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DUALIZING COMPLEXES AND PERVERSE SHEAVES ON

NONCOMMUTATIVE RINGED SCHEMES

AMNON YEKUTIELI AND JAMES J. ZHANG

Abstract. Let (X,A) be a separated differential quasi-coherent ringed schemeof finite type over a field k. We prove that there exists a rigid dualizing com-

plex over A. The proof consists of two main parts. In the algebraic part westudy differential filtrations on rings, and use the results obtained to show thata rigid dualizing complex exists on every affine open set in X. In the geometricpart of the proof we construct a perverse t-structure on the derived categoryof bimodules, and this allows us to glue the affine rigid dualizing complexes toa global dualizing complex.

Contents

0. Introduction 11. Background and Summary of Results 42. Dualizing Complexes over Rings 83. Quasi-Coherent Ringed Schemes and Localization 114. Products of Quasi-Coherent Ringed Schemes 165. Dualizing Complexes over Ringed Schemes 206. Localization of Dualizing Complexes 267. Local Dualizing Complexes 298. Perverse Modules and the Auslander Condition 339. Perverse Coherent Sheaves on Ringed Schemes 3710. Filtrations of Rings 4311. Differential k-Algebras of Finite Type 4712. The Rigid Dualizing Complex of a Differential k-Algebra 5313. Differential Quasi-Coherent Ringed Schemes of Finite Type 60References 67

0. Introduction

The “classical” Grothendieck Duality theory, dealing with dualizing complexesover commutative schemes, was developed in the book Residues and Duality byHartshorne [RD]. Various refinements, generalizations and explicit reformulations

Date: 19 November 2002.Key words and phrases. Noncommutative algebraic geometry, perverse sheaves, dualizing

complexes.Mathematics Subject Classification 2000. Primary: 14A22; Secondary: 14F05, 14J32, 16E30,

16D90, 18E30.This research was supported by the US-Israel Binational Science Foundation. The second

author was partially supported by the US National Science Foundation.

1

http://arxiv.org/abs/math/0211309v1

2 AMNON YEKUTIELI AND JAMES J. ZHANG

of Grothendieck’s theory have appeared since. A partial list of recent papers is[Ne], [AJL], [Ye5] and [Co].

A noncommutative affine theory was introduced in [Ye1]. By “affine” we meanthat this theory deals with noncommutative algebras over a base field k. In thedecade since its introduction the theory of noncommutative dualizing complexeshas progressed in several directions. See the papers [VdB1], [Jo], [WZ], [Ye4] and[Ch2].

The aim of this paper is to study Grothendieck Duality on noncommutativespaces. As motivation one should consider the role Grothendieck Duality playsboth in commutative algebraic geometry and in noncommutative ring theory. Du-alizing sheaves and complexes have for years been key ingredients in classificationof varieties, construction of moduli spaces and resolution of singularities. Recentdevelopments, mainly surrounding homological mirror symmetry (cf. [BO]) tell usthat there is a profound interplay between algebraic geometry, noncommutativealgebra and derived categories. Noncommutative duality sits right in the middle ofthese areas of research.

Actually there is one type of non-affine noncommutative space for which dualityhas already been studied. These are the noncommutative projective schemes ProjAof Artin-Zhang [AZ]. Here A is a noetherian connected graded k-algebra satisfyingthe χ-condition. A global duality theory for ProjA was discussed in [YZ1] and[KKO]. However in the present paper we choose to stay closer to the [RD] paradigm,namely to develop a theory that has both local and global aspects.

Here are a few issues one should consider before proposing a theory of Grothen-dieck Duality on noncommutative spaces. The first is to decide what is meant by anoncommutative space – there are several reasonable choices in current literature.The second is to find a suitable formulation of duality, which shall include in anatural way the established commutative and noncommutative theories. The thirdissue is whether this duality theory applies to a wide enough class of spaces.

The noncommutative spaces we shall concentrate on are the quasi-coherentringed schemes. A quasi-coherent ringed scheme is a pair (X,A), where X is ascheme over k, and A is a (possibly noncommutative) quasi-coherent OX -ring.This type of geometric object includes schemes (A = OX), noncommutative rings(X = Spec k) and rings of differential operators (A = DX). Our definition ofdualizing complex extends the established definitions in all these cases. A dualiz-ing complex R over (X,A) is an object of the derived category D(ModAe), where(X2,Ae) is the product of (X,A) and its opposite (X,Aop). The duality functorD : Db

c (ModA) → Dbc (ModAop) is a contravariant Fourier-Mukai transform with

respect to R.Actually we allow the slightly more general situation of a dualizing complex

over two distinct ringed schemes (X,A) and (Y,B). Thus our definition also in-cludes contravariant versions of the following operations: Fourier-Mukai transformsbetween two schemes, as in [Mu] and [BO]; Beilinson equivalences induced by excep-tional sequences, as in [Be1]; and derived equivalences of rings induced by two-sidedtilting complexes.

We shall restrict our attention to dualizing complexes that have a “local” be-havior, namely those that are supported on the diagonal ∆(X) ⊂ X2. Our mainobjective is to prove existence of such dualizing complexes (under suitable assump-tions).

DUALIZING COMPLEXES AND PERVERSE SHEAVES 3

Fix a quasi-coherent ringed scheme (X,A). Assume that for any affine open setU ⊂ X , writing A := Γ(U,A), the rigid dualizing complex RA of A exists and issupported on ∆(U) ⊂ U2. Then one can show that RA localizes in the Zariskitopology of U , giving rise to a complex of sheaves RA|U on U2. Because the rigiddualizing complex RA is unique up to a unique isomorphism, it follows that aswe vary U we get effective gluing data for the complexes RA|U in the appropriatederived categories. But at this stage we run into a problem: objects in triangulatedcategories usually cannot be glued!

Our solution to this problem is to use perverse coherent sheaves. For everyaffine open set U we have the rigid perverse t-structure on Db

c (ModA|U ), whichis induced by the rigid dualizing complex RA. We prove that as U varies these t-structures agree on intersections, and hence give rise to a global perverse t-structureon Db

c (ModA). Again this relies on the fact that the rigid dualizing complex RAlocalizes in the Zariski topology of U . The heart of the rigid perverse t-structure,called the category of perverse coherent sheaves on (X,A), is a stack of abeliancategories. This means that perverse coherent sheaves can be glued.

The gluing just described solves only “half of the problem,” since we really wantto glue the affine bimodule complexes RA|U to obtain a global bimodule complex

RA ∈ Dbc (ModAe). Fortunately this difficulty can be easily overcome, as we explain

below.A class of noncommutative spaces for which our construction works is that of

separated differential quasi-coherent ringed schemes of finite type over k. Suppose(X,A) is such a space. Roughly this says that X is a separated finite type k-scheme, and A has a filtration such that the graded ring grA is a finite type quasi-coherent OX -algebra with a big center. This class of spaces includes the followingprototypical examples:

(1) A is a coherent OX -algebra (e.g. an Azumaya algebra);(2) A is the ring DX of differential operators on a smooth scheme X in char-

acteristic 0; and(3) A is the universal enveloping algebra U(OX ,L) of a coherent Lie algebroidL on X .

In (1) and (3) there are no regularity assumptions on X , A or L. Using Van denBergh’s criterion for existence of rigid dualizing complexes, and a delicate analysisof filtrations, we show that on every affine open set U the ring A := Γ(U,A)admits a rigid dualizing complex RA, and moreover this complex is supported onthe diagonal. Hence as explained above we have perverse coherentA-modules. Nowthe product (X2,Ae) is also a separated differential quasi-coherent ringed schemeof finite type over k, so by the same token we obtain the category of perverse Ae-modules. It turns out that each complex RA|U is itself a perverse Ae|U2 -module

supported on ∆(U) ⊂ U2. Therefore these affine pieces can be glued to a perverseAe-module RA, which is the rigid dualizing complex of (X,A).

Our results on perverse coherent sheaves are new even in commutative algebraicgeometry. Perverse t-structures have appeared recently at the juncture of bira-tional algebraic geometry (blow-ups, flops, McKay correspondence) and theoreticalphysics (homological mirror symmetry, string theory). It will be interesting to seehow our approach might contribute to this area of research.


Acknowledgments. The authors wish to thank Eitan Bachmat, Joseph Bernstein,Sophie Chemla, Maxim Kontsevich, Thierry Levasseur, Joseph Lipman, ZinovyReichstein, Pramathanath Sastry, Paul Smith and Michel Van den Bergh for helpfulconversations.

1. Background and Summary of Results

1.1. Existing Theories of Dualizing Complexes. Throughout we shall beworking over a base field k. In Grothendieck’s original theory in [RD] a dualizingcomplex over a finite type k-scheme X is a complex R of OX -modules such that thefunctor RHomOX (−,R) is an auto-duality of the derived category Db

c (ModOX).To any morphism f : X → Y there is attached a functor f ! : D+

c (ModOX) →D+

c (ModOY ) called the twisted inverse image; and it sends dualizing complexes todualizing complexes. If f is proper then there is a nondegenerate trace morphismTrf : Rf∗f

!R→ R. Of particular importance is the dualizing complex π!Xk, where

πX : X → Spec k is the structural morphism. When X is integral of dimension nthen the sheaf ωX := H−nπ!

Xk is called the dualizing sheaf of X , and this is thesheaf occurring in Serre Duality. If furthermore X is smooth then ωX ∼= ΩnX/k, the

sheaf of top degree differentials.If X = SpecA is an affine commutative scheme then Grothendieck Duality takes

on the following form. A dualizing complex over A is a complex R ∈ Dbf (ModA)

such that the functor RHomA(−, R) is an auto-duality of this category.Now consider a noncommutative noetherian k-algebra A. By default we shall

work with left A-modules, and by finite module we mean finitely generated. Wedenote by ModA the category of A-modules, and by Modf A the subcategory offinite modules. The opposite algebra is Aop, and the enveloping algebra is Ae :=A ⊗k A

op. Thus A-bimodules are Ae-modules. A dualizing complex over A is acomplex R ∈ Db(ModAe) such that the contravariant functors

RHomA(−, R) : Dbf (ModA)→ Db

f (ModAop)

RHomAop(−, R) : Dbf (ModAop)→ Db

f (ModA)

are a duality, i.e. a contravariant equivalence. There are also some technical condi-tions that are stated in Definition 2.1.

Among the features of noncommutative dualizing complexes we wish to singleout one that shall play a central role in this paper. This is the rigidity propertyintroduced by Van den Bergh [VdB1]. One says that R is a rigid dualizing complex

if there exists a rigidifying isomorphism

ρ : R≃−→ RHomAe(A,R⊗ R)

in D(ModAe). It is known that a rigid dualizing complex (R, ρ) over A is uniqueup to a unique isomorphism. As for existence, perhaps the most effective criterionis the one due to Van den Bergh, which involves filtrations and local duality forgraded algebras.

If A is a commutative finitely generated k-algebra with spectrum X := SpecAand structural morphism πX : X → Spec k, then the complex RA := RΓ(X, π!

Xk)is a rigid dualizing complex over A.

1.2. Quasi-Coherent Ringed Schemes. The noncommutative spaces we con-centrate on in this paper are of the following sort: Y is a noncommutative space


that is an “affine fibration” over a commutative schemeX . To be precise, Y is a pair(X,A), where X is a noetherian k-scheme, and A is a noetherian quasi-coherentOX -ring. Such a pair (X,A) is called a noetherian quasi-coherent ringed k-scheme

(see Section 3 for full details). On any affine open set U ⊂ X we get a noetheriank-algebra Γ(U,A), that could be very noncommutative; the topology of X controlshow the rings Γ(U,A) fit together.

As examples indicate, dualizing complexes over (X,A) ought to be complexesof sheaves on a suitable product (X2,Ae) = (X,A) × (X,Aop). In Definition 4.1we require that this product shall also be a quasi-coherent ringed scheme. Unique-ness of such a product is not hard to prove; yet existence depends on certain Oreconditions being satisfied (Theorem 4.4), and there are counterexamples. A similarnotion of product appears in recent work of Lunts [Lu].

1.3. Dualizing Complexes over Quasi-Coherent Ringed Schemes. Asmentioned above a dualizing complex R over A should live in the derived categoryD(ModAe). So when talking about dualizing complexes we shall tacitly assumethat the product (X2,Ae) exists. The definition of dualizing complex (Definition5.9) is global, and it involves the two projection morphisms p1 : (X2,Ae)→ (X,A)and p2 : (X2,Ae)→ (X,Aop). The duality is a pair of contravariant functors

D : Dbc (ModA)→ Db

c (ModAop)

Dop : Dbc (ModAop)→ Db

f (ModA)

each being a “contravariant Fourier-Mukai transform” with respect to R. We re-quire the adjunction morphisms 1 → DopD and 1 → DDop to be isomorphisms offunctors. There are some technical complications, mainly due to the fact that wedo not understand the structure and sheaf-theoretical behavior of injective objectsin the category QCohA of quasi-coherent A-modules. This is in contrast to thecommutative situation, where all was worked out by Grothendieck in [RD].

Definition 5.9 is pretty loose, and admits certain exotic examples. For instanceif X is an elliptic curve then the Poincare bundle R ∈ Db

c (ModOX2) is a dualizingcomplex over OX in the noncommutative sense; see Proposition 5.11. Here is an-other exotic example (it is Proposition 5.12 in the body of the paper). In Definition5.9 we allow dualizing complexes over two distinct ringed schemes (X,A) and (Y,B).Take X := Pn

k, E :=

⊕ni=0OX(i) and B := EndOX (E)

op. The two ringed schemeswe obtain are (X,OX) and (Spec k, B). It turns out that E ∈ D(Mod (OX ⊗Bop))is a dualizing complex over these ringed schemes. This is a reinterpretation ofBeilinson’s equivalence [Be1].

We prefer to study dualizing complexes that have a local behavior (relative tothe topology of X). This is accomplished by requiring R to be supported on thediagonal in X2. Indeed, let us assume that X is separated (we shall keep thisassumption throughout this section). Then a complex R ∈ Db

c (ModAe) that issupported on the diagonal is dualizing if and only if for any affine open set U ⊂ Xthe complex RΓ(U2,R) ∈ Db

f (ModAe) is a dualizing complex over A := Γ(U,A).See Theorem 7.4.

By a rigid dualizing complex over (X,A) we mean a pair (R,ρ) where R ∈Db

c (ModAe) is a dualizing complex supported on the diagonal ∆(X) ⊂ X2, andρ = ρU is a collection of rigidifying isomorphisms indexed by the affine open setsU ⊂ X and satisfying obvious compatibility conditions (Definition 7.6).


Assume for a moment that for each affine open set U ⊂ X we can find a dualizingcomplex RA|U ∈ Db

c (ModAe|U2) supported on the diagonal in U2, together withgluing data. That is not too hard to do in many cases (cf. Subsection 1.5 below).Still it remains to glue the local pieces RA|U to a global object in Db

c (ModAe).This is a genuine problem. In the commutative setup the solution by Grothendieckwas to use Cousin complexes. However for noncommutative ringed schemes thissolution will seldom apply; and this is where perverse sheaves enter the picture.

1.4. Perverse Coherent Sheaves. Let us open this subsection with a quickreview of Cousin complexes, following [RD]. Given a complex M ∈ D+(ModOX)there is the niveau spectral sequence Ep,q1 ⇒ Hp+qM which comes from the fil-tration of ModOX by dimension of support. The Cousin complex EM is the rowq = 0 in the E1 page of this spectral sequence. In this way one obtains a functorE : D+(ModOX) → C+(ModOX) into the abelian category of complexes. If Ris a dualizing complex with the correct dimension shift then there is an isomor-phism R ∼= ER in D+(ModOX). Therefore, given an open covering X =

⋃Ui,

dualizing complexes Ri on each Ui and gluing data φi,j : Ri|Ui∩Uj

≃−→ Rj |Ui∩Uj

in D+(ModOUi∩Uj ), we obtain gluing data E(φi,j) : ERi|Ui∩Uj

≃−→ ERj |Ui∩Uj in

C+(ModOUi∩Uj ). Since the objects of C+(ModOX) are local there is no longer any

difficulty to glue.Now let us move to the noncommutative setup. Here too it is possible to define

Cousin complexes, as explained in [Ye2] and [YZ3]. Suppose R is an Auslanderdualizing complex over the ring A with associated canonical dimension functiondim := CdimR (see Definitions 2.6 and 2.9). Then there is a filtration on ModA bydimension, namely ModA =

⋃iMi(dim), where Mi(dim) is the subcategory con-

sisting of all modules M with dimM ≤ i. There is a corresponding niveau spectralsequence, and a Cousin functor E : D+(ModA) → C+(ModA). The problem isthat, due to known obstructions, it is rarely the case that R ∼= ER. Hence Cousincomplexes are ruled out as a general solution for gluing in the noncommutativesetting.

