since we think that for thisthesis is more useful the last interpretation, we decided togive the details for it.

We can feature a partitionλ as a(Young) diagram, that we will still denote byλ ,namely:

λ := (i, j) ∈N\ 0×N\ 0 : i ≤ k and j ≤ λi.

It is convenient to think at a diagram as a sequence of rows of boxes, for instance thediagram associated to the partitionλ = (6,5,5,3,1) features as

λ =

Obviously we can also recover the partitionλ from its diagram, that is why we willspeak indifferently about diagrams or partitions. Sometimes will be useful to usethe partial order on diagram given by inclusion. That is, given two partitionsγ =(γ1, . . . ,γh) andλ = (λ1, . . . ,λk), byγ ⊆ λ we meanh≤ k andγi ≤ λi for all i = 1, . . . ,h.We will denote by|λ | the number of boxes of a diagramλ , that is|λ | := λ1+ . . .+λk.Soλ ⊢ |λ |. Given a diagramλ , a (Young) tableuΛ of shapeλ on [r] := 1, . . . , r isa filling of the boxes ofλ by letters in the alphabet[r]. For instance the following is a


tableu of shape(6,5,5,3,1) on1, . . . , r, providedr ≥ 7 .

Λ =

3 5 4 3 2 7

2 1 7 6 4

2 2 3 1 2

5 6 7

1

Formally, a tableuΛ of shapeλ on [r] is a mapΛ : λ → [r]. Thecontentof Λ is thevectorc(Λ) = (c(Λ)1, . . . ,c(Λ)r) ∈ Nr such thatc(Λ)p := |(i, j) : Λ(i, j) = p|. Inorder to define the Schur modules we need to introduce the Young symmetrizers: Letλ be a partition ofm, andΛ be a tableu of shapeλ such thatc(Λ) = (1,1, . . . ,1) ∈Nm.Let Sm be the symmetric group onm elements, and let us define the following subsetsof it:

CΛ := σ ∈ Sm : σ preserves each column ofΛ,RΛ := τ ∈ Sm : τ preserves each row ofΛ.

The symmetric groupSm acts on⊗mV extending byk-linearity the rule

σ(v1⊗ . . .⊗ vm) := vσ(1)⊗ . . .⊗ vσ(m), σ ∈ Sm, vi ∈V.

With these notation, theYoung symmetrizeris the following map:

e(Λ) :m⊗

V →m⊗

Vv= v1⊗ . . .⊗ vm 7→ ∑σ∈CΛ ∑τ∈RΛ sgn(τ)στ(v)

.

One can check that the image ofe(Λ) is a GL(V)-subrepresenation of⊗mV. More-

over, up to GL(V)-isomorphism, it just depends from the partitionλ , and not from theparticular chosen tableu. Thus we define the Schur moduleLλV as

LλV := e(Λ)(m⊗

V).

To see that this definition coincides with the one given at thebeginning of this sub-section see [107, Lemma 2.2.13 (b)]. (Actually in [107] the definition of the Youngsymmetrizers is different from the one given here, and the statement of [107, Lemma2.2.13 (b)] is wrong: However, the argument of its proof is the correct one).

Example D.2.6. The Schur modules somehow fill the gap between exterior and sym-metric powers. This example should clarify what we mean.

(i) Let λ := (m) andΛ(1, j) := j for any j= 1, . . . ,m. In this caseCΛ = id[m] andRΛ = Sm. Therefore, if v= v1⊗ . . .⊗ vm ∈⊗mV, then

e(Λ)(v) = ∑τ∈Sm

sgn(τ)τ(v).

The image of such a map is exactly∧mV ⊆⊗mV. So L(m)V ∼=∧mV.

(ii) Let λ := (1m) ∈ Nm andΛ( j,1) := j for any j = 1, . . . ,m. This is the oppositecase of the above one, in fact we haveCΛ = Sm andRΛ = id[m]. Therefore, ifv= v1⊗ . . .⊗ vm ∈⊗mV, then

e(Λ)(v) = ∑σ∈Sm

σ(v).

The image of such a map is exactlySymmV ⊆⊗mV. So L(1m)V ∼= SymmV.