The discovery at the heart of our present paper is that perverse coherent sheaves

can be used instead of Cousin complexes for gluing dualizing complexes.The concept of t-structure is due to Bernstein, Beilinson and Deligne [BBD], and

we recall the definitions in Section 8. In our situation suppose that for any affineopen set U ⊂ X we have some t-structure

TU =(pDb

c (ModA)≤0, pDbc (ModA)≥0

)

on the derived category Dbf (ModA), where A := Γ(U,A). We say that TU is a

local collection of t-structures if a rather obvious locality condition is satisfied (seeDefinition 9.1). Let pDb

c (ModA)≤0 be the full subcategory of Dbc (ModA) consisting

of all complexesM such that RΓ(U,M) ∈ pDbc (ModΓ(U,A))≤0 for all affine open

sets U . Likewise define pDbc (ModA)≥0. The intersection of these subcategories is

denoted by pDbc (ModA)0.

Theorem 1.4.1. Let (X,A) be a quasi-coherent ringed scheme and let TU be a

local collection of t-structures on it.

(1) The pair of subcategories(pD

bc (ModA)≤0, pDb

c (ModA)≥0)


induced by TU is a t-structure on Dbc (ModA).

(2) The assignment V 7→ pDbc (ModA|V )0, for V ⊂ X open, is a stack of abelian

categories on X.

Part (2) of the theorem says that pDbc (ModA)0 behaves like the category of

sheaves ModA; hence its objects are called perverse coherent sheaves. The theoremis a compendium of Theorems 9.9 and 9.11.

It should be noted that gluing of t-structures as originally done in [BBD] doesnot seem to work for coherent sheaves. See Remark 9.12.

1.5. Differential Quasi-Coherent Ringed Schemes. A class of noncommu-tative spaces for which our theory works is that of differential quasi-coherent ringedk-schemes of finite type. In such a ringed scheme (X,A) by definition the under-lying scheme X is of finite type over k. We assume the sheaf of rings A admitssome exhaustive nonnegative filtration F = FiAi∈Z such that grFA is a coherentmodule over its center Z(grFA), and Z(grFA) is a finite type quasi-coherent OX -algebra. The prototypical examples are listed in the introduction. We say (X,A)is separated if X is separated.

The ring theoretic counterpart of this type of space is a differential C-ring of

finite type, where C is a finitely generated commutative k-algebra. Indeed if wetake an affine open set U ⊂ X then A := Γ(U,A) is a differential Γ(U,OX)-ring offinite type. We also call A a differential k-algebra of finite type.

An important result regarding such algebras is the following “theorem on the twofiltrations,” which is Theorem 11.11 in the body of the paper. A slightly weakerresult was already obtained by McConnel-Stafford [MS].

Theorem 1.5.1. Let A be a k-algebra. Assume A has a nonnegative exhaustive

filtration F = FiAi∈Z such that grFA is a finite module over its center Z(grFA),and Z(grFA) is a finitely generated k-algebra. Then there is a nonnegative exhaus-

tive filtration G = GiAi∈Z such that grGA is a commutative, finitely generated,

connected graded k-algebra.

We remind the reader that a graded k-algebra B is called connected if B =⊕i∈N

Bi, B0 = k and each Bi is a finite k-module. The importance of Theorem1.5.1 is that having the filtration G on A guarantees that A has an Auslander rigiddualizing complex RA, see [VdB1] and [YZ2].

Now suppose C is a finitely generated commutative k-algebra and A is differentialC-ring of finite type. By Theorem 1.5.1 we know that the rigid dualizing complexRA exists. In Theorem 13.1 we prove that for every i the C-bimodule HiRA issupported on the diagonal in SpecCe = (SpecC)2. This implies that the complexRA localizes on SpecC.

For A as above let D : Dbf (ModA) → Db

f (ModAop) be the duality functorRHomA(−, RA). The rigid perverse t-structure on Db

c (ModA) is defined by

pD

bc (ModA)≤0 := M | HiDM = 0 for all i < 0

pD

bc (ModA)≥0 := M | HiDM = 0 for all i > 0.

The fact that the rigid dualizing complex RA localizes on SpecC implies thenext result (Theorem 9.13 in the body of the paper).


Theorem 1.5.2. Let (X,A) be a differential quasi-coherent ringed k-scheme of

finite type. For any affine open set U ⊂ X let TU be the rigid perverse t-structure

on Dbf (ModΓ(U,A)). Then TU is a local collection of t-structures on (X,A).

Now by Theorem 1.4.1 we obtain a t-structure on Dbc (ModA); we call it the rigid

perverse t-structure.One can show that the product (X2,Ae) exists, and is also a differential quasi-

coherent ringed k-scheme of finite type. According to Theorem 12.9 for everyaffine open set U ⊂ X the rigid dualizing complex RA is a perverse bimodule,namely RA ∈ pDb

c (ModAe)0. The sheafification on U2 is then a perverse sheafRA|U ∈

pDbc (ModAe|U2)0 supported on ∆(U). We see that the perverse sheaves

RA|U can be glued along ∆(X) to a perverse Ae-module RA. Thus we have arrivedat the main result of the paper (it is repeated as Theorem 13.6).

Theorem 1.5.3. Let (X,A) be a separated differential quasi-coherent ringed k-

scheme of finite type. Then A has a rigid dualizing complex (RA,ρ). It is unique

up to a unique isomorphism in Dbc (ModAe).

With a little extra effort we prove that rigid traces exists for finite centralizingmorphisms (Theorem 13.9). In Examples 13.10 - 13.14 we make a detailed studyof several interesting cases.

1.6. Perverse Coherent Sheaves on Schemes. Two aspects of this paper maybe of interest to (commutative) algebraic geometers. The first is our treatment ofperverse coherent sheaves. Perverse coherent sheaves on schemes (with respect toother kinds of t-structures) have appeared lately in papers on birational geometry(e.g. [Br] and [VdB2]). There is a profound connection to homological mirrorsymmetry and other aspects of string theory (cf. the survey talks [BO] and [Do]).For the reader interested in this aspect of the paper we recommend turning toSections 8, 9 and 13, keeping in mind that in the commutative setup A = OX .

The second aspect is a totally new approach to dualizing complexes on schemes.Indeed, using rigid dualizing complexes and perverse sheaves, we have an essentiallyself-contained proof of the existence of dualizing complexes on finite type k-schemes,and of traces for finite morphisms; cf. Theorem 13.16. Applications, including caseswhere the base ring k is not a field, shall be discussed in a future paper.

Finally we should say something about the relation between perverse coherentsheaves and Cousin complexes in the commutative setup. We prove that the per-verse coherent sheaves (for the rigid perverse t-structure) are precisely the Cohen-Macaulay complexes, i.e. those complexes M ∈ Db

c (ModA) satisfying M ∼= EM;see Theorem 13.19.

2. Dualizing Complexes over Rings

Throughout the paper k denotes a base field. By a k-algebra we mean an associa-tive unital algebra. The unadorned tensor product ⊗ will mean ⊗k. Unless specifiedotherwise all rings that appear in the paper are k-algebras, all ring homomorphismsare over k and all bimodules are k-central.

For a ring A we denote by Aop the opposite algebra, and by Ae := A⊗Aop theenveloping algebra. By default an A-module will mean a left A-module. With thisconvention a right A-module is an Aop-module, and an A-bimodule is an Ae-module.


In this section we review the definition of dualizing complexes over rings andrelated concepts.

Let ModA be the category of A-modules, and let Modf A be the full subcate-gory of finite (i.e. finitely generated) modules. The latter is abelian when A is leftnoetherian. Let D(ModA) be the derived category of A-modules. The full sub-category of bounded complexes is denoted by Db(ModA), the full subcategory ofcomplexes with finite cohomologies is denoted by Df(ModA), and their intersectionis Db

f (ModA).

Definition 2.1 ([Ye1], [YZ2]). Let A be a left noetherian k-algebra and B a rightnoetherian k-algebra. A complex R ∈ Db(Mod (A ⊗ Bop)) is called a dualizing

complex over (A,B) if it satisfies the following three conditions:

(i) R has finite injective dimension over A and over Bop.(ii) R has finite cohomology modules over A and over Bop.(iii) The canonical morphisms B → RHomA(R,R) in D(ModBe), and A →

RHomBop(R,R) in D(ModAe), are both isomorphisms.

In the case A = B we say R is a dualizing complex over A.

Whenever we refer to a dualizing complex over (A,B) in the paper we tacitlyassume that A is left noetherian and B is right noetherian.

Remark 2.2. There are many non-isomorphic dualizing complexes over a givenk-algebra A. The isomorphism classes of dualizing complexes are parameterized bythe derived Picard group DPic(A), whose elements are the isomorphism classes oftwo-sided tilting complexes. See [Ye4] and [MY].

Here are two easy examples.

Example 2.3. Suppose A is a Gorenstein noetherian ring, namely the bimoduleR := A has finite injective dimension as left and right module. Then R is a dualizingcomplex over A.

Example 2.4. If A is a finite k-algebra then the bimodule A∗ := Homk(A, k) isa dualizing complex over A. In fact it is a rigid dualizing complex (see Definition2.10).

Definition 2.5. Let R be a dualizing complex over (A,B). The duality functors

induced by R are the contravariant functors

D := RHomA(−, R) : D(ModA)→ D(ModBop)

andDop := RHomBop(−, R) : D(ModBop)→ D(ModA).

By [YZ2, Proposition 1.3] the functors D and Dop are a duality (i.e. an anti-equivalence) of triangulated categories between Df(ModA) and Df(ModBop), re-stricting to a duality between Db

f (ModA) and Dbf (ModBop).

Definition 2.6 ([Ye2], [YZ2]). Let R be a dualizing complex over (A,B). We saythat R has the Auslander property, or that R is an Auslander dualizing complex, ifthe conditions below hold.

(i) For every finite A-module M , every integers p > q, and every Bop-sub-module N ⊂ ExtpA(M,R), one has ExtqBop(N,R) = 0.

(ii) The same holds after exchanging A and Bop.


Rings with Auslander dualizing complexes can be viewed as a generalization ofAuslander regular rings (cf. [Bj] and [Le]).

Example 2.7. If A is either the nth Weyl algebra or the universal envelopingalgebra of a finite dimensional Lie algebra, then A is Auslander regular, and thebimodule R := A is an Auslander dualizing complex.

Definition 2.8. An exact dimension function on ModA is a function

dim : ModA→ −∞ ∪R ∪ infinite ordinals,

satisfying the following axioms:

(i) dim 0 = −∞.(ii) For every short exact sequence 0→M ′ →M →M ′′ → 0 one has dimM =

maxdimM ′, dimM ′′.(iii) If M =

⋃αMα then dimM = supdimMα.

The basic examples of dimension functions are the Gelfand-Kirillov dimension,denoted by GKdim, and the Krull dimension, denoted by Kdim. See [MR, Section6.8.4]. Here is another dimension function.

Definition 2.9 ([Ye2], [YZ2]). Let R be an Auslander dualizing complex over(A,B). Given a finite A-module M the canonical dimension of M with respect to

R isCdimR;AM := − infq | ExtqA(M,R) 6= 0 ∈ Z ∪ −∞.

For any A-module M we define

CdimR;AM := supdimM ′ |M ′ ⊂M is finite.

Likewise we define CdimR;Bop N for a Bop-module N .

Often we shall abbreviate CdimR;A by dropping subscripts when no confusioncan arise. According to [YZ2, Theorem 2.1] Cdim is an exact dimension functionon ModA and ModBop.

The following concept is due to Van den Bergh. Since R is a complex of Ae-modules, the complex R ⊗ R consists of modules over Ae ⊗ Ae ∼= (Ae)e. In thedefinition below RHomAe(A,R ⊗ R) is computed using the “outside” Ae-modulestructure of R⊗R, and the resulting complex retains the “inside” Ae-module struc-ture.

Definition 2.10 ([VdB1, Definition 8.1]). Let R be a dualizing complex over A.If there is an isomorphism

ρ : R→ RHomAe(A,R ⊗R)

in D(ModAe) then we call (R, ρ), or just R, a rigid dualizing complex. The isomor-phism ρ is called a rigidifying isomorphism.

A rigid dualizing complex, if it exists, is unique up to isomorphism, by [VdB1,Proposition 8.2].

A ring homomorphism A → B is said to be finite if B is a finite A-module onboth sides.

Definition 2.11. [YZ2, Definition 3.7] Let A → B be a finite homomorphism ofk-algebras. Assume the rigid dualizing complexes (RA, ρA) and (RB , ρB) exist. LetTrB/A : RB → RA be a morphism in D(ModAe). We say TrB/A is a rigid trace ifit satisfies the following two conditions:


(i) TrB/A induces isomorphisms

RB ∼= RHomA(B,RA) ∼= RHomAop(B,RA)

in D(ModAe).(ii) The diagram

RBρB−−−−→ RHomBe(B,RB ⊗RB)

Tr

yyTr⊗Tr

RAρA

−−−−→ RHomAe(A,RA ⊗RA)

in D(ModAe) is commutative.

Often we shall say that TrB/A : (RB, ρB)→ (RA, ρA) is a rigid trace morphism.

By [YZ2, Theorem 3.2], a rigid trace TrB/A is unique (if it exists). In particulartaking the identity map A → A and any two rigid dualizing complexes (R, ρ) and

(R′, ρ′) overA, it follows there is a unique isomorphismR≃−→ R′ that is a rigid trace;

see [YZ2, Corollary 3.4]. Given another finite homomorphism B → C such that therigid dualizing complex (RC , ρC) and the rigid trace TrC/B exist, the compositionTrC/A := TrC/B TrB/A is a rigid trace.

Finally we mention that by [YZ2, Corollary 3.6] the cohomology bimodules HiRAof the rigid dualizing complex are central Z(A)-bimodules, where Z(A) is the centerof A.

Example 2.12. Suppose A is a finitely generated commutative k-algebra. Choose afinite homomorphism k[t]→ A where k[t] = k[t1, . . . , tn] is the polynomial algebra.Define RA := RHomk[t](A,Ω

nk[t]/k[n]), and consider this as an object of Db(ModAe).

By [Ye4, Proposition 5.7] the complex RA is an Auslander rigid dualizing complex,and in fact it is equipped with a canonical rigidifying isomorphism ρA.

3. Quasi-Coherent Ringed Schemes and Localization

Let (X,A) be a ringed space over k. Thus X is a topological space and A is asheaf of (possibly noncommutative) k-algebras on X . By an A-bimodule we meana sheafM of k-modules on X together with a left A-module structure and a rightA-module structure that commute with each other. In other wordsM is a moduleover the sheaf of rings A ⊗kX A

op, where kX is the constant sheaf k on X . AnA-ring is a sheaf B of rings on X together with a ring homomorphism A → B.Note that B is an A-bimodule.

Definition 3.1. (1) A ringed scheme over k is a pair (X,A) consisting of ak-scheme X and an OX -ring A.

(2) We say A is a quasi-coherent OX-ring, and the pair (X,A) is a quasi-

coherent ringed scheme, if the OX -bimodule A is a quasi-coherent OX -module on both sides.

(3) A quasi-coherent ringed scheme (X,A) is called separated (resp. affine) ifX is a separated (resp. affine) k-scheme.

Definition 3.2. Let (X,A) be a quasi-coherent ringed scheme over k. Supposethat X is noetherian and that for every affine open set U ⊂ X the ring Γ(U,A)is left noetherian. Then we call A a left noetherian quasi-coherent OX-ring, andthe pair (X,A) is called a left noetherian quasi-coherent ringed scheme. We say A


is right noetherian if Aop is left noetherian. If A is both left and right noetherianthen the pair (X,A) is called a noetherian quasi-coherent ringed scheme.

When we speak of a left or right noetherian quasi-coherent OX -ring A we tacitlyassume that X itself is noetherian.

Definition 3.3. Let (X,A) be a quasi-coherent ringed scheme and let M be anA-module.

(1) M is called a quasi-coherent A-module if it is quasi-coherent as OX -module.(2) Suppose A is left noetherian. M is called a coherent A-module if it is

quasi-coherent and locally finitely generated over A.

We shall denote the categories of quasi-coherent (resp. coherent) A-modules byQCohA (resp. CohA).

Proposition 3.4. Let (X,A) be a quasi-coherent ringed scheme, let U ⊂ X be an

affine open set and A := Γ(U,A).

(1) The functor Γ(U,−) is an equivalence of categories QCohA|U → ModA.(2) If A is left noetherian then Γ(U,−) restricts to an equivalence of categories

CohA|U → Modf A.

Proof. This is a slight generalization of [EGA I, Corollary 1.4.2 and Theorem 1.5.1].See also [Ha, Corollary II.5.5].

Given an A-module M we shall usually denote the corresponding quasi-coherentA|U -module by A|U ⊗AM .

The following definition is due to Silver [Si, p. 47].

Definition 3.5. Let A be a ring. An A-ring A′ is called a localization of A if A′

is a flat A-module on both sides, and if the multiplication map A′ ⊗A A′ → A′ isbijective.

Example 3.6. Let A be a ring and S ⊂ A a (left and right) denominator set.The ring of fractions AS of A with respect to S is the prototypical example of alocalization of A. For reference we call such a localization an Ore localization.

We remind the reader that a denominator set S is a multiplicatively closed subsetof A satisfying the left and right Ore conditions and the left and right torsionconditions (see [MR, Section 2.1]). The left Ore condition is that for all a ∈ A ands ∈ S there exist a′ ∈ A and s′ ∈ S such that as′ = a′s. The left torsion conditionis

a ∈ A | as = 0 for some s ∈ S ⊂ a ∈ A | sa = 0 for some s ∈ S.

The right Ore and torsion conditions for A are the respective left conditions forAop.

Not all localizations are Ore, as we see in Example 3.9Here is a list of some nice descent properties enjoyed by localization, that are

proved in [Si, Section 1].

Lemma 3.7. Let A be a ring and let A′ be a localization of A.

(1) For any A′-module M ′ the multiplication A′ ⊗AM ′ →M ′ is bijective.

(2) Let M ′ be an A′-module and M ⊂M ′ an A-submodule. Then the multipli-

cation A′ ⊗AM →M ′ is injective.


(3) Let M be an A-module and φ :M → A′⊗AM the homomorphism φ(m) :=1 ⊗ m. Then for any A′-submodule N ′ ⊂ A′ ⊗A M the multiplication

A′ ⊗A φ−1(N ′)→ N ′ is bijective.

(4) In the situation of part (3) the A-submodule φ(M) ⊂ A′⊗AM is essential.

(5) Localization of a left noetherian ring is left noetherian.

Proposition 3.8. Let (X,A) be a quasi-coherent ringed scheme, and let V ⊂ Ube affine open sets in X. Define A := Γ(U,A) and A′ := Γ(V,A). Then A′ is

a localization of A. Furthermore, if M is a quasi-coherent A-module then the

multiplication map

A′ ⊗A Γ(U,M)→ Γ(V,M)

is bijective.

Proof. Define C := Γ(U,OX) and C′ := Γ(V,OX). We first show that C′ is alocalization of C, namely that φ : C → C′ is flat and ψ : C′⊗CC′ → C′ is bijective.This can be checked locally on V . Choose an affine open covering V =

⋃i Vi with

Vi = SpecCsi for suitable elements si ∈ C. We note that Csi∼= C′

si∼= Γ(Vi,OX)

for all i. Hence restricting φ and ψ to Vi, namely applying Csi ⊗C − to them, weobtain bijections.

LetM be any quasi-coherent A-module. By [Ha, Proposition 5.1] multiplication

C′ ⊗C Γ(U,M)→ Γ(V,M)

is bijective.Because A is a quasi-coherent left and right OX -module, the previous formula

implies that C′⊗CA→ A′ and A⊗C C′ → A′ are both bijective. In addition, sincewe now know that

A′ ⊗A Γ(U,M) ∼= C′ ⊗C Γ(U,M),

we may conclude thatA′ ⊗A Γ(U,M)→ Γ(V,M)

is bijective.Finally we have a sequence of isomorphisms, all compatible with the multiplica-

tion homomorphisms into A′:

A′ ⊗A A′ ∼= A′ ⊗A (A⊗C C

′) ∼= A′ ⊗C C′

∼= (A⊗C C′)⊗C C

′ ∼= A⊗C (C′ ⊗C C′)

∼= A⊗C C′ ∼= A′.

Example 3.9. Let X be an elliptic curve over C and O ∈ X the zero element forthe group structure. Let P ∈ X be any non-torsion point. Define U := X−O andV := X − O,P, which are affine open sets, and C := Γ(U,OX), C′ := Γ(V,OX).By the previous proposition C → C′ is a localization. We claim this is not an Orelocalization. If it were then there would be some non-invertible nonzero functions ∈ C that becomes invertible in C′. Hence the divisor of s on X would be(s) = n(O−P ) for some positive integer n. In the group structure this would meanthat P is a torsion point, and this is a contradiction.

Definition 3.10. Let A be a ring, let M be an A-bimodule and let A′ be a local-ization of A. If the canonical homomorphisms

A′ ⊗AM → A′ ⊗AM ⊗A A′


and

M ⊗A A′ → A′ ⊗AM ⊗A A

′

are bijective then M is said to be evenly localizable to A′.

Trivially the bimodule M := A is evenly localizable to A′. The next lemma isalso easy and we omit its proof.

Lemma 3.11. Let M be an A-bimodule. Suppose A′ is a localization of A.

(1) If A′ = AS is an Ore localization of A for some denominator set S ⊂ A,and if sm = ms for all s ∈ S and m ∈ M , then M is evenly localizable to

A′.

(2) If there is a short exact sequence of A-bimodules 0 → L → M → N → 0with L and N evenly localizable to A′, then M is also evenly localizable to

A′.

(3) Suppose M ∼= limi→Mi for some directed system of A-bimodules Mi. If

each Mi is evenly localizable to A′ then so is M .

Proposition 3.12. Let C be a commutative ring and let M be a C-bimodule.

Define U := SpecC. The following conditions are equivalent.

(i) For any multiplicatively closed subset S ⊂ C, with localization CS , M is

evenly localizable to CS.(ii) M is evenly localizable to C′ := Γ(V,OU ) for every affine open set V ⊂ U .

(iii) There is a sheaf of OU -bimodules M, quasi-coherent on both sides, with

M ∼= Γ(U,M). SuchM is unique up to a unique isomorphism.

Proof. (i) ⇒ (ii): Let us write

φ : C′ ⊗C M → C′ ⊗C M ⊗C C′.

As in the proof of Proposition 3.8 we choose an affine open covering V =⋃i Vi with

Vi = SpecCsi and C′si∼= Csi . It suffices to show that the homomorphism φi gotten

by applying C′si ⊗C′ − to φ (localizing on the left) is bijective for all i. Using the

hypothesis (i) with S := slil∈N and the fact that Csi → C′si is bijective we get

C′si ⊗C′ (C′ ⊗C M) ∼= Csi ⊗C M ⊗C Csi

andC′si ⊗C′ (C′ ⊗C M ⊗C C

′) ∼= Csi ⊗C M ⊗C Csi ⊗C C′

∼= Csi ⊗C M ⊗C Csi .

So φi is bijective.Similarly one shows that

M ⊗C C′ → C′ ⊗C M ⊗C C

′

is bijective.

(ii) ⇒ (i): For any element s ∈ S let V := SpecCs ⊂ U . By assumption

Cs ⊗C M ∼= Cs ⊗C M ⊗C Cs ∼=M ⊗C Cs.

Taking direct limit over s ∈ S we get

CS ⊗C M ∼= CS ⊗C M ⊗C CS ∼=M ⊗C CS .

(ii) ⇒ (iii): LetM := OU ⊗C M be the sheafification of the (left) C-module M toU . By definitionM is a quasi-coherent left OU -module.


Given an affine open set V ⊂ U write C′ := Γ(V,OU ). By Proposition 3.8 themultiplication map C′⊗CM → Γ(V,M) is a bijection. Therefore Γ(V,M) ∼= C′⊗CM ⊗C C′. Since M is evenly localizable to C′ it follows that M ⊗C C′ → Γ(V,M)is also bijective. We conclude thatM is also a quasi-coherent right OU -module.

Regarding the uniqueness, suppose N is another OU -bimodule quasi-coherenton both sides such that Γ(U,N ) ∼= M as bimodules. For any affine open set V asabove we get an isomorphism of C′-bimodules

Γ(V,M) ∼= C′ ⊗C M ⊗C C′ ∼= Γ(V,N )

which is functorial in V . ThereforeM∼= N as OU -bimodules.

(iii) ⇒ (ii): Since M is quasi-coherent on both sides, for any affine open set V =SpecC′ we have

Γ(V,M) = C′ ⊗C M =M ⊗C C′,

so M is evenly localizable to C′.

The relation between even localization and Ore localization of a ring is explainedin the next theorem.

Theorem 3.13. Let C be a commutative ring, let A be a C-ring and S ⊂ C a

multiplicatively closed subset. Denote by CS the ring of fractions of C with respect

to S. Then the following two conditions are equivalent.

(i) The image S of S in A is a denominator set, with ring of fractions AS.(ii) The C-bimodule A is evenly localizable to CS .

When these conditions hold the multiplication map

CS ⊗C A⊗C CS → AS

is bijective.

Proof. (i) ⇒ (ii): Since AS is the left ring of fractions of A with respect to S (see[MR, Section 2.1.3]), it follows that the homomorphism CS⊗CA→ AS is bijective.On the other hand, since AS is also the right ring of fractions, A ⊗C CS → AS isbijective.

(ii) ⇒ (i): WriteQ := CS ⊗C A⊗C CS

andφ : A→ Q, φ(a) := 1⊗ a⊗ 1.

The assumption that A is evenly localizable to CS implies that

Ker(φ) = a ∈ A | as = 0 for some s ∈ S

= a ∈ A | sa = 0 for some s ∈ S,

verifying the torsion conditions.The even localization assumption also implies that given a1 ∈ A and s1 ∈ S

there are a2 ∈ A and s2 ∈ S such that

s−11 ⊗ a1 ⊗ 1 = 1⊗ a2 ⊗ s

−12 ∈ Q.

Multiplying this equation by s1 on the left and by s2 on the right we obtain 1 ⊗a1s2 ⊗ 1 = 1⊗ s1a2 ⊗ 1. Therefore φ(s1a2 − a1s2) = 0. Since Q ∼= A⊗C CS thereexists some s3 ∈ S such that (s1a2 − a1s2)s3 = 0 in A, i.e.

s1(a2s3) = a1(s2s3) ∈ A.


We have verified the right Ore condition. The left Ore condition is verified thesame way.

Remark 3.14. The theorem applies to any ring A and any commutative multi-plicatively closed subset S ⊂ A, since we can take C := Z[S] ⊂ A.

We will need a geometric interpretation of Theorem 3.13.

Corollary 3.15. Let C be a commutative ring, let U := SpecC and let A be a

C-ring. The following conditions are equivalent:

(i) For every multiplicatively closed set S ⊂ C the C-bimodule A is evenly

localizable to CS .(ii) For every multiplicatively closed set S ⊂ C its image S ⊂ A is a denomi-

nator set.

(iii) There is a quasi-coherent OU -ring A such that Γ(U,A) ∼= A as C-rings.

When these conditions hold the quasi-coherent OU -ring A is unique up to a unique

isomorphism.

Proof. (i) and (ii) are equivalent by Theorem 3.13. The implication (iii) ⇒ (i) is aspecial case of Proposition 3.12. It remains to show that (i) ⇒ (iii).

By Proposition 3.12 there is an OU -bimodule A, quasi-coherent on both sides,such that A ∼= Γ(U,A) as C-bimodules. The bimodule A is unique up to a uniqueisomorphism. Next by Theorem 3.13, for any s ∈ C, letting S := sii∈N, theimage S ⊂ A is a denominator set. Therefore on V := SpecCs we have canonicalisomorphisms

Γ(V,A) ∼= Cs ⊗C A⊗C Cs ∼= AS ,

where AS is the ring of fractions of A with respect to S. Hence A has a uniquestructure of quasi-coherent OX -ring.

In Example 4.12 we show a commutative ring C and a C-ring A which fail tosatisfy the conditions of the corollary.

4. Products of Quasi-Coherent Ringed Schemes

Suppose we are given two quasi-coherent ringed schemes (X,A) and (Y,B) overk. Let us denote by X × Y := X ×k Y the usual product of schemes, and byp1 : X × Y → X and p2 : X × Y → Y the projections. We obtain a few sheavesof kX×Y -algebras: p

−11 OX , p−1

2 OY , p−11 A and p−1

2 B, and there are canonical ringhomomorphisms between them:

p−11 OX ⊗ p−1

2 OY −−−−→ OX×Yy

p−11 A⊗ p−1

2 B,

where ⊗ := ⊗kX×Y .

Definition 4.1. Let (X,A) and (Y,B) be two quasi-coherent ringed schemes overk. Their product is a quasi-coherent ringed scheme (X × Y,A⊠ B), together witha ring homomorphism

φ : p−11 A⊗ p−1

2 B → A⊠ B,

satisfying the conditions below.


(i) The diagram

p−11 OX ⊗ p−1

2 OY −−−−→ OX×Yyy

p−11 A⊗ p−1

2 Bφ

−−−−→ A⊠ B

commutes.(ii) For every pair of affine open sets U ⊂ X and V ⊂ Y the homomorphism

Γ(U,A)⊗ Γ(V,B)→ Γ(U × V,A⊠ B)

induced by φ is bijective.

Observe that given a quasi-coherent OX -ring A the opposite ring Aop is also aquasi-coherent OX -ring.

Definition 4.2. Let (X,A) be a quasi-coherent ringed scheme over k. We denoteby

(X2,Ae) := (X ×X,A⊠Aop),

the product of the quasi-coherent ringed schemes (X,A) and (X,Aop).

Example 4.3. Let (X,A) be any quasi-coherent ringed scheme. Let B be a k-algebra, which we consider as a quasi-coherent ringed scheme (Y,B) := (Spec k, B).Then the product exists, and it is

(X × Y,A⊠ B) = (X,A⊗B).

The existence of products turns out to be more complicated in general, as wesee in the next Theorem.

Theorem 4.4. Let (X,A) and (Y,B) be quasi-coherent ringed schemes over k.

(1) A product (X × Y,A⊠ B) is unique up to isomorphism.

(2) The product (X × Y,A ⊠ B) exists if and only if the following condition

holds: for every pair of affine open sets U ⊂ X and V ⊂ Y , writing

C := Γ(U × V,OX×Y ), A := Γ(U,A) and B := Γ(V,B), the C-ring A ⊗Bis evenly localizable to CS for any multiplicatively closed set S ⊂ C.

Proof. Suppose a product A ⊠ B exists. Consider affine open sets U ⊂ X andV ⊂ Y . On the affine scheme U ×V = SpecC we have a quasi-coherent OU×V -ring(A⊠ B)|U×V , so by Corollary 3.15 the C-ring

Γ(U × V,A⊠ B) ∼= Γ(U,A)⊗ Γ(V,B)

is evenly localizable to CS for every multiplicatively closed set S ⊂ C. Thus the“necessary” half of part 2 is proved. According to Corollary 3.15 the quasi-coherentOU×V -ring (A⊠B)|U×V is unique up to a unique isomorphism. Since the open setsU × V cover X × Y it follows that A⊠ B is unique. So part 1 is also proved.