Before stating the following crucial theorem, let us recallthat thetranspose parti-tion of a partitionλ = (λ1, . . . ,λk) is the partition

tλ = (t λ1, . . . ,tλλ1

) wheretλi := | j : λ j ≥ i|.

Notice that|λ |= |tλ |, tλ has ht(λ ) parts,λ has ht(tλ ) parts andt(tλ ) = λ .

Theorem D.2.7. The Schur module LλV is an irreducible polynomial representationwith highest weighttλ . So all the irreducible polynomial representations are of thiskind.

From TheoremD.2.7 it is quite simple to get all the irreducible rational GL(V)-representations. Fork∈ Z, let us define thekth determinant representation Dk to be the1-dimensional representation GL(V)→ k∗ which to a matrixg∈ GL(V) associates thekth power of its determinant, namely det(g)k. It urns out thatD :=D1 is the determinantrepresentation

∧nV.

Remark D.2.8. In this remark we list some basic properties of the determinant repre-sentations.

(i) Clearly Dk is a rational representation for allk ∈ Z. Moreover it is polynomialprecisely whenk∈ N.

(ii) Being 1-dimensional,Dk is obviously irreducible. Furthermore it is straightfor-ward to check that its highest weight is(k,k, . . . ,k) ∈ Zn. Therefore, ifk ≥ 0TheoremD.2.7implies thatDk ∼= LλV whereλ = (n,n, . . . ,n) ∈ Zk.

(iii) If E is a irreducible rational representation with highest weight (α1, . . . ,αn), onecan easily verify thatE⊗Dk is a rational irreducible representation with highestweight(α1+ k, . . . ,αn+ k).

Eventually we are able to state the result which underlies the representation theoryof the general linear group.

Theorem D.2.9. A rational representation E is irreducible if and only if there exists apartition λ and k∈ Z such that E∼= LλV ⊗Dk. In particular:

(i) For each vectorα = (α1, . . . ,αn) ∈ Zn with α1 ≥ . . .≥ αn the unique irreduciblerational representation of weightα is given by LtλV⊗Dk whereλi = αi −k≥ 0for i = 1, . . . ,n.

(ii) Two irreducible rational representations LλV⊗Dk and LγV⊗Dh are isomorphicif and only iftλi + k= tγi +h for all i = 1, . . . ,n.

Remark D.2.10. If E is an irreducible rational representation with highest weightα = (α1, . . . ,αn), then it easy to check that its dual representationE∗ is an irre-ducible rational representation with highest weight(−αn, . . . ,−α1). In particular, ifλ = (λ1, . . . ,λk) is a partition, we denote byλ ∗ = (λ ∗

1 , . . . ,λ ∗k ) the partition such that

λ ∗i = n−λk−i+1. By TheoremD.2.9we have a GL(V)-isomorphism

(LλV)∗ ∼= Lλ ∗ ⊗D−k,

Moreover one can show that(LλV)∗ ∼= Lλ (V∗) (see the book of Procesi [90, Chapter

9, Section 7.1]). For this reason, from now on we will writeLλV∗ for (LλV)∗: Eachinterpretation of such a notation is correct!

We want to end this subsection describing ak-basis of the Schur modulesLλVin terms of the tableux of shapeλ . A tableuΛ is said to bestandardif its rows areincreasing (Λ(i, j) < Λ(i, j + 1)) and its columns are not decreasing (Λ(i, j) ≤ Λ(i +


1, j)). Among the standard tableux of fixed shapeλ one plays a crucial role: Thecanonical tableu cλ . For anyi, j such thatj ≤ λi, we havecλ (i, j) := j. For instancethe canonical tableu of shapeλ = (6,5,5,3,1) is:

Λ =

1 2 3 4 5 6

1 2 3 4 5

1 2 3 4 5

1 2 3

1

Notice that the content of the canonical tableucλ is the transpose partitiontλ of λ . Ifnot already guessed, the reason of the importance of the canonical tableux will soon beclear. Fixed ak-basise1, . . . ,en of V, it turns out that there is a 1-1 correspondencebetween standard tableux of shapeλ on [n] and ak-basis ofLλV. The correspondenceassociates toΛ the equivalence class of the element

(eΛ(1,1)∧ . . .∧eΛ(1,λ1))⊗ . . .⊗ (eΛ(k,1)∧ . . .∧eΛ(k,λk)).

By meaning of the Young symmetrizers, once fixed a tableuΓ of shapeλ on [|λ |] ofcontent(1,1, . . . ,1) ∈ Z|λ |, the correspondence is given by

Λ 7→ e(Γ)(eΛ(1,1)⊗ . . .⊗eΛ(1,λ1)⊗ . . .⊗eΛ(k,1)⊗ . . .⊗eΛ(k,λk)).

With respect to both the correspondences above, any tableuΛ is a weight vector ofweightc(Λ). Moreover the canonical tableucλ corresponds to the highest weight vec-tor of LλV.

D.2.2 Plethysms

What said up to now implies that, given a finite dimensional rationalGL(V)-representationE, there is a unique decomposition of it in irreducible representations, namely

E ∼=⊕

ht(λ)<nk∈Z

(LλV ⊗Dk)m(λ ,k). (D.1)

The numbersm(λ ,k) are themultiplicitiesof the irreducible representationLλV ⊗Dk

appears inE with. To find these numbers is a fascinating problem, still open evenfor some very natural representations. When the representation E is polynomial, thedecomposition, rather than as in (D.1), is usually written as

E ∼=⊕

ht(λ )≤n

LλVm(λ ) (D.2)

For instance, the decomposition of Symp(∧qV) is unknown in general, and to find

it falls in the so-called plethysm’s problems. Unfortunately, this fact caused someobstructions to our investigations in Chapter3. At the contrary, an help from represen-tation theory came from Pieri’s formula, a special case of the Littlewood-Richardsonrule.

D.3 Minors of a matrix and representations 103

Theorem D.2.11.(Pieri’s formula) Letλ =(λ1, . . . ,λk)⊢ r andλ ( j)= (λ1+ j,λ1,λ2,λ3, . . . ,λk).Then

LλV ⊗j∧

V ∼=⊕

µ⊢r+ j, ht(µ)≤nλ⊆µ⊆λ( j)

LµV

and

LλV∗⊗j∧

V∗ ∼=⊕

µ⊢r+ j, ht(µ)≤nλ⊆µ⊆λ( j)

LµV∗

Proof. For the proof of the first formula see [90, Chapter 9, Section 10.2]. The dualformula is straightforward to get from the first one.

D.3 Minors of a matrix and representations

Let k be a field of characteristic 0,m andn two positive integers such thatm≤ n and

X :=

x11 x12 · · · · · · x1n

x21 x22 · · · · · · x2n...

.... . .

. . ....

xm1 xm2 · · · · · · xmn

a m×n matrix of indeterminates overk. Moreover let

R := k[xi j : i = 1, . . . ,m, j = 1, . . . ,n]

be the polynomial ring inm· n variables overk. Let W andV bek-vector spaces ofdimensionm andn, and setG := GL(W)×GL(V). The rule

(A,B)∗X := A ·X ·B−1

induces an action of the groupG on R1: Namely (A,B) ∗ xi j = x′i j where x′i j de-notes the(i j )th entry of the matrix(A,B) ∗ X. Extending this actionR becomes aG-representation. Actually, each graded componentRd of R is a (finite dimensional)rationalG-representation.

In Chapter3 we dealt with the above action ofG = GL(W)×GL(V): Thereforewe need to introduce some notation aboutG-representations of this kind. LetF be aGL(W)-representation, andE be a GL(V)-representation. ThenT := F ⊗E becomesa G-representation. Furthermore, ifF andE are irreducible,T is irreducible as well.More generally, two potential decompositions in irreducible representations

F =⊕

i

Fi and E =⊕

j

E j ,

yield a decomposition in irreducibleG-representation ofF ⊗E, namely

T =⊕

i, j

Fi ⊗E j .