Finally suppose the condition in part 2 holds. The uniqueness in Corollary 3.15guarantees that the sheaves of OU×V -rings (A⊠B)|U×V can be glued on X×Y .

Definition 4.5. A morphism f : (Y,B)→ (X,A) of quasi-coherent ringed schemesover k is a morphism of schemes f : Y → X , together with a homomorphismf∗ : A → f∗B of OX -rings.


Proposition 4.6. Let (X1,A1) and (X2,A2) be quasi-coherent ringed schemes

over k, and assume the product (X1 × X2,A1 ⊠ A2) exists. Then the projections

pi : X1 ×X2 → Xi extend to morphisms

pi : (X1 ×X2,A1 ⊠A2)→ (Xi,Ai).

Proof. Let us look at the the first projection p1 (the second projection is done thesame way). We must produce an OX1 -ring homomorphism A1 → p1∗(A1⊠A2). Byadjunction this is the same as a (p−1

1 OX1)-ring homomorphism p−11 A1 → A1 ⊠A2.

However we already have ring homomorphisms

p−11 A1 → p−1

1 A1 ⊗ p−12 A2

φ−→ A1 ⊠A2,

and condition (i) of Definition 4.1 says these are (p−11 OX1)-ring homomorphisms.

Proposition 4.7. Let (X1,A1), (X2,A2), (Y1,B1) and (Y2,B2) be quasi-coherent

ringed schemes over k, and let fi : (Yi,Bi) → (Xi,Ai) be morphisms. Assume the

products (X1 ×X2,A1 ⊠A2) and (Y1 × Y2,B1 ⊠ B2) exist. Then there is a unique

morphism

f1 × f2 : (Y1 × Y2,B1 ⊠ B2)→ (X1 ×X2,A1 ⊠A2)

that is compatible with the projections and with the morphisms fi.

Proof. Let Ui ⊂ Xi and Vi ⊂ Yi be arbitrary affine open sets such that fi(Vi) ⊂ Ui.Condition (ii) says that

Γ(U1 × U2,A1 ⊠A2) ∼= Γ(U1,A1)⊗ Γ(U2,A2)

and

Γ(V1 × V2,B1 ⊠ B2) ∼= Γ(V1,B1)⊗ Γ(V2,B2).

Therefore we obtain a unique ring homomorphism

(4.8) Γ(U1 × U2,A1 ⊠A2)→ Γ(V1 × V2,B1 ⊠ B2)

that is compatible with the projections. The uniqueness implies that the formationof the homomorphisms (4.8) is compatible with restriction to smaller affine opensets. Hence we can glue the various homomorphisms (4.8) to a homomorphism ofsheaves

(f1 × f2)∗ : A1 ⊠A2 → (f1 × f2)∗(B1 ⊠ B2).

Let M be a k-module. By a filtration on M we mean an ascending filtrationF = FiMi∈Z by k-submodules. F is exhaustive ifM =

⋃i FiM , and it is bounded

below if Fi0−1M = 0 for some i0. More on filtrations in Section 9.

Definition 4.9. Let C be a commutative k-algebra and M a C-bimodule. Adifferential C-filtration onM is an exhaustive, bounded below filtration F = FiMwhere each FiM is a C-sub-bimodule, and grFM is a central C-bimodule. If Madmits some differential C-filtration then we call M a differential C-bimodule.

The name “differential filtration” signifies the similarity to Grothendieck’s defi-nition of differential operators; see [EGA IV].

Localization of a ring was defined in Definition 3.5, and even localization of abimodule was introduced in Definition 3.10.


Lemma 4.10. Let C be a commutative k-algebra and let M be a differential C-bimodule. If C′ is a localization of C, then M is evenly localizable to C′.

Proof. If M is a central C-bimodule then according to Lemma 3.11(1)M is evenlylocalizable to C′.

Now let M be a C-bimodule equipped with a differential C-filtration F . SayFi0−1M = 0. We prove by induction on i ≥ i0 that FiM is evenly localizable toC′. First Fi0M is central, so the above applies to it. For any i there is an exactsequence

0→ Fi−1M → FiM → grFi M → 0.

By the previous paragraph and by the induction hypothesis Fi−1M and grFi M areevenly localizable to C′. The flatness of C → C′ extends this to FiM , see Lemma3.11(2).

Proposition 4.11. Let C be a commutative k-algebra, let U := SpecC, let M be

a Ce-module and let M := OU2 ⊗Ce M , the quasi-coherent OU2-module associated

to M . Assume Ce is noetherian. Then the following conditions are equivalent:

(i) M is a differential C-bimodule.

(ii) M is supported on the diagonal ∆(U) ⊂ U2.

Proof. (i) ⇒ (ii): Denote by I := Ker(Ce։ C) and I := OU2 ⊗Ce I. So I is an

ideal defining the diagonal ∆(U). Suppose F = FiM is a differential C-filtrationof M , with Fi0−1M = 0. Then for all i ≥ i0 we have

Ii−i0+1 · FiM = 0.

It follows that the OU2 -module FiM := OU2 ⊗Ce FiM is supported on ∆(U). ButM =

⋃FiM.

(ii)⇒ (i): Let Mα be the set of coherent OU2 -submodules ofM, soM =⋃Mα.

Now Mα is a coherent OU2 -module supported on the diagonal ∆(U), so there issome integer iα ≥ 0 such that Iiα+1 · Mα = 0. It follows that the Ce-moduleMα := Γ(U2,Mα) satisfies I

iα+1 ·M = 0. And M =⋃Mα.

Define a filtration F on M by FiM := HomCe(Ce/Ii+1,M) for i ≥ 0, andF−1M := 0. Then Mα ⊂ FiαM , and this implies that M =

⋃FiM . Finally

I · FiM ⊂ Fi−1M , and hence grFi M is a central C-bimodule.

Here is a (somewhat artificial) example of a quasi-coherent ringed scheme whoseproduct with its opposite does not exist.

Example 4.12. Let C := Q[t] with t a variable, and let X := SpecC. TakeA := Q(t)[u;σ], an Ore extension of the field Q(t), where σ is the automorphismσ(t) = −t. Since every nonzero element s ∈ C is invertible in A, the C-ring A isevenly localizable to CS for any multiplicatively closed subset S ⊂ C. Hence thereis a quasi-coherent ringed scheme (X,A) with Γ(X,A) ∼= A as C-rings. (In fact Ais a constant sheaf on X .)

We claim that the product of (X,A) and (X,Aop) doesn’t exist. By Theorems4.4(2) and 3.13 it suffices to exhibit a multiplicatively closed subset S ⊂ Ce that isnot a denominator set in Ae. Consider the element s := t⊗ 1− 1⊗ t ∈ Ce and theset S := snn∈N. Let µ : Ae → A be the multiplication map µ(a1 ⊗ a2) := a1a2,which is a homomorphism of (left) Ae-modules, and denote by I the left idealKer(µ). Then Ae · s ⊂ I. On the other hand s(u ⊗ 1) = tu ⊗ 1 − u ⊗ t, soµ(s(u ⊗ 1)) = tu − ut = 2tu, and by induction µ(sn(u ⊗ 1)) = (2t)nu 6= 0 for


all n ≥ 0. We conclude that sn(u ⊗ 1) /∈ Ae · s, so S fails to satisfy the left Orecondition.

Example 4.13. The quasi-coherent ringed scheme (X,A) of the previous examplealso has the following peculiarity: the Ce-module A is not supported on the diagonal∆(X) ⊂ X2. Indeed, for every n ≥ 0 one has snu = (2t)nu 6= 0, so In · u 6= 0.

5. Dualizing Complexes over Ringed Schemes

Let (X,A) be a ringed space over k (cf. Section 2). We denote by ModA thecategory of left A-modules, and by D(ModA) its derived category.

The category ModA is abelian and it has enough injectives. Given an injectiveA-module I its restriction I|U to an open subset U is an injective A|U -module.Any complex N ∈ D+(ModA) has an injective resolution N → I, namely a quasi-isomorphism to a bounded below complex I of injective A-modules. This allows usto define the derived functor

RHomA(−,−) : D−(ModA)op × D+(ModA)→ D+(ModkX),

where kX is the constant sheaf k on X . The formula is

RHomA(M,N ) := HomA(M, I)

for any injective resolutionN → I. SinceHomA(M, I) is a bounded below complexof flasque sheaves it follows that

RHomA(M,N ) ∼= RΓ(X,RHomA(M,N )),

which is a functor

RHomA(−,−) : D−(ModA)op × D+(ModA)→ D+(Modk).

For more details regarding derived categories of sheaves see [RD], [KS] or [Bor].

Remark 5.1. One can of course remove some boundedness restrictions using K-injective resolutions, but we are not going to worry about this. Already there areenough delicate issues regarding injective resolutions of quasi-coherent A-modules;see Remark 5.3.

Assume now that (X,A) is a quasi-coherent ringed scheme over k. Recall thatQCohA denotes the full subcategory of ModA consisting of quasi-coherent A-modules. If furthermore (X,A) is left noetherian then we also consider the fullsubcategory CohA of coherent A-modules. There are corresponding full subcate-gories Dqc(ModA) and Dc(ModA) of D(ModA).

Proposition 5.2. Suppose (X,A) is a quasi-coherent ringed scheme.

(1) The category QCohA is a thick abelian subcategory of ModA, closed under

direct limits.

(2) If (X,A) is left noetherian then QCohA is a locally noetherian category,

whose noetherian objects are the coherent modules.

Proof. (1) This follows from Proposition 3.4.

(2) By Proposition 3.4 QCohA has exact direct limits, and also the coherent A-modules are the noetherian objects in QCohA. Given a quasi-coherent A-module


M let Lα be the set of its coherent OX -submodules. Since X is noetherian weknow thatM =

⋃α Lα; cf. [EGA I, Corollary 6.9.9]. Hence we get a surjection

⊕α(A⊗OX Lα)→M

in QCohA. But each A⊗OX Lα is a coherent A-module.

Remark 5.3. We do not know if an injective object I ∈ QCohA is also injectivein the bigger category ModA, nor if it is a flasque sheaf. Also we do not know if therestriction to an open set I|U is injective in QCohA|U . This is in contrast to thecommutative noetherian case A = OX in which the answers to all three questionsare positive, see [RD, Proposition II.7.17].

We now recall a theorem of Bernstein about equivalences of derived categoriesof A-modules (see [Bor, Theorem VI.2.10 and Proposition VI.2.11]).

Theorem 5.4. Let (X,A) be a quasi-coherent ringed scheme.

(1) The inclusion functor Db(QCohA)→ Dbqc(ModA) is an equivalence.

(2) If in addition A is left noetherian then the inclusion functor Db(CohA)→Db

c (ModA) is an equivalence.

Corollary 5.5. Assume (X,A) is a quasi-coherent ringed scheme and U ⊂ X is

an affine open set. Write A := Γ(U,A). Then

RΓ(U,−) : Dbqc(ModA|U )→ D

b(ModA)

is an equivalence with inverse M 7→ A|U ⊗AM . If A is left noetherian then we get

an equivalence

RΓ(U,−) : Dbc (ModA|U )→ Db

f (ModA).

Proof. By Proposition 3.4 any quasi-coherent OU -module is acyclic for the functorΓ(U,−). Hence if M ∈ Db(QCohA|U ), and if M → I is a quasi-isomorphismwith I a bounded below complex of injective A|U -modules (not necessarily quasi-coherent), then Γ(U,M) → Γ(U, I) is a quasi-isomorphism. It follows that thecomposed functor

Db(QCohA|U )→ Dbqc(ModA|U )

RΓ(U,−)−−−−−→ Db(ModA)

is an equivalence. Now use Theorem 5.4.

Lemma 5.6. Let (X,A) be a quasi-coherent ringed scheme and let V ⊂ U be two

affine open sets. Let A := Γ(U,A) and A′ := Γ(V,A). Then the diagram

D+qc(ModA|U )

RΓ(U,−)−−−−−−→ D+(ModA)

rest

y A′⊗A−

y

D+qc(ModA|V )

RΓ(V,−)−−−−−→ D+(ModA′)

is commutative.

Proof. GivenM ∈ D+qc(ModA|U ) take a resolutionM→ I where I is a bounded

below complex of injective A|U -modules. ThenM|V → I|V is an injective resolu-tion. We get a natural morphism

RΓ(U,M) = Γ(U, I)→ Γ(V, I) = RΓ(V,M),


and hence a morphism

A′ ⊗A RΓ(U,M)→ RΓ(V,M).

To show the latter is an isomorphism it suffices to check for a single quasi-coherentA|U -module M – since these are way-out functors, cf. [RD, Section I.7]. But forsuchM we have RΓ(U,M) = Γ(U,M) and RΓ(V,M) = Γ(V,M), so Proposition3.8 applies.

Lemma 5.7. Let (X,A) and (Y,B) be two quasi-coherent ringed schemes over k.

Assume the product (X × Y,A⊠ B) exists. Then:

(1) The homomorphisms of sheaves of rings

p−11 A → (p−1

1 A)⊗ (p−12 B)

and

(p−11 A)⊗ (p−1

2 B)→ A⊠ B

are flat on both sides.

(2) An injective (A⊠ B)-module is also an injective (p−11 A)-module.

(3) Given a (p−11 A)-moduleM and an injective (A⊠B)-module I, the (p−1

2 B)-module Homp−1

1 A(M, I) is injective.

These statements hold also if we exchange p−11 A with p−1

2 B, or p−11 A with p−1

1 Aop,

etc.

Proof. (1) Flatness can be checked on stalks at points of X ×Y . Take an arbitrarypoint z ∈ X × Y , and let x := p1(z) ∈ X and y := p2(z) ∈ Y . The stalks satisfy(p−1

1 A)z∼= Ax, (p

−12 B)z

∼= By and((p−1

1 A)⊗ (p−12 B)

)z∼= (p−1

1 A)z ⊗ (p−12 B)z

∼= Ax ⊗ By.

Hence the homomorphism

(p−11 A)z →

((p−1

1 A)⊗ (p−12 B)

)z

is flat on both sides.Choose affine open sets U ⊂ X and V ⊂ Y such that z ∈ U × V . By definition

Γ(U × V,A⊠ B) ∼= Γ(U,A)⊗ Γ(V,B).

Now the stalk Ax is an Ore localization of Γ(U,A) with respect to a suitable mul-tiplicatively closed set S ⊂ Γ(U,OX). Likewise the stalk By is an Ore localizationof Γ(V,B) with respect to a suitable multiplicatively closed set T ⊂ Γ(V,OY ). Onthe other hand the stalk (A ⊠ B)z is an Ore localization of Γ(U × V,A ⊠ B) withrespect to some multiplicatively closed set in Γ(U×V,OX×Y ) containing p∗1(S) andp∗2(T ). Hence (A⊠ B)z is an Ore localization of (Ax ⊗ By), so it’s flat.

(2) LetM be a (p−11 A)-module and I an injective (A⊠B)-module. The adjunction

formula for the ring homomorphism p−11 A → A⊠B (cf. [KS, Proposition 2.2.9] and

[Ro, Theorem 2.11])

Homp−11 A(M, I) ∼= HomA⊠B((A⊠ B)⊗p−1

1 AM, I)

together with the flatness from part (1) show that this is an exact functor ofM.


(3) Let N be any (p−12 B)-module. Then by adjunction (twice)

Homp−12 B

(N ,Homp−1

1 A(M, I))

∼= Hom(p−11 A⊗p−1

2 B)(M⊗N , I)

∼= HomA⊠B

((A⊠ B)⊗(p−1

1 A⊗p−12 B) (M⊗N ), I

)

and by part (1) this is an exact functor of N .

Lemma 5.8. Let (X,A) and (Y,B) be quasi-coherent ringed schemes over k. As-

sume that the product (X×Y,A⊠Bop) exists. Let R ∈ D+(Mod (A⊠Bop)) be some

complex.