Let us assume thatF andE are both rational. Iff ∈ F is a weight vector of weightβ ∈ Zm ande∈ E is a weight vector of weightα ∈ Zn, then we say thatt := f ⊗e is


a bi-weight vectorof bi-weight(β |α). This is equivalent to say that, for any diag(y) ∈GL(W) and diag(x) ∈ GL(V), we have

(diag(y),diag(x)) · t = yβ xα t.

Let f and e be highest weight vectors of, respectively,F and E. For questions ofnotation we suppose that they are invariant, respectively,with respect toB−(W) and toB+(V). SettingB= B−(W)×B+(V), we have that

B · t ⊆< t > .

We callt anhighest bi-weight vectorof T.Let us consider the symmetric algebra

Sym(W⊗V∗) =⊕

d∈NSymd(W⊗V∗).

Let e1, . . . ,em be a basis ofW and f1, . . . , fn be a basis ofV. Denoting the dualbasis of f1, . . . , fn by f ∗1 , . . . , f ∗n, consider the isomorphism of gradedk-algebras

φ : R → Sym(W⊗V∗)xi j 7→ ei ⊗ f ∗j

.

As one can check,φ is G-equivariant. Thus, it is an isomorphism ofG-representations.Furthermore theG-representation Symd(W⊗V∗) can be decomposed in irreduciblerepresentations: There is an explicit formula for such a decomposition, known as theCauchy formula[90, Chapter 9, Section 7.1]:

Symd(W⊗V∗)∼=⊕

λ⊢dht(λ)≤m

LλW⊗LλV∗. (D.3)

SinceRd and Symd(W⊗V∗) are isomorphicG-representations, we will describe whichpolynomials belong to the isomorphic copy ofLλW⊗ LλV∗ in R, underlining whichone isU-invariant. We need a notation do denote the minors of the matrix X. Giventwo sequences 1≤ i1 < .. . < is ≤ m and 1≤ j1 < .. . < js ≤ n, we write

[i1, . . . , is| j1, . . . , js]

for the s-minor which insists on the rowsi1, . . . , is and the columnsj1, . . . , js of X.Since an element ofG takes ans-minor in a linear combination ofs-minors, thek-vector space spanned by thes-minors is a finite dimensionalG-representation. So itis plausible to expect a description of the irreducible representationsLλW⊗LλV∗ interms of minors. For any pair of standard tableuΛ andΓ of shapeλ = (λ1, . . . ,λk),respectively on[m] and on[n], we define the following polynomial ofR:

[Λ|Γ] := δ1 · · ·δk

whereδi := [Λ(i,1), . . . ,Λ(i,λi)|Γ(i,1), . . . ,Γ(i,λi)] ∀ i = 1, . . . ,k

We say that the product of minors[Λ|Γ] hasshapeλ . One can show that the product ofminors[cλ |cλ ] is a highest bi-weight vector of bi-weight((tλ1, . . . ,

tλm)|(−tλn, . . . ,−tλ1)),so there is an isomorphism ofG-representations

Mλ := G∗ [cλ |cλ ]∼= LλW⊗LλV∗.

D.3 Minors of a matrix and representations 105

Actually, it turns out that the set

[Λ|Γ] : Λ andΓ are tableu of shapeλ , respectively on[m] and on[n]

is ak-basis ofMλ (see the paper of DeConcini, Eisenbud and Procesi [28] or the bookof Bruns and Vetter [15, Section 11]).

In Chapter3 we were especially interested in the study of thek-subalgebra ofR

At = At(m,n)

generated by thet-minors ofX, wheret is a positive integer smaller than or equal tom.These algebras are known asalgebras of minors. It turns out that the algebra of minorsAt is aG-subrepresentation ofR. Therefore it is natural to ask about its decompositionin irreducibles. Equivalently, whichMλ are in At? The answer to this question isknown. Before describing it, we fix a definition.

Definition D.3.1. Given a partitionλ = (λ1, . . . ,λk) we say that it isadmissibleif tdivides|λ | andtk≤ |λ |. Furthermore we say thatλ is d-admissibleif |λ |= td.

Let us denote by[At ]d thek-vector space consisting in the elements ofAt of de-greetd in R. This way we are defining a new grading overAt such that thet-minorshave degree 1. The below result follows at once by the description of the powers ofdeterminantal ideals got in [28].

Theorem D.3.2. For each natural number d and for each1≤ t ≤ m we have

[At ]d =⊕

λ is d-admissibleht(λ)≤ m

Mλ ∼=⊕

λ is d-admissibleht(λ)≤ m

LλW⊗LλV∗.

Appendix E

Combinatorial commutativealgebra

Below we recall some basic facts concerning Stanley-Reisner rings. Standard refer-ences for this topic are Bruns and Herzog [13, Chapter 5], Stanley [100] or Miller andSturmfels [81]

Let k be a field,n a positive integer andS:= k[x1, . . . ,xn] the polynomial ring onnvariables overk. Moreover, let us denote bym := (x1, . . . ,xn) the maximal irrelevantideal ofS. We write [n] for 1, . . . ,n. By a simplicial complex∆ on [n] we mean acollection of subsets of[n] such that for anyF ∈ ∆, if G⊆ F thenG∈ ∆. An elementF ∈ ∆ is called afaceof ∆. The dimension of a faceF is dimF := |F | − 1 and thedimension of∆ is dim∆ := maxdimF : F ∈ ∆. The faces of∆ which are maximalunder inclusion are calledfacets. We denote the set of the facets of∆ by F (∆). Ob-viously a simplicial complex on[n] is univocally determined by its set of facets. Asimplicial complex∆ is pure if all its facets have the same dimension. It isstronglyconnectedif for any two facetsF andG there exists a sequenceF = F0,F1, . . . ,Fs = Gsuch thatFi ∈ F (∆) and|Fi \ (Fi ∩Fi−1)|= |Fi−1\ (Fi ∩Fi−1)|= 1 for anyi = 1, . . . ,s.For a simplicial complex∆ we can consider a square-free monomial ideal, known astheStanley-Reisner idealof ∆,

I∆ := (xi1 · · ·xis : i1, . . . , is /∈ ∆). (E.1)

Such a correspondence turns out to be one-to-one between simplicial complexes andsquare-free monomial ideals, its inverse being

I 7→ ∆(I) := F ∈ [n] : ∏i /∈F

xi ∈ I

for any square-free monomial idealI ⊆ S. Thek-algebrak[∆] := S/I∆ is called theStanley-Reisner ringof ∆, and it turns out that

dim(k[∆]) = dim∆+1.

More precisely, with the convention of denoting by℘A := (xi : i ∈ A) the prime idealof Sgenerated by the variables correspondent to a given subsetA⊆ [n], we have

I∆ =⋂

F∈F (∆)℘[n]\F .

108 Combinatorial commutative algebra

Remark E.0.3. Thanks to the above interpretation and to LemmaB.0.7, we have thata simplicial complex∆ on [n] is strongly connected if and only ifk[∆] is connected incodimension 1.

We will use another correspondence between simplicial complexes and square-freemonomial ideals, namely

∆ 7→ J(∆) :=⋂

F∈F (∆)℘F . (E.2)

The idealJ(∆) is called thecover idealof ∆. The name “cover ideal” comes fromthe following fact: A subsetA⊆ [n] is called avertex coverof ∆ if A∩F 6= /0 for anyF ∈ F (∆). Then it is easy to see that

J(∆) = (xi1 · · ·xis : i1, . . . , is is a vertex cover of∆).

Let ∆c be the simplicial complex on[n] whose facets are[n] \F such thatF ∈ F (∆).Clearly we haveI∆c = J(∆) and I∆ = J(∆c). Furthermore(∆c)c = ∆, therefore alsothe correspondence (E.2) is one-to-one between simplicial complexes and square-freemonomial ideals.

E.1 Symbolic powers

If R is a ring, anda⊆ R is an ideal, then themth symbolic powerof a is the ideal ofR

a(m) := (amRW)∩R⊆ R,

where the multiplicative systemW is the complement inRof the union of the associatedprime ideals ofa. If a =℘ is a prime ideal, then℘(m) = (℘mR℘)∩R. One can showthat℘(m) is the℘-primary component of℘m, so that℘(m) =℘m if and only if ℘m isprimary. Furthermore, one can show the following:

Proposition E.1.1. Let a be an ideal of R with no embedded primes, that isAss(a) =Min(a). If a=

⋂ki=1qi is a minimal primary decomposition ofa, then

a(m) =k⋂

i=1

q(m)i .