(1) There are functors

D : Db(ModA)op → D(ModBop)

and

Dop : Db(ModBop)op → D(ModA)

defined by the formulas

DM := Rp2∗RHomp−11 A(p

−11 M,R)

and

DopN := Rp1∗RHomp−12 Bop(p

−12 N ,R).

(2) If HiDM = 0 for i ≫ 0 then DopDM is well defined, and there is a

morphismM→ DopDM in D(ModA), which is functorial inM. Similarly

for N and DDopN .

Proof. (1) Choose an injective resolution R→ I in C+(Mod (A⊠ Bop)). Since I isa complex of injective (p−1

1 A)-modules, we get

RHomp−11 A(p

−11 M,R) = Homp−1

1 A(p−11 M, I) ∈ D(Modp−1

2 Bop).

Because Homp−11 A(p

−11 M, I) is a bounded below complex of injective (p−1

2 Bop)-

modules (see Lemma 5.7(3)) we get

Rp2∗RHomp−11 A(p

−11 M,R) = p2∗Homp−1

1 A(p−11 M, I) ∈ D(ModBop).

Likewise for Dop.

(2) Let R→ I be as above. If HiDM = 0 for i≫ 0 then DM∼= L in D(ModBop),where L ∈ Db(ModBop) is a truncation of DM. So DopDM := DopL is welldefined. Adjunction gives a morphism

p−12 L

∼= p−12 p2∗Homp−1

1 A(p−11 M, I)→ Homp−1

1 A(p−11 M, I)

in D(Mod p−11 A). Applying Homp−1

2 Bop(−, I) and a second adjunction we get a

morphism

p−11 M→Homp−1

2 Bop

(Homp−1

1 A(p−11 M, I), I

)→ Homp−1

2 Bop(p−12 L, I).

Finally a third adjunction gives a morphism

M→ p1∗p−11 M→ p1∗Homp−1

2 Bop(p−12 L, I) = DopL.

All morphisms occurring above (including the truncation DM∼= L) are functorialinM.


Definition 5.9. Let (X,A) and (Y,B) be quasi-coherent ringed schemes over k.Assume A and Bop are left noetherian, and the product (X × Y,A ⊠ Bop) exists.A complex R ∈ Db

qc(Mod (A ⊠ Bop)) is called a dualizing complex over (A,B) ifconditions (i)-(iii) below hold for the functors D and Dop from Lemma 5.8(1).

(i) The functors D and Dop have finite cohomological dimensions when re-stricted to CohA and CohBop respectively.

(ii) The functor D sends CohA into Dc(ModBop), and the functor Dop sendsCohBop into Dc(ModA).

(iii) The adjunction morphisms 1 → DopD in Dbc (ModA), and 1 → DDop in

Dbc (ModBop), are both isomorphisms.

Note that conditions (i)-(ii) imply that there are well defined functors

D : Dbc (ModA)op → Db

c (ModBop)

and

Dop : Dbc (ModBop)op → D

bc (ModA).

By Lemma 5.8(2) we have the adjunction morphisms appearing in condition (iii).

Convention 5.10. When we talk about a dualizing complex R over (A,B) wetacitly assume that we are given a left noetherian quasi-coherent ringed k-scheme(X,A) and a right noetherian quasi-coherent ringed k-scheme (Y,B), and the prod-uct (X × Y,A⊠ Bop) exists.

We end this section with a couple of examples, that in fact digress from themain direction of our paper, yet are quite interesting on their own. The first is acontravariant Fourier-Mukai transform on elliptic curves.

Proposition 5.11. Let X be an elliptic curve over C and let R be the Poincare

bundle on X2. Then (X2,OX2) is the product of the quasi-coherent ringed scheme

(X,OX) with itself, and R ∈ Dbc (ModOX2) is a dualizing complex over (OX ,OX)

in the sense of Definition 5.9.

Proof. Conditions (i)-(ii) are clearly verified in this case. To check (iii) it sufficesto prove that the adjunction morphisms φ1 : OX → DopDOX and φ2 : OX →DDopOX are isomorphisms.

We begin by showing that φ1 and φ2 are nonzero. Since OX ⊠ OX = OX2 iscommutative we get

DOX = Rp2∗RHomp−11 OX

(p−11 OX ,R)

∼= Rp2∗RHomOX2 (p∗1OX ,R)

∼= Rp2∗R,

and soDopDOX ∼= DopRp2∗R

∼= Rp1∗RHomOX2 (p∗2Rp2∗R,R).

Therefore

HomD(ModOX )(OX ,DopDOX) ∼= H0

(X,Rp1∗RHomOX2 (p

∗2Rp2∗R,R)

)

∼= H0(X2,RHomOX2 (p

∗2Rp2∗R,R)

)

∼= HomOX2 (p∗2Rp2∗R,R)

∼= HomOX (Rp2∗R,Rp2∗R).


Under these isomorphisms the morphism φ1 corresponds to the identity 1 ∈HomOX (Rp2∗R,Rp2∗R), which is nonzero. Similarly φ2 is shown to be nonzero.

Next for anyM ∈ Dbc (ModOX) define

FM := Rp2∗(p∗1M⊗

LOX2R),

F opM := Rp1∗(p∗2M⊗

LOX2R)

and

EM := RHomOX (M,OX).

We haveDM = Rp2∗RHomOX2 (p

∗1M,R)

∼= Rp2∗((p∗1RHomOX (M,OX))⊗L

OX2R)

= FEM.

Likewise we get DopM∼= F opEM.Let O ∈ X be the zero for the group structure and Z := Ored. According to

[Mu] the Fourier-Mukai transform satisfies FOX ∼= F opOX ∼= OZ [−1] and FOZ ∼=F opOZ ∼= OX . Because EOX ∼= OX and EOZ ∼= OZ [−1] we obtain

DDopOX ∼= DopDOX ∼= OX .

Finally we use the facts that HomD(ModOX)(OX ,OX) ∼= k and that φ1 and φ2 arenonzero to deduce that φ1 and φ2 are isomorphisms.

The previous result can certainly be extended to higher dimensional abelianvarieties. The next example is based on a result of Beilinson [Be1], [Be2].

Proposition 5.12. Consider projective space X := Pnk. Let E :=

⊕ni=0OX(i) and

B := EndOX (E)op. Then E ∈ Mod (OX⊗Bop) is a dualizing complex over (OX , B).

Proof. One has

DM = Rp2∗RHomOX (M, E) ∼= RHomOX (M, E)

for everyM ∈ Dbc (ModOX). Since X is smooth and proper over k, this shows that

D satisfies conditions (i) and (ii).On the other hand

DopN = Rp1∗RHomp−12 Bop(p

−12 N, E)

for every N ∈ Dbf (ModBop). It is known that B is a finite k-algebra of global

dimension n. So any N ∈ Modf Bop has a resolution P → N with each P i a

finite projective Bop-module, and P i = 0 unless −n ≤ i ≤ 0. Because DopN ∼=Homp−1

2 Bop(p−12 P, E) we see that the functor Dop satisfies conditions (i) and (ii).

It remains to verify that the adjunctions M → DopDM and N → DDopN areisomorphisms.

Choose an injective resolution E → J over OX ⊗Bop. Then

DM∼= HomOX (M,J )

and

DopN ∼= Homp−12 Bop(p

−12 N,J ).


Because B and E generate the categories Dbf (ModBop) and Db

c (ModOX) re-spectively, it suffices to check that φ1 : E → DopDE and φ2 : B → DDopB areisomorphisms. Now the fact that ExtiOX

(E , E) = 0 for all i 6= 0 implies that

B → HomOX (J ,J )→ HomOX (E ,J )

are quasi-isomorphisms. Therefore

Eφ1−→ Homp−1

2 Bop

(p−12 HomOX (E ,J ),J

)

→ Homp−12 Bop(p

−12 B,J

)∼= J

and

Bφ2−→ HomOX

(Homp−1

2 Bop(p−12 B,J ),J

)

→ HomOX (J ,J )

are quasi-isomorphisms.

The crucial fact in Proposition 5.12 is that OX ,OX(1), . . . ,OX(n) is an excep-tional sequence in the sense of [BO]. This result could most likely be extended toother smooth complete varieties that admit exceptional sequences.

6. Localization of Dualizing Complexes

In this section we study the behavior of rigid dualizing complexes over rings withrespect to localization (cf. Definition 3.10).

Definition 6.1. Let A → A′ be a localization homomorphism between two noe-therian k-algebras. Suppose the rigid dualizing complexes (R, ρ) and (R′, ρ′) of Aand A′ respectively exist. A rigid localization morphism is a morphism

qA′/A : R→ R′

in D(ModAe) satisfying the conditions below.

(i) The morphisms A′ ⊗A R → R′ and R ⊗A A′ → R′ induced by qA′/A areisomorphisms.

(ii) The diagram

Rρ

−−−−→ RHomAe(A,R ⊗R)

q

yyq⊗ q

R′ ρ′

−−−−→ RHom(A′)e(A′, R′ ⊗ R′)

in D(ModAe) is commutative.

We shall sometimes express this by saying that qA′/A : (R, ρ) → (R′, ρ′) is a rigidlocalization morphism.

Here is a generalization of [YZ3, Theorem 3.8].

Theorem 6.2. Let A be a noetherian k-algebra and let A′ be a localization of A.Assume A has a dualizing complex R such that the cohomology bimodules HiR are

evenly localizable to A′. Then:

(1) The complex

R′ := A′ ⊗A R⊗A A′

is a dualizing complex over A′.


(2) If R is an Auslander dualizing complex over A then R′ is an Auslander

dualizing complex over A′.

(3) Suppose R is a rigid dualizing complex over A with rigidifying isomorphism

ρ, and Ae is noetherian. Then R′ is a rigid dualizing complex over A′.

Furthermore R′ has a unique rigidifying isomorphism ρ′ such that the mor-

phism qA′/A : R → R′ defined by r 7→ 1 ⊗ r ⊗ 1 is a rigid localization

morphism.

(4) In the situation of part (3) the rigid localization morphism qA′/A : (R, ρ)→(R′, ρ′) is unique.

Proof. (1) This follows essentially from the proof of [YZ2, Theorem 1.13]. ThereA was commutative and A′ the localization of A at some prime ideal, but thearguments are valid for an arbitrary localization A′.

(2) To check the Auslander property for R′ let M ′ be any finite A′-module. ByLemma 3.7(2) there is a finite A-module M such that M ′ ∼= A′ ⊗AM . For any i,[YZ3, Lemma 3.7(1)] implies that

ExtiA′(M ′, R′) ∼= ExtiA(M,R′) ∼= ExtiA(M,R)⊗A A′

as (A′)op-modules. Given any (A′)op-submodule N ′ ⊂ ExtiA′(M ′, R′), Lemma

3.7(3) tells us that there is an Aop-submodule N ⊂ ExtiA(M,R) such that N ′ ∼=N ⊗A A′. For such N we have

Extj(A′)op(N′, R′) ∼= A′ ⊗A ExtjAop(N,R)

which is 0 for all j < i. By symmetry we get the other half of the Auslanderproperty for R′.

(3) As in the proof of [YZ3, Theorem 3.8(2)] we have a canonical isomorphism

A′ ⊗A RHomAe(A,R⊗R)⊗A A′ ≃−→ RHom(A′)e(A

′, R′ ⊗R′)

in D(Mod (A′)e). This defines a rigidifying isomorphism ρ′ that respects qA′/A asdepicted in the diagram in Definition 6.1. Given any other morphism

ρ : R′ → RHom(A′)e(A′, R′ ⊗R′)

that renders the diagram commutative, applying the functor A′⊗A−⊗A A′ to thewhole diagram we deduce that ρ = ρ′.

(4) Write q1 := qA/A′ . Suppose q2 : (R, ρ) → (R′, ρ′) is another rigid localizationmorphism. Consider the commutative diagrams

Rρ

−−−−→ RHomAe(A,R ⊗R)

qi

yyqi⊗qi

R′ ρ′

−−−−→ RHom(A′)e(A′, R′ ⊗ R′)

in D(ModAe). Applying the base change functor (A′)e⊗Ae − to these diagrams weobtain diagrams

A′ ⊗A R⊗A A′ 1⊗ρ⊗1−−−−→ A′ ⊗A RHomAe(A,R⊗R)⊗A A′

1⊗qi⊗1

yy1⊗(qi⊗qi)⊗1

R′ ρ′

−−−−→ RHom(A′)e(A′, R′ ⊗R′)


consisting of isomorphisms in D(Mod (A′)e); cf. proof of [YZ3, Theorem 3.8(2)]. Weobtain an isomorphism τ : R′ → R′ such that

1⊗ q2 ⊗ 1 = τ (1⊗ q1 ⊗ 1) : A′ ⊗A R⊗A A′ → R′.

But then τ : (R′, ρ′)→ (R′, ρ′) is a rigid trace morphism. By [YZ3, Theorem 3.2] τhas to be the identity. This implies 1⊗q2⊗1 = 1⊗q1⊗1 and therefore q2 = q1.

The next proposition guarantees that under suitable assumptions the rigid tracelocalizes.

Proposition 6.3. LetA −−−−→ A′

yy

B −−−−→ B′

be a commutative diagram of k-algebras, where the horizontal arrows are localiza-

tions, the vertical arrows are finite, and the multiplication maps A′ ⊗A B → B′

and B ⊗A A′ → B′ are bijective. Assume A,A′, Ae, B,B′ and Be are all noether-

ian. Also assume the rigid dualizing complexes (RA, ρA) and (RB, ρB) exist, and so

does the rigid trace morphism TrB/A : RB → RA. By Theorem 6.2 the complexes

RA′ := A′⊗ARA⊗AA′ and RB′ := B′⊗B RB ⊗B B′ are rigid dualizing complexes

over A′ and B′ respectively, with induced rigidifying isomorphisms ρA′ and ρB′ .

Then the morphism

TrB′/A′ := 1⊗ TrB/A⊗1 : RB′ → RA′

is a rigid trace.

Proof. We begin by showing that the morphism ψ′ : RB′ → RHomA′(B′, RA′)induced by TrB′/A′ is an isomorphism. Let’s recall how ψ′ is defined: one chooses aquasi-isomorphismRA′ → I ′ where I ′ is a bounded below complex of injective (A′)e-modules. Then TrB′/A′ is represented by an actual homomorphism of complexesτ ′ : RB′ → I ′. The formula for ψ′ : RB′ → HomA′(B′, I ′) is ψ′(β′)(b′) = τ ′(b′β′)for β′ ∈ RB′ and b′ ∈ B′.

Let RA → I be a quasi-isomorphism where I is a bounded below complex ofinjective Ae-modules, and let τ : RB → I be a homomorphism of complexes repre-senting TrB/A. We know that the homomorphism ψ : RB → HomA(B, I) given bythe formula ψ(β)(b) = τ(bβ) is a quasi-isomorphism.

Since RA′∼= A′ ⊗A I ⊗A A′ there is a quasi-isomorphism A′ ⊗A I ⊗A A′ → I ′,

and using it we can assume that τ ′ = 1⊗ τ ⊗ 1 as morphisms

RB′ = A′ ⊗A RB ⊗A A′ → A′ ⊗A I ⊗A A

′ → I ′.

Thus we get a commutative diagram

RBψ

−−−−→ RHomA(B,RA)yy

RB′

ψ′

−−−−→ RHomA′(B′, RA′)

in D(ModAe). Applying the base change −⊗Ae (A′)e to the diagram we concludethat ψ′ = 1⊗ ψ ⊗ 1. So it is an isomorphism.

By symmetry RB′ → RHom(A′)op(B′, RA′) is also an isomorphism.


Next we have to show that the diagram

(6.4)

RB′

ρB′

−−−−→ RHom(B′)e(B′, RB′ ⊗RB′)

Tr

yyTr⊗Tr

RA′

ρA′

−−−−→ RHom(A′)e(A′, RA′ ⊗RA′)

is commutative. This is true since (6.4) gotten by applying − ⊗Ae (A′)e to thecommutative diagram

RBρB−−−−→ RHomBe(B,RB ⊗RB)

Tr

yyTr⊗Tr

RAρA

−−−−→ RHomAe(A,RA ⊗RA).

If a k-algebra A has an Auslander rigid dualizing complex R then we writeCdimA := CdimR;A for this preferred dimension function.

We finish this section with a digression from our main theme, to present thiscorollary to Theorem 6.2.

Corollary 6.5. Suppose chark = 0, A is the nth Weyl algebra over k and Dis its total ring of fractions, i.e. the nth Weyl division ring. Then D[2n] is an

Auslander rigid dualizing complex over D, and hence the canonical dimension of Dis CdimDD = 2n.

Proof. By [Ye3], A[2n] is a rigid Auslander dualizing complex over A. Now useTheorem 6.2 with A′ := D.

We see that unlike the Gelfand-Kirillov dimension GKdim, that cannot distin-guish between the various Weyl division rings (since GKdimD =∞), the canonicaldimension is an intrinsic invariant of D that does recover the number n. Moreoverthis fact can be expressed as a “classical” formula, namely

ExtiD⊗Dop(D,D ⊗D) ∼=

D if i = 2n,

0 otherwise.

7. Local Dualizing Complexes

We are mainly interested in dualizing complexes on ringed schemes that have alocal behavior – as opposed to, say, the dualizing complex occurring in Proposition5.11.

Definition 7.1. Let (X,A) and (Y,B) be separated quasi-coherent ringed schemesover k, and let R ∈ Db

qc(Mod (A ⊠ Bop)) be a dualizing complex over (A,B). If

the support of R (i.e. the union of the supports of the cohomology sheaves HiR)

is contained in the graph of an isomorphism of schemes X≃−→ Y then we call R a

local dualizing complex.

Lemma 7.2. Let (X,A) be a left noetherian quasi-coherent ringed scheme, U ⊂ Xan open set and M a coherent A|U -module. Then M extends to a coherent A-module.


Proof. Let g : U → X be the inclusion. The sheaf g∗M is a quasi-coherent OX -module, hence it is a quasi-coherent A-module. Also g∗M =

⋃α Lα where Lα

is the set of its coherent OX -submodules (cf. [Ha, Exercise II.5.25] or [EGA I,Corollary 6.9.9]). For any α the image

Nα := Im(A⊗OX Lα → g∗M)

is a coherent A-module. Now

M = (g∗M)|U =⋃

α

(Nα|U ),

and becauseM is a noetherian object we getM = Nα|U for some α.

Proposition 7.3. Let (X,A) and (Y,B) be separated quasi-coherent ringed schemes

over k. Assume that A and Bop are left noetherian and that the product (X × Y,A ⊠ Bop) exists. Let R ∈ Db

qc(Mod (A ⊠ Bop)) be a complex, and let X =⋃Ui be

any open covering of X. Then the following two conditions are equivalent.

(i) R is a local dualizing complex over (A,B).

(ii) R is supported on the graph of some isomorphism f : X≃−→ Y , and for

every i, defining Vi := f(Ui), the restriction

R|Ui×Vi ∈ Dbqc(Mod (A⊠ Bop)|Ui×Vi)

is a dualizing complex over (A|Ui ,B|Vi).

Proof. Assume R has support in the graph of an isomorphism f : X → Y . Weshall write p(1,i) : Ui × Vi → Ui and p(2,i) : Ui × Vi → Vi for the projections onthe open sets. Also we denote by Di and Dop

i the duality functors determined byR|Ui×Vi . For anyM ∈ Db(ModA) we have

(DM)|Vi = (Rp2∗RHomp−11 A(p

−11 M,R))|Vi

∼= Rp(2,i)∗(RHomp−11 A(p

−11 M,R)|X×Vi)

∼= Rp(2,i)∗(RHomp−11 A(p

−11 M,R)|Ui×Vi)

∼= Rp(2,i)∗(RHomp−1(1,i)

A(p−1(1,i)M|Ui ,R)|Ui×Vi)

= Di(M|Ui).

Likewise (DopN )|Ui∼= Dop

i (N|Vi ) for N ∈ Db(ModBop).Any coherent A-module restricts to a coherent A|Ui -module, and the same for

coherent Bop-modules. On the other hand by Lemma 7.2 any coherentA|Ui -moduleextends to a coherent A-module; and of course the same for coherent B|Vi-modules.The upshot is that the three conditions of Definition 5.9 are satisfied for R if andonly if they are satisfied for all the complexes R|Ui×Vi .

It is trivial that given k-algebras A and B, and viewing them as quasi-coherentringed schemes (Spec k, A) and (Spec k, B), then Definitions 2.1 and 5.9 becomeequivalent for complexes R ∈ Db(Mod (A ⊗Bop)). As the next theorem shows the“same” is true for all affine quasi-coherent ringed schemes.

Theorem 7.4. Let (X,A) and (Y,B) be affine quasi-coherent ringed schemes over

k. AssumeA and Bop are left noetherian and the product (X×Y,A⊠Bop) exists. LetR ∈ Db

qc(Mod (A⊠Bop)) be some complex. Define A := Γ(X,A) and B := Γ(Y,B).Then the following two conditions are equivalent.


(i) R is a dualizing complex over (A,B) in the sense of Definition 5.9.(ii) The complex

R := RΓ(X × Y,R) ∈ D(Mod (A⊗Bop))

is a dualizing complex over (A,B) in the sense of Definition 2.1.

Proof. Let us write Dglob := RHomA(−, R) and Dopglob := RHomBop(−, R). Accord-

ing to Corollary 5.5 the sheafification functors

A⊗A − : Dbf (ModA)→ Db

c (ModA)

and

−⊗B B : Dbf (ModBop)→ Db

c (ModBop)

are equivalences, with inverses RΓ(X,−) and RΓ(Y,−) respectively. Thus it sufficesto show that the diagram

Dbf (ModA)op

Dglob−−−−→ Db

f (ModBop)

A⊗A−

y RΓ(Y,−)

x

Dbc (ModA)op

D−−−−→ Db

c (ModBop)

and the “opposite” diagram (the one involving Dop and Dopglob) are commutative.

By symmetry it suffices to check only one of them, say the one displayed.We can assume that R = (A⊠ Bop)⊗(A⊗Bop) R. Choose an injective resolution

R→ J in C+(Mod (A⊗Bop)), and let J := (A⊠Bop)⊗(A⊗Bop)J . ThenR → J is aquasi-isomorphism. Now J is a complex of injectives in QCoh (A⊠Bop), but it mightnot be a complex of injectives in Mod (A⊠ Bop); cf. Remark 5.3. So we choose aninjective resolution J → K in C+(Mod (A⊠Bop)). LetM := A⊗AM ∈ Db

c (ModA).Then

DM = Rp2∗RHomp−11 A(p

−11 M,R)

∼= p2∗Homp−11 A(p

−11 M,K).

Since the latter is a complex of flasque Bop-modules on Y , we get

RΓ(Y,DM) = Γ(Y, p2∗Homp−1

1 A(p−11 M,K)

)

∼= Γ(X × Y,Homp−1

1 A(p−11 M,K)

)

∼= Homp−11 A(p

−11 M,K).

Now choose a bounded above resolution P → M by finite free A-modules, andlet P := A⊗AP . Then p−1

1 P → p−11 M is a quasi-isomorphism of (p−1

1 A)-modules,and so

Homp−11 A(p

−11 M,K)→ Homp−1

1 A(p−11 P ,K)

is a quasi-isomorphism of Bop-modules. Each J p and Kp is acyclic for the functorΓ(X × Y,−), since J p is quasi-coherent and Kp is injective. Therefore

Γ(X × Y,J )→ Γ(X × Y,K)

is a quasi-isomorphism. Now p−11 P is a bounded above complex of finite free

(p−11 A)-modules, and thus

Homp−11 A(p

−11 P ,J )→ Homp−1

1 A(p−11 P ,K)


is a quasi-isomorphism of Bop-modules. But

Homp−11 A(p

−11 P ,J )

∼= HomA(P, J) = RHomA(M,R) = DglobM.

Definition 7.5. Let (X,A) be a separated noetherian quasi-coherent ringed schemeover k. Suppose the product (X2,Ae) = (X × X,A ⊠ Aop) exists. If R ∈Db

qc(ModAe) is a dualizing complex over (A,Aop), then we say R is a dualizing

complex over A.

Let us denote by AffX the set of affine open subsets of X . Suppose R is alocal dualizing complex over A supported on the diagonal ∆(X) ⊂ X2. By Propo-sition 7.3 and Theorem 7.4, for every U ∈ AffX the complex RΓ(U2,R) is adualizing complex over A := Γ(U,A). If V ⊂ U is another affine open set andA′ := Γ(V,A) then A → A′ is a localization (see Proposition 3.8). Moreover therestriction Db

c (ModAe|U2 ) → Dbc (ModAe|V 2) induces a morphism qA′/A : R → R′

in D(ModAe); cf. Lemma 5.6.

Definition 7.6. Let (X,A) be a separated noetherian quasi-coherent ringed schemeover k. Assume the product (X2,Ae) exists, and is also noetherian. A rigid dual-

izing complex over A is a pair (R,ρ), where:

(1) R ∈ Dbc (ModAe) is a local dualizing complex over A supported on the

diagonal ∆(X) ⊂ X2.(2) ρ = ρUU∈AffX is a collection of rigidifying isomorphisms, namely for each

U ∈ AffX , letting A := Γ(U,A) and R := RΓ(U2,R), the pair (R, ρU ) is arigid dualizing complex over A (Definition 2.10).

The following compatibility condition is required of the data (R,ρ):

(∗) Given V ⊂ U in AffX , write A′ := Γ(V,A) and R′ := RΓ(V 2,R). LetqA′/A : R → R′ be the morphism in D(ModAe) coming from restriction.Then

qA′/A : (R, ρU )→ (R′, ρV )

is a rigid localization morphism (Definition 6.1).

Example 7.7. Let X be a separated finite type k-scheme, smooth of dimension n.There is a canonical isomorphism

ρ : ∆∗ΩnX/k

≃−→ ExtnOX2

(∆∗OX ,Ω2nX2/k),

and ExtiOX2(∆∗OX ,Ω2n

X2/k) = 0 for i 6= n (see [RD, Proposition III.7.2]). Therefore

on any affine open set U = SpecA ⊂ X we get an isomorphism

ρU : ΩnA/k[n]≃−→ RHomAe

(A,ΩnA/k[n]⊗ ΩnA/k[n]

)

in D(ModAe), and these ρU are compatible with localization. We see that(∆∗Ω

nA/k[n],ρ) is a rigid dualizing complex over X in the sense of Definition 7.6.

See Theorem 13.16 for more about rigid dualizing complexes over (commutative)schemes.

Definition 7.8. A morphism f : (Y,B) → (X,A) between noetherian quasi-coherent ringed schemes is called finite if f : Y → X is finite and f∗B is a coherentA-module on both sides.


Given a morphism f : (Y,B)→ (X,A) of ringed schemes then fop : (Y,Bop)→(X,Aop) is also a morphism. According to Proposition 4.7, if the products existthen there is a morphism

f e := f × fop : (Y 2,Be)→ (X2,Ae).

Definition 7.9. Let f : (Y,B)→ (X,A) be a finite morphism between noetherianseparated quasi-coherent ringed k-schemes. Assume both (X,A) and (Y,B) haverigid dualizing complexes (RA,ρA) and (RB,ρB) respectively. A rigid trace is amorphism

Trf : Rf e∗RB →RA

in D(ModAe) satisfying the following condition. Let U ⊂ X be any affine open set,V := f−1(U), A := Γ(U,A) and B := Γ(V,B). Let

RA := RΓ(U2,RA)

and

RB := RΓ(V 2,RB) ∼= RΓ(U2,Rf e∗RB).

Then

RΓ(U2,Trf ) : (RB , ρV )→ (RA, ρU )

is a rigid trace morphism (see Definition 2.11).

8. Perverse Modules and the Auslander Condition

In this section we discuss t-structures on the derived category Dbf (ModA). We

begin by recalling the definition of a t-structure and its basic properties, following[KS, Chapter X].

Definition 8.1. Suppose D is a triangulated category and D≤0,D≥0 are two full

subcategories. Let D≤n := D≤0[−n] and D

≥n := D≥0[−n]. We say (D≤0,D≥0) is a

t-structure on D if:

(i) D≤−1 ⊂ D

≤0 and D≥1 ⊂ D

≥0.

(ii) HomD(M,N) = 0 for M ∈ D≤0 and N ∈ D

≥1.(iii) For any M ∈ D there is a distinguished triangle

M ′ →M →M ′′ →M ′[1]

in D with M ′ ∈ D≤0 and M ′′ ∈ D

≥1.

When these conditions are satisfied we define the heart of D to be the full subcat-egory D

0 := D≤0 ∩D≥0.

Given a t-structure there are truncation functors τ≤n : D → D≤n and τ≥n :

D → D≥n, and functorial morphisms τ≤nM → M , M → τ≥nM and τ≥n+1M →

(τ≤nM)[1] such that

τ≤nM →M → τ≥n+1M → (τ≤nM)[1]

is a distinguished triangle in D. One shows that the heart D0 is an abelian category,and the functor

H0 := τ≤0τ≥0 ∼= τ≥0τ≤0 : D→ D0

is a cohomological functor.


Example 8.2. Let A be a left noetherian ring. The standard t-structure onDb

f (ModA) is

Dbf (ModA)≤0 := M ∈ Db

f (ModA) | HjM = 0 for all j > 0

and

Dbf (ModA)≥0 := M ∈ Db

f (ModA) | HjM = 0 for all j < 0.

For a complex

M = (· · · →Mn dn

−−→Mn+1 → · · · )

the truncations are

τ≤nM = (· · · →Mn−2 →Mn−1 → Ker(dn)→ 0→ · · · )

and

τ≥nM = (· · · → 0→ Coker(dn−1)→Mn+1 →Mn+2 → · · · ).

The heart Dbf (ModA)0 is equivalent to Modf A.

Other t-structures on Dbf (ModA) shall be referred to as perverse t-structures,

and the notation(pDb

f (ModA)≤0, pDbf (ModA)≥0

)shall be used. The letter “p”

stands for “perverse”, but often it will also signify a specific perversity function(see below).

Now suppose for i = 1, 2 we are given triangulated categories Di, endowed with t-

structures (D≤0i ,D≥0

i ). An exact functor F : D1 → D2 is called t-exact if F (D≤01 ) ⊂

D≤02 and F (D≥0

1 ) ⊂ D≥02 . The functor F : D0

1 → D02 between these abelian categories

is then exact. To apply this definition to a contravariant functor F we note that((D≥0)op, (D≤0)op) is a t-structure on the opposite category D

op. A contravariant

triangle functor F : D1 → D2 is called t-exact if F (D≤01 ) ⊂ D

≥02 and F (D≥0

1 ) ⊂ D≤02 .

Example 8.3. Let A be a left noetherian k-algebra and let B be a right noetheriank-algebra. Suppose we are given a dualizing complex R over (A,B), and let Dand Dop be the duality functors R induces; see Definitions 2.1 and 2.5. Put onDb

f (ModBop) the standard t-structure (see Example 8.2). Define subcategories

pDbf (ModA)≤0 := M ∈ Db

f (ModA) | DM ∈ Dbf (ModBop)≥0

andpDb

f (ModA)≥0 := M ∈ Dbf (ModA) | DM ∈ Db

f (ModBop)≤0.

Since D : Dbf (ModA)→ Db

f (ModBop) is a duality it follows that(pDb


)

is a t-structure on Dbf (ModA), which we call the perverse t-structure induced by R.

The functors D and Dop are t-exact, and

D : pDbf (ModA)0 → Db

f (ModBop)0 ≈ Modf Bop

is a duality of abelian categories.

Definition 8.4. Suppose A is a noetherian k-algebra with rigid dualizing complexRA. The perverse t-structure induced on Db

f (ModA) by RA is called the rigid

perverse t-structure. An object M ∈ pDbf (ModA)0 is called a perverse A-module.