In the case in whicha is a power of a prime monomial ideal of the polynomial ringS, i.e. there existsF ⊆ [n] andk∈ N such thata=℘k

F , then it is easy to show that

am = a(m) ∀ m∈N.

Therefore if∆ is a simplicial complex on[n] andI =⋂

F∈F (∆)℘ωFF , whereωF are some

positive integers, then PropositionE.1.1yields

I (m) =⋂

F∈F (∆)℘mωF . (E.3)

In particular

I (m)∆ =

⋂

F∈F (∆)℘m

[n]\F and J(∆)(m) =⋂

F∈F (∆)℘m

F .

E.2 Matroids 109

E.2 Matroids

A simplicial complex∆ on [n] is amatroid if, for any two facetsF andG of ∆ and anyi ∈ F, there exists aj ∈ G such that(F \ i)∪ j is a facet of∆.

Example E.2.1. The following is the most classical example of matroid. Let Vbe ak-vector space and let A:= v1, . . . ,vn a set of distinct vectors of V . We define asimplicial complex on[n] as follows: A subset F⊆ [n] is a face of∆ if and only ifdimk(< vi : i ∈ F >) = |F |. This way the facets of∆ are the subsets of[n] correspond-ing to the bases in A of thek-vector subspace< v1, . . . ,vm>⊆V. Actually, the conceptof matroid was born as an “abstraction of the bases of ak-vector space”.

A matroid∆ has very good properties. For an exhaustive account the reader can seethe book of Welsh [106] or the one of Oxley [88]. We use matroids in Chapter4 and,essentially, we need three results about them. The first one,the easier to show, is that amatroid is a pure simplicial complex ([88, Lemma 1.2.1]). The second one, known astheexchange propertyof matroids, states that for any matroid∆, we have

∀ F,G∈ F (∆), ∀ i ∈ F, ∃ j ∈ G : (F \ i)∪ j, (G\ j)∪i ∈ F (∆), (E.4)

(this result is of Brualdi [11], see also [88, p. 22, Exercise 11]). The last fact is a basicresult of matroid theory, which is a kind of duality. For the proof see [88, Theorem2.1.1].

Theorem E.2.2. A simplicial complex∆ on [n] is a matroid if and only if∆c is amatroid.

If ∆ is a matroid,∆c is called itsdual matroid.

E.3 Polarization and distractions

Sometimes, problems regarding (not necessarily square-free) monomial idealsI ⊆ Scan be faced passing to a square-free monomial ideal associated toI , namely itspolar-izationI , which preserves many invariants ofI , such as its minimal free resolution.

More precisely, the polarization of a monomialt = xa11 · · ·xan

n is

t :=a1

∏j=1

x1, j ·a2

∏j=1

x2, j · · ·an

∏j=1

xn, j ⊆ S,

where thexi, j ’s are new variables overk and S := k[xi, j : i = 1, . . . ,n j = 1, . . . ,ai ].Actually it is not so important thatS is generated by variablesxi, j with j ≤ ai . Thesignificant thing is that it contains all such variables, in such a way thatt ∈ S. Forexample, it is often convenient to considerS= k[xi, j : i = 1, . . . ,n j = 1, . . . ,d] whered is the degree oft.

The concept of polarization fits in a more general context, that of distractions(seeBigatti, Conca and Robbiano [8]), which is useful to introduce. LetP := k[y1, . . . ,yN]be a polynomial ring inN variables overk. An infinite matrixL = (Li, j )i∈[N], j∈N withentriesLi, j ∈ P1 is called adistraction matrixif < L1, j1, . . . ,LN, jN >= P1 for all j i ∈ Nand there existsk∈N such thatLi, j = Li,k for any j > k. If t = ya1

1 · · ·yaNN is a monomial

of P, theL -distractionof t is the monomial

DL (t) :=a1

∏j=1

L1, j ·a2

∏j=1

L2, j · · ·aN

∏j=1

LN, j ⊆ P.

110 Combinatorial commutative algebra

Remark E.3.1. In this remark we want to let the reader noticing how the polarizationis a particular distraction. Lett = xa1

1 · · ·xann be a monomial ofS and t is polarization

in the polynomial ringS= k[xi, j : i ∈ [n], j ∈ [d]], whered is the degree oft. Let usthink atSas the polynomial ringP given for the definition of distractions. SoN = nd.Moreover, we can look att as an element ofP, namely

t = xa11,1 · · ·x

ann,1.

Let us consider the following matrixL :

Li, j :=

xi, j if i ≤ n and j ≤ d

xr,q+ xr,q+1 if n< i ≤ N, j ≤ d and i = qn+ r with 0< r ≤ n

Li,d if j > d

One can easily check thatL is a distraction matrix, and that

DL (t) = t.

By RemarkE.3.1, we can state the results we need during the thesis in the moregeneral context of distractions, even if we will actually use them just for the case of thepolarization.

Fixed a distraction matrixL , we can extendk-linearlyDL , getting ak-linear map:

DL : P−→ P.

Of courseDL is not a ring homomorphism, however we have thatDL (I) is an ideal ofP for any monomial idealI ⊆ P ([8, Corollary 2.10 (a)]). Moreover, ifI = (t1, . . . , tm),thenDL (I) = (DL (t1), . . . ,DL (tm)) ([8, Corollary 2.10 (b)]) and ht(DL (I)) = ht(I)([8, Corollary 2.10 (c)]). Furthermore, [8, Proposition 2.9 (d)] implies that, ifI =∩p

i=1Ii , where theIi ’s are monomial ideals ofP, then:

DL (I) =p⋂

i=1

DL (Ii). (E.5)

Remark E.3.2. If I = (t1, . . . , tm)⊆ S, the polarization ofI is defined to be:

I = (t1, . . . , tm)⊆ S= k[xi, j : i ∈ [n], j ∈ [d]],

whered is the maximum of the degrees of theti ’s. By the discussion previous to theremark we deduce thatI andDL (I) are basically the same object. The equality (E.5)can be interpreted as

I =p⋂

i=1

Ii ⊆ S. (E.6)

where all the ideal involved are polarized in the same polynomial ring S.

Remark E.3.3. As it already got out by RemarkE.3.2, among the things which distin-guish polarization and distractions, is that the ambient ring, contrary to what happensfor the former, does not change under the latter operation. Of course, this is just a super-ficial problem, since we can add variables before polarizing, as we did in RemarkE.3.1.However the reader should play attention to this fact: IfJ is a monomial ideal ofP andL is a distraction matrix, then we have the equality HFP/J = HFP/DL (J) ([8, Corollary

E.3 Polarization and distractions 111

2.10 (c)]). Of course, it is not true the same fact for polarization: Namely, in general, ifI is a monomial ideal ofS, it might happen that HFS/I 6= HFS/I . Actually, even dimS/I

is in general different from dimS/I . However, in view of RemarkE.3.1, one shouldexpect that the properties of distractions hold true also for polarization. This is actuallythe case, but the right way to think is “for codimension”: In fact, we have ht(I) = ht(I);moreover, suppose that the Hilbert series ofS/I is HSS/I (z) = h(z)/(1− z)d, whereh(z) ∈ Z[z] is such thath(1) 6= 0 andd is the dimension ofS/I (for instance see thebook of Bruns and Herzog [13, Corollary 4.1.8]). Then, HSS/I(z) = h(z)/(1− z)e,

wheree= dim(S/I).

In [8, Theorem 2.19], the authors showed that the minimal free resolution of amonomial idealI ⊆ P can be carried to a minimal free resolution ofDL (I). In partic-ular, we get the following:

Theorem E.3.4. Given a distraction matrixL , the graded Betti numbers of P/I andthose of P/DL (I) are the same. Particularly, P/I is Cohen-Macaulay if and only ifP/DL (I) is Cohen-Macaulay.