Example 8.5. Suppose the triangulated category D admits a semi-orthogonal de-composition (D1,D2) with six functors

D1i∗,i∗,i

!

←−−−−−→ Dj!,j

∗,j∗←−−−−−→ D2

satisfying adjunction relations (cf. [BBD]). Given any t-structures on D1 and D2

they can be glued to a t-structure on D. See [Br], [BO] and [VdB2] for recentapplications of this procedure.

In the remainder of this section we concentrate on yet another method of produc-ing t-structures on Db

f (ModA). This method is of a geometric nature, and closelyresembles the t-structures that originally appeared in [BBD].

A perversity is a function p : Z→ Z satisfying p(i)− 1 ≤ p(i+1) ≤ p(i). We callthe function p(i) = 0 the trivial perversity, and the function p(i) = −i is called theminimal perversity.

Let A be a ring. Fix an exact dimension function dim on ModA (see Definition2.8). For an integer i let Mi(dim) be the full subcategory of ModA consisting ofthe modules M with dimM ≤ i. The subcategory Mi(dim) is localizing, and thereis a functor

ΓMi(dim) : ModA→ ModA

defined by

ΓMi(dim)M := m ∈M | dimAm ≤ i ⊂M.

The functor ΓMi(dim) has a derived functor

RΓMi(dim) : D+(ModA)→ D+(ModA)

calculated using injective resolutions. For M ∈ D+(ModA) the jth cohomology ofM with supports in Mi(dim) is defined to be

HjMi(dim)M := HjRΓMi(dim)M.

The definition above was introduced in [Ye2] and [YZ3]. It is based on the followinggeometric paradigm.

Example 8.6. If A is a commutative noetherian ring of finite Krull dimension andwe set dimM := dimSuppM for a finite module M , then

HjMi(dim)M

∼= lim→

HjZM

where Z runs over the closed sets in SpecA of dimension ≤ i.

Definition 8.7. Let A be a left noetherian ring. Given an exact dimension functiondim on ModA and a perversity p, define subcategories

pDbf (ModA)≤0 := M ∈ Db

f (ModA) | dimHjM < i for all i, j with j > p(i)

and

pDbf (ModA)≥0 := M ∈ Db

f (ModA) | HjMi(dim)M = 0 for all i, j with j < p(i)

of Dbf (ModA).

Example 8.8. Suppose dim is any exact dimension function such that dimM >−∞ for all M 6= 0. Take the trivial perversity p(i) = 0. Then

pD

bf (ModA)≤0 = D

bf (ModA)≤0


andpD

bf (ModA)≥0 = D

bf (ModA)≥0,

namely the standard t-structure on Dbf (ModA).

The following lemma is straightforward.

Lemma 8.9. In the situation of Definition 8.7 let p be the minimal perversity,

namely p(i) = −i, and let M ∈ Dbf (ModA). Then:

(1) M ∈ pDbf (ModA)≤0 if and only if dimH−iM ≤ i for all i.

(2) M ∈ pDbf (ModA)≥0 if and only if Hj

Mi(dim)M = 0 for all j < −i, if and

only if RΓMi(dim)M ∈ Dbf (ModA)≥−i for all i.

Recall that if R is an Auslander dualizing complex over the rings (A,B) thenthe canonical dimension CdimR (Definition 2.9) is an exact dimension function onModA.

Theorem 8.10. Let A be a left noetherian k-algebra and B a right noetherian k-

algebra. Suppose R is an Auslander dualizing complex over (A,B). Let dim be the

canonical dimension function CdimR on ModA, and let p be the minimal perversity

p(i) = −i. Then:

(1) The pair (pD

bf (ModA)≤0, pDb

f (ModA)≥0)

from Definition 8.7 is a t-structure on Dbf (ModA).

(2) Put on Dbf (ModBop) the standard t-structure. Then the duality functors D

and Dop determined by R (see Definition 2.5) are t-exact.

Proof. In the proof we shall use the abbreviations D(A) := Dbf (ModA) etc.

If M ∈ pD(A)≤−1 then M [−1] ∈ pD(A)≤0. By Lemma 8.9(1) we getdimH−i(M [−1]) ≤ i. Changing indices we get dimH−iM ≤ i − 1 ≤ i. Againusing Lemma 8.9(1) we see that the first part of condition (i) of Definition 8.1 isverified. The second part of condition (i) is verified similarly using Lemma 8.9(2).

By the Auslander condition dimH−j R = dimExt−jBop(B,R) ≤ j for all j. There-fore according to Lemma 8.9(1) we get R ∈ pD(A)≤0. On the other hand since

HjMi(dim)R

∼= lim−→a∈F

ExtjA(A/a, R),

where F is the Gabriel filter of left ideals corresponding to Mi(dim), the Auslandercondition and Lemma 8.9(2) imply that R ∈ pD(A)≥0.

Let M ′ →M → M ′′ → M ′[1] be a distinguished triangle in D(A). Since dim isexact, and using the criterion in Lemma 8.9(1), we see that if M ′ and M ′′ are inpD(A)≤0 then so is M . Likewise applying the functor Hj

Mi(dim) to this triangle and

using Lemma 8.9(2) it follows that if M ′ and M ′′ are in pD(A)≥0 then so is M .Suppose we are given M ∈ pD(A)≤0, N ∈ pD(A)≥1 and a morphism φ : M →

N . In order to prove that φ = 0 we first assume M is a single finite module,concentrated in some degree −l, with l ≥ 0 and dimM ≤ l. Then φ factors

through Mφ′

−→ RΓMl(dim)N → N . Now M ∈ D(A)≤−l and, by Lemma 8.9(2),

RΓMl(dim)N ∈ D(A)≥−l+1; hence φ′ = 0. Next let us consider the general case. Let

H−lM be the lowest nonzero cohomology of M . We have a distinguished triangle

T :=((H−lM)[l]→M →M ′′ → (H−lM)[l + 1]

)


where M ′′ is the standard truncation of M . According to Lemma 8.9(1) we have

dimH−lM ≤ l, so by the previous argument the composition (H−lM)[l]→Mφ−→ N

is zero. So applying HomD(ModA)(−, N) to the triangle T we conclude that φ comes

from some morphism φ′′ :M ′′ → N . Since M ′′ ∈ pD(A)≤0 and by induction on thenumber of nonvanishing cohomologies we have φ′′ = 0. Therefore condition (ii) isverified.

Next suppose M ∈ D(Bop)≤0. In order to prove that DopM ∈ pD(A)≥0 we canassume M is a single finite Bop-module, concentrated in degree −l for some l ≥ 0.By [YZ3, Proposition 5.2] and its proof we deduce

RΓMi(dim)DopM = RΓMi(dim) RHomBop(M,R)

∼= RHomBop(M,RΓMi(dim)R).

As we saw above, the Auslander condition implies that RΓMi(dim)R ∈ D(Bop)≥−i.

But M ∈ D(Bop)≤−l, and hence

RHomBop(M,RΓMi(dim)R) ∈ D(A)≥l−i ⊂ D(A)≥−i.

Now the criterion in Lemma 8.9(2) tells us that DopM ∈ pD(A)≥0.Let M ∈ D(Bop)≥0. We wish to prove that DopM ∈ pD(A)≤0. To do so we

can assume M is a single finite module, concentrated in some degree l ≥ 0. Thenfor every i, H−iDopM = Ext−iBop(M,R) has dimH−iDopM ≤ i. Now apply Lemma8.9(1).

At this point we know that Dop(D(Bop)≤0) ⊂ pD(A)≥0 and Dop(D(Bop)≥0) ⊂pD(A)≤0. Let M ∈ D(A) be an arbitrary complex, and consider the distinguishedtriangle

τ≤−1DM → DM → τ≥0DM → (τ≤−1DM)[1]

in D(Bop) gotten from the standard t-structure there. Applying Dop we obtain adistinguished triangle

M ′ →M →M ′′ →M ′[1]

in D(A), where M ′ := Dopτ≥0DM and M ′′ := Dopτ≤−1DM . This proves thatcondition (iii) is fulfilled, so we have a t-structure on D(A), and also the functorDop is t-exact.

To finish the proof we invoke [KS, Corollary 10.1.18] which tells us that D is alsot-exact.

Remark 8.11. The idea for Definition 8.7 comes from [KS, p. 438 Exercise X.2].Strictly speaking Theorem 8.10 is not needed for the main result of our paper(namely Theorem 13.6), since we use the rigid perverse t-structure in its proof.Nonetheless we have included Theorem 8.10 because we think it provides an in-teresting linkage between t-structures and the Auslander property (cf. Theorem13.19). Also see the following question.

Question 8.12. Let A be a left noetherian ring. Find necessary and sufficientconditions on a dimension function dim on ModA, and on a perversity function p,such that part (1) of Theorem 8.10 holds.

9. Perverse Coherent Sheaves on Ringed Schemes

In this section we study perverse coherent sheaves on noetherian quasi-coherentringed schemes.


Definition 9.1. Let (X,A) be a left noetherian, quasi-coherent ringed scheme overk. Suppose that for every affine open set U ⊂ X we are given a t-structure

TU =(pDb

c (ModA|U )≤0, pDb

c (ModA|U )≥0

)

on Dbc (ModA|U ). Furthermore suppose this collection of t-structures TUU∈AffX

satisfies the following condition.

(loc) Let U =⋃Ui be any affine open covering, and let “⋆” denote either

“≤ 0” or “≥ 0”. Then for anyM ∈ Dbc (ModA|U ) the following are equiv-

alent:(i) M∈ pDb

c (ModA|U )⋆.(ii) M|Ui ∈

pDbc (ModA|Ui)

⋆ for all i.

Then we call TUU∈AffX a local collection of t-structures on (X,A).

Definition 9.2. Let (X,A) be a ringed scheme and let TUU∈AffX be a localcollection of t-structures on it. Define full subcategories

pD

bc (ModA)≤0 := M ∈ D

bc (ModA) | M|U ∈

pD

bc (ModA|U )

≤0

for every affine open set U ⊂ X

and

pDbc (ModA)≥0 := M ∈ Db

c (ModA) | M|U ∈pDb

c (ModA|U )≥0

for every affine open set U ⊂ X.

The next lemmas are modifications of material in [KS, Section 10.2]. They referto a quasi-coherent ringed scheme (X,A) endowed with a a local collection of t-structures TUU∈AffX .

Lemma 9.3. LetM ∈ pDbc (ModA)≤0 and N ∈ pDb

c (ModA)≥0.

(1) The sheaf HiRHomA(M,N ) vanishes for all i ≤ −1.(2) The assignment

U 7→ HomD(ModA|U )(M|U ,N|U ),

for open sets U ⊂ X, is a sheaf on X.

(3) HomD(ModA)(M,N [i]) = 0 for all i ≤ −1.

Proof. (1) We note that

HiRHomA(M,N ) ∼= H0RHomA(M,N [i]),

and N [i] ∈ pDbc (ModA)≥1, because i ≤ −1.

Now H0RHomA(M,N [i]) is isomorphic to the sheaf associated to the presheaf

U 7→ H0 RHomA|U (M|U ,N [i]|U ) ∼= HomD(ModA|U )(M|U ,N [i]|U ).

Thus it suffices to prove that

(9.4) HomD(ModA|U )(M|U ,N [i]|U ) = 0

for all affine open sets U . Since by Definition 9.2 we haveM|U ∈ pDbc (ModA|U )≤0

and N [i]|U ∈ pDbc (ModA|U )≥1, the assertion follows from condition (ii) in Defini-

tion 8.1.


(2) Let us write L := RHomA(M,N ) ∈ D(ModkX). By part (1) we know thatHiL = 0 for all i < 0. So after truncation we can assume Li = 0 for all i < 0.Hence H0RΓ(U,L) ∼= Γ(U,H0L). But

H0RΓ(U,L) ∼= H0 RHomA|U (M|U ,N|U )

∼= HomD(ModA|U )(M|U ,N|U ).

We see that the presheaf

U 7→ HomD(ModA|U )(M|U ,N|U )

is actually a sheaf, namely the sheaf H0L.

(3) Here N [i] ∈ pDbc (ModA)≥1 ⊂ pDb

c (ModA)≥0, so equation (9.4) holds forall affine open sets. Applying part (2) of the lemma to M and N [i] we getHomD(ModA)(M,N [i]) = 0.

Lemma 9.5. Suppose the two rows in the diagram below are distinguished triangles

with M′1 ∈

pDbc (ModA)≤0 and M′′

2 ∈pDb

c (ModA)≥1. Given any morphism g :M1 → M2 there exist unique morphisms f : M′

1 → M′2 and h : M′′

1 → M′′2

making the diagram commutative. If g is an isomorphism then so are f and h.

M′1

α1−−−−→ M1β1

−−−−→ M′′1

γ1−−−−→ M′

1[1]

f

y g

y h

y f [1]

y

M′2

α2−−−−→ M2β2

−−−−→ M′′2

γ2−−−−→ M′

2[1]

Proof. By assumption we haveM′1 ∈

pDbc (ModA)≤0 and M′′

2 [i] ∈pDb

c (ModA)≥1

for all i ≤ 0. So by to Lemma 9.3(3) we get HomD(ModA)(M′1,M

′′2 [i]) = 0 for

i ≤ 0. According to [BBD, Proposition 1.1.9] there exist unique morphisms f andh making the diagram commutative. The assertion about isomorphisms is also in[BBD, Proposition 1.1.9].

If V ⊂ X is any open subset then TUU∈Aff V is a local collection of t-structureson the ringed scheme (V,A|V ); hence using Definition 9.2 we get subcategoriespDb

c (ModA|V )⋆.

Lemma 9.6. Let V ⊂ X be an open set, and let V =⋃j Vj be some open covering

(where V and Vj are not necessarily affine). Write “⋆” for either “≤ 0” or “≥ 0”.Then the following are equivalent for M ∈ Db

c (ModA|V ).

(i) M ∈ pDbc (ModA|V )⋆.

(ii) M|Vj ∈pDb

f (ModA|Vj )⋆ for all j.

Proof. The implication (i) ⇒ (ii) is immediate from Definition 9.2. For the reverseimplication use condition (loc) of Definition 9.1 twice with suitable affine opencoverings of V and of Vj .

Lemma 9.7. Let U ⊂ X be an open set, and let U =⋃ni=1 Ui be some covering by

open sets (U and Ui not necessarily affine). Let M ∈ Dbc (ModA|U ), and suppose

there are distinguished triangles

(9.8) Ti =(M′

iαi−→M|Ui

βi−→M′′

iγi−→M′

i[1])


in Dbc (ModA|Ui) withM

′i ∈

pDbc (ModA|Ui)

≤0 andM′′i ∈

pDbc (ModA|Ui)

≥1. Then

there exists a distinguished triangle

M′ α−→M

β−→M′′ γ

−→M′[1]

in Dbc (ModA|U ) whose restriction to Ui is isomorphic to Ti.

Proof. First assume n = 2. Denote by gi : Ui → U the inclusions, for i = 1, 2. Alsowrite U(1,2) := U1 ∩ U2 and g(1,2) : U(1,2) → U . For any open immersion g let g! beextension by zero, which is an exact functor.

The restriction of the triangles T1 and T2 to U(1,2) and Lemma 9.5 give rise to an

isomorphism f :M′1|U(1,2)

≃−→M′

2|U(1,2)in Db

c (ModA|U(1,2)) satisfying α2 f = α1.

Therefore we get a morphism

δ : g(1,2)!(M′1|U(1,2)

)→ g1!M′1 ⊕ g2!M

′2

in Db(ModA|U ), whose components are extensions by zero of the identity and frespectively. (Note that the complexes g1!M′

1 etc. might not have coherent coho-mologies.)

DefineM′ to be the cone of δ. So there is a distinguished triangle

g(1,2)!(M′1|U(1,2)

)δ−→ g1!M

′1 ⊕ g2!M

′2 →M

′ → (g(1,2)!(M′1|U(1,2)

))[1]

in D(ModA|U ). Applying the cohomological functor HomD(ModA|U )(−,M|U ) tothis triangle we get an exact sequence

HomD(ModA|U )(M′,M|U )→ HomD(ModA|U )(g1!M

′1 ⊕ g2!M

′2,M|U )

→ HomD(ModA|U )

(g(1,2)!(M

′1|U(1,2)

),M|U).