Index

a-invariant,xix–xx, 52–56affine schemes,2, 8–9algebra of basic covers,69algebra of minors,45, 105Algebra with Straightening Laws,23–25,

29, 32, 75arithmetical rank,xvii , 24, 25, 32–42, 75–

76, 82homogeneous,xvii , 24, 25, 32–42,

82ASL, seeAlgebra with Straightening Laws

Betti numbers,xx, 12graded,xxtopological,12

bi-diagram,47admissible,48asymmetric,49symmetric,48

bi-weight,104vector,104

highest,104

Castelnuovo-Mumford regularity,xviii , xx,27–30, 44, 52–56

Cauchy formula,104Cech complex,xivcohomological dimension,xvi, 1–21, 30,

34–35, 41, 81connected,2–11, 23–32, 85–89

in codimension 1,23–32, 85–89in codimensiond, 85–89r-, 2–11, 23–32, 85–89

sequence,87strongly,seesimplicial complex

constant sheaf,81cover ideal,108

De Rham cohomology,84algebraic,84

depth,xvi, 3, 17–19, 88–89

diagram,seeYoung diagramdistractions,109

etale cohomological dimension,3, 15–16,19, 81

etale cohomology,80

Flat Base Change,xii

G-equivariant map,95G-representation,seerepresentationGAGA, 82–83general linear group,97generic matrix,43–64

minors of,43–64Grobner basis,92Grassmannian,44, 46

h-vector,27, 31Hilbert function,xx, 27Hilbert polynomial,xx, 69–72, 75Hilbert quasi-polynomial,xxHilbert series,27, 111Hodge numbers,12–15homogenization,seeω-homogenization

Independence of the Base,xiiinitial algebra,27, 52, 61–63, 91–94

with respect to a monomial order,91with respect to a weight,92

initial form, 92initial ideal,23–32, 91–94

with respect to a monomial order,91with respect to a weight,92

initial monomial,91

k-cover,seesimplicial complex

local cohomology,xi–xxfunctor,ximodule,xi

INDEX 113

of the polynomial ring with supportin the maximal irrelevant,xv

local duality,xix

matroid,65–77, 109dual,109exchange property of,109

Mayer-Vietories sequence,xiiimonomial,91monomial ideal,107

good-weighted,67square-free,107weighted,67

monomial order,91multiplicity, 27–31, 75

minimal,31

ω-graduation,92–94ω-homogenization,92–94

partition,99admissible,105d-admissible,105difference sequence,58height of,99parts of,99transpose,101

partition identity,61homogeneous,62

primitive,62primitive,61with entries in[q], 61

Pieri’s formula,58, 102plethysm,46, 102polarization,28, 72–74, 109predecessor,47, 58–61presheaf,80punctured spectrum,86

r-connected,seeconnected,r-relations,43–64

between minors,43–64determinantal,44, 50Plucker,43, 46shape,49, 58–61

representation,95–105decomposable,96determinant,101indecomposable,96irreducible,95

minimal,48multiplicities,47, 102polynomial,97rational,97theory,95–105theory of the general linear group,97–

103ring of invariants,45, 63–64

Sagbi bases,44, 61–63, 91Schur module,99Segre product,24, 25, 32–42set-theoretic complete intersection,xvii , 24,

25, 32–42, 75–76sheaf,80sheaf cohomology,79–82simplicial complex,23–32, 107

k-cover,68basic,68

faces of,107facets of,107good weighted,67pure,107strongly connected,23–32, 107vertex cover,108weighted,67

k-cover,68Singular cohomology,83Singular homology,83site,79

etale,80Zariski,80

space of initial monomials,91–94Stanley-Reisner ideal,107Stanley-Reisner ring,107straightening relations,29symbolic power,65–77, 108

tableu,seeYoung tableutorsion functor,xi

U-invariant,45, 50, 63–64universal coefficient theorem for cohomol-

ogy,83universal coefficient theorem for homol-

ogy,83

vertex cover,seesimplicial complexvertex cover algebra,69

weight,98

114 INDEX

space,98vector,98

highest,98weighted function,67

canonical,67

Young diagram,99Young symmetrizer,100Young tableu,99

canonical,102content of,100standard,101

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