The pair (α1,−α2) in the middle term goes to zero, and hence it comes from somemorphism α :M′ → M|U . By construction the restriction M′|Ui

∼=M′i, so with

the help of Lemma 9.6 we deduce thatM′ ∈ pDbc (ModA|U )

≤0. Also the restriction

ofM′ α−→M|U to Ui isM′

iαi−→M|Ui .

DefineM′′ to be the cone of α. So we have a distinguished triangle

T =(M′ α−→M|U

β−→M′′ γ

−→M′[1])

in Db(ModA|U ). By Lemma 9.5 there is a (unique) isomorphism M′′|Ui

≃−→

M′′i such that T |Ui

≃−→ Ti is an isomorphism of triangles. Therefore M′′|Ui ∈

pDbc (ModA|Ui)

≥1. Using Lemma 9.6 we see thatM′′ ∈ pDbc (ModA|U )≥1.

When n > 2 the statement follows from induction and the case n = 2.

Theorem 9.9. Let (X,A) be a left noetherian, quasi-coherent ringed scheme with

a local collection of t-structures TUU∈AffX . Then the pair(pD

bc (ModA)≤0, pDb

c (ModA)≥0)

from Definition 9.2 is a t-structure on Dbc (ModA).

Proof. Condition (i) of Definition 8.1 is trivially verified.Let M ∈ pDb

c (ModA)≤0 and N ∈ pDbc (ModA)≥1. So N [1] ∈ pDb

c (ModA)≥0,and by Lemma 9.3(3) with i = −1 we have

HomD(ModA)(M,N ) = 0.

This verifies condition (ii).


It remains to prove condition (iii). Let a complex M ∈ Dbc (ModA) be given.

Choose a covering X =⋃ni=1 Ui by affine open sets. For every i the t-structure TUi

gives rise to a distinguished triangle

Si =(M′

iαi−→M|Ui

βi−→M′′

iγi−→M′

i[1])

in Dbc (ModA|Ui) with M′

i ∈pDb

c (ModA|Ui)≤0 and M′′

i ∈pDb

c (ModA|Ui)≥1. By

Lemma 9.7 there is a triangle

M′ α−→M

β−→M′′ γ

−→M[1]

in Dbc (ModA) whose restriction to each Ui is isomorphic to Si. Therefore by Lemma

9.6 one hasM′ ∈ pDbc (ModA)≤0 andM′′ ∈ pDb

c (ModA)≥1.

The next definition if based on [KS, Section X.10].

Definition 9.10. Let X be a topological space. By a stack on X we mean a “sheafof categories” C. To be precise, for every open set U ⊂ X there is a category C(U),with restriction functors C(U) → C(V ), M 7→ M|V for every inclusion V ⊂ U .The following axioms are required.

(a) Given an open covering U =⋃Vi, objectsMi ∈ C(Vi) and isomorphisms

φi,j :Mi|Vi∩Vj

≃−→Mj |Vi∩Vj

satisfying the cocycle condition

φi,k = φj,k φi,j :Mi|Vi∩Vj∩Vk

≃−→Mk|Vi∩Vj∩Vk

,

there exists an objectM ∈ C(U) and isomorphisms φi :M|Vi

≃−→Mi such

that

φi,j φi = φj :M|Vi∩Vj

≃−→Mj |Vi∩Vj .

(b) Given two objects M,N ∈ C(U), an open covering U =⋃Vi and mor-

phisms ψi :M|Vi → N|Vi such that ψi|Vi∩Vj = ψj |Vi∩Vj , there is a uniquemorphism ψ :M→ N such that ψ|Vi = ψi.

Theorem 9.11. Let (X,A) be a left noetherian, quasi-coherent ringed scheme

with a local collection of t-structures TUU∈AffX . For any open set V ⊂ X letpDb

c (ModA|V )0 be the heart of the t-structure from Theorem 9.9. Then V 7→pDb

c (ModA|V )0 is a stack on X.

Proof. Axiom (b) is Lemma 9.3(2). Let us prove axiom (a). Suppose we are givenopen sets V =

⋃i∈I Vi ⊂ X , complexes Mi ∈ pDb

c (ModA|Vi)0 and isomorphisms

φi,j :Mi|Vi∩Vj →Mj|Vi∩Vj satisfying the cocycle condition. Since X is noetherian

we may assume I = 1, . . . , n. Let us define Wi :=⋃ij=1 Vj . By induction on i we

will construct an object Ni ∈ pDbc (ModA|Wi)

0 with isomorphisms ψi,j : Ni|Vj

≃−→

Mj for all j ≤ i that are compatible with the φj,k. Then M := Nn will be thedesired global object on V =Wn.

So assume i < n and Ni has already been defined. For any j ≤ i we have anisomorphism

φj,i+1 ψi,j : Ni|Vj∩Vi+1

≃−→Mj |Vj∩Vi+1

≃−→Mi+1|Vj∩Vi+1 ,


and these satisfy the cocycle condition. According to Lemma 9.3(2) there is anisomorphism

ψi,i+1 : Ni|Wi∩Vi+1

≃−→Mi+1|Wi∩Vi+1

in pDbc (ModA|Wi∩Vi+1)

0. Denote by fi+1 : Wi → Wi+1, gi+1 : Vi+1 → Wi+1 andhi+1 : Wi ∩ Vi+1 → Wi+1 the inclusions. Define Ni+1 ∈ D(ModA|Wi+1) to be thecone on the morphism

h(i+1)!(Ni|Wi∩Vi+1)(γ,ψi,i+1)−−−−−−→ f(i+1)!Ni ⊕ g(i+1)!Mi+1

where γ is the canonical morphism. We obtain a distinguished triangle

h(i+1)!(Ni|Wi∩Vi+1)→ f(i+1)!Ni ⊕ g(i+1)!Mi+1 → Ni+1 → h(i+1)!(Ni|Wi∩Vi+1)[1]

in D(ModA|Wi+1). Upon restriction to Wi we get an isomorphism Ni ∼= Ni+1|Wi ;

and upon restriction to Vi+1 we get an isomorphism Ni+1|Vi+1

≃−→Mi+1 which we

call ψi+1,i+1. Finally from Lemma 9.6 we see that Ni+1 ∈ pDbc (ModA|Wi+1)

0.

Remark 9.12. Let (X,A) be a ringed space. Suppose X =∐Si is a finite

stratification by locally closed subsets. For each i let fi : Si → X be the inclusionand A|Si := f−1

i A. Suppose we are given a t-structure on every D+(ModA|Si). Thegluing procedure of [BBD] (see Example 8.5) gives a t-structure on D+(ModA).However it seems this procedure does not apply to coherent sheaves, because thetruncation functors τ≤0 and τ≥1 do not preserve quasi-coherence.

The next theorem shows how rigid dualizing complexes on affine open set caninduce a local collection of t-structures.

Theorem 9.13. Let (X,A) be a noetherian quasi-coherent ringed scheme over k.

Assume that for every affine open set U ⊂ X the following three conditions hold.

(i) The k-algebra A := Γ(U,A) has a rigid dualizing complex RA.(ii) Let C := Γ(U,OX). For each i the cohomology bimodule HiRA, when con-

sidered as Ce-module, is supported on the diagonal ∆(U) ⊂ U2 = SpecCe.

(iii) The enveloping algebra Ae is noetherian.

Let (pDb


)

be the rigid perverse t-structure on Dbf (ModA); see Definition 8.4. By the equiva-

lence Dbf (ModA) ≈ Db

c (ModA|U ) in Corollary 5.5 we get a t-structure

TU =(pDb

c (ModA|U )≤0, pDb

c (ModA|U )≥0

)

on Dbc (ModA|U ). Then TUU∈AffX is a local collection of t-structures.

Proof. Let U ⊂ X be any affine open set and let U =⋃Uk be any affine open

covering. In the notation of Definition 9.1 let A := Γ(U,A), M := RΓ(U,M),Ak := Γ(Uk,A) and Mk := RΓ(Uk,M). We must show that M ∈ pDb

f (ModA)⋆ ifand only if Mk ∈ pDb

f (ModAk)⋆ for all k.

Denote by RA and RAkthe rigid dualizing complexes of the rings A and Ak

respectively. By Proposition 3.8 Ak is a localization of A. According to Proposition4.11 and Lemma 4.10 each A-bimodule HiRA is evenly localizable to Ak. HenceTheorem 6.2 tells us that RAk

∼= RA ⊗A Ak in D(Mod (A ⊗ Aopk )), and likewise

RAk∼= Ak ⊗A RA in D(Mod (Ak ⊗A

op)).


Define complexes N := RHomA(M,RA) and Nk := RHomAk(Mk, RAk

). Ac-cording to Lemma 5.6 we have Mk

∼= Ak ⊗AkMk. By [YZ3, Lemma 3.7]

N ⊗A Ak = RHomA(M,RA)⊗A Ak∼= RHomA(M,RA ⊗A Ak)

∼= RHomA(M,RAk)

∼= RHomAk(Ak ⊗AM,RAk

)

∼= RHomAk(Mk, RAk

) = Nk.

Let us consider the case where “⋆” is “≥”. SupposeM ∈ pDbf (ModA)≥0. By def-

inition of the perverse rigid t-structure we have N ∈ Dbf (ModAop)≤0, i.e. HjN = 0

for all j > 0. Hence HjNk ∼= (HjN)⊗AAk = 0, implying thatMk ∈ Dbf (ModAk)

≥0.Conversely suppose Mk ∈ pDb

f (ModAk)≥0 for all k. We may assume that U =⋃n

k=1 Uk. Let C := Γ(U,OX) and Ck := Γ(Uk,OX). The ring homomorphismC →

∏nk=1 Ck is faithfully flat (cf. Proposition 3.8 and [Ma, p. 28 Theorem 3]).

Applying A⊗C − it follows that A→∏nk=1 Ak is faithfully flat (on both sides), so

we get an injection

HjN →⊕

k(HjN)⊗A Ak.

As above we conclude that N ∈ Dbf (ModAop)≤0, and hence M ∈ pDb

f (ModA)≥0.The case where “⋆” is “≤ 0” is handled similarly. The only difference is that we

have to verify the vanishing of HjN and HjNk for j < 0.

Definition 9.14. Suppose the conditions in Theorem 9.13 hold for (X,A). Theresulting t-structure

(pDbc (ModA)≤0, pDb

c (ModA)≥0)

on Dbc (ModA) is called the rigid perverse t-structure. An objectM ∈ pDb

c (ModA)0

is called a perverse coherent A-module.

Example 9.15. If X is a finite type k-scheme and A = OX then conditions (i)-(iii)in Theorem 9.13 hold (cf. Example 2.12). Hence Db

c (ModOX) has the rigid perverset-structure.

10. Filtrations of Rings

In this section we establish notation to be used in Section 11, and also prove sev-eral auxiliary results. The reader may want to consult [MR] or [LV] for backgroundon filtrations.

By a filtration (or Z-filtration) of a k-algebra A we mean an ascending filtrationF = FiAi∈Z by k-submodules such that 1 ∈ F0A and FiA · FjA ⊂ Fi+jA. Weshall call (A,F ) a filtered k-algebra; often we shall just say that A is a filteredalgebra and leave F implicit.

Suppose (A,F ) is a filtered k-algebra. Given an A-module M , by an (A,F )-filtration of M we mean an ascending filtration F = FiMi∈Z of M by k-submod-ules such that FiA · FjM ⊂ Fi+jM for all i and j. We call (M,F ) a filtered(A,F )-module, and allow ourselves to drop reference to F when no confusion mayarise.

We say the filtration F on M is exhaustive if M =⋃i FiM , F is separated

if 0 =⋂i FiM , F is bounded below if Fi0−1A = 0 for some integer i0, and F is

nonnegative if F−1A = 0. The trivial filtration on M is F−1M := 0, F0M :=M .


Let us recall some facts about associated graded modules, and establish somenotation. It shall be convenient to use the ordered semigroup Z ∪ −∞ where−∞ < i for every i ∈ Z, and i+ j := −∞ if either i = −∞ or j = −∞.

Let (M,F ) be an exhaustive filtered module. The associated graded module is

gr (M,F ) = grFM =⊕

i∈ZgrFi M :=

⊕i∈Z

FiM

Fi−1M.

Given an element m ∈M the F -degree of m is

degF (m) := infi | m ∈ FiM ∈ Z ∪ −∞.

The F -symbol of m is

symbF (m) := m+ Fi−1M ∈ grFi M

if i = degF (m) ∈ Z; and symbF (m) := 0 if degF (m) = −∞. Thus the (nonzero)homogeneous elements of grFM are the symbols.

Recall that the product on the graded algebra grFA is defined on symbols asfollows. Given elements a1, a2 ∈ A let di := degF (ai) and ai := symbF (ai). If bothdi > −∞ then

a1 · a2 := a1 · a2 + Fd1+d2−1A ∈ grFd1+d2A.

Otherwise a1 · a2 := 0. Similarly one defines a graded (grFA)-module structure ona filtered module M .

If A =⊕

i∈ZAi is a graded algebra then A is also filtered, where

FiA :=⊕

j≤iAj .

The filtration F is exhaustive and separated. Moreover A ∼= grFA as graded alge-bras. The isomorphism sends a ∈ Ai to its symbol symbF (a) ∈ grFi A.

Lemma 10.1. Suppose the k-algebra A is generated by elements a1, . . . , an. Given

numbers d1, . . . , dn ∈ N there is a unique nonnegative exhaustive filtration F =FiA such that:

(i) For every d, FdA is the k-linear span of the products aj1 · · · ajm such that

dj1 + · · ·+ djm ≤ d.(ii) The graded algebra grFA is generated by n elements a1, . . . , an, where either

ai is homogeneous of degree di, and is the F -symbol of ai; or ai = 0.

Proof. Let x1, . . . , xn be indeterminates and let k〈x〉 = k〈x1, . . . , xn〉 be the freeassociative algebra on these generators. Define φ : k〈x〉 → A to be the surjectionsending xi 7→ ai. Put on k〈x〉 the Z-grading such that deg(xi) = di. This inducesa filtration F = Fik〈x〉i∈Z where

Fik〈x〉 :=⊕

j≤ik〈x〉j .

This filtration can now be transferred to A by setting FiA := φ(Fik〈x〉). Clearly(A,F ) is exhaustive and nonnegative, and also condition (i) holds. This conditionalso guarantees uniqueness.

As for condition (ii) consider the surjective graded algebra homomorphism

grF (φ) : grFk〈x〉 → grFA.

Because of the way the filtration on k〈x〉 was constructed the graded algebra

grFk〈x〉 is a free algebra on the symbols xi := symbF (xi). Define ai := grF (φ)(xi).These elements have the required properties.


Conversely we have the following two lemmas, whose standard proofs we leaveout.

Lemma 10.2. Let F = FiA be an exhaustive nonnegative filtration of A, and

let a1, . . . , an ∈ A be some elements. Denote by ai := symbF (ai). Suppose that

a1, . . . , an generate grFA as k-algebra. Then:

(1) A is generated by a1, . . . , an as k-algebra.

(2) Let di := max0, degF (ai). Then F coincides with the filtration from

Lemma 10.1.

Lemma 10.3. Let (A,F ) be a nonnegative exhaustive filtered k-algebra and let

(M,F ) be a bounded below exhaustive filtered (A,F )-module. Suppose ai, bi ⊂A and ci ⊂M are sets satisfying:

(i) The set of symbols ai ∪ bi generates grFA as k-algebra.

(ii) The set of symbols ci generates grFM as (grFA)-module.

(iii) For every i, j the symbols ai and bj commute.

Then for every integer d the k-module FdM is generated by the set of products

ai1 · · · aikbj1 · · · bjlcm | degF (ai1) + · · ·+ degF (bj1) + · · ·+ degF (cm) ≤ d.

The base field k is of course trivially filtered. The filtered k-modules (M,F )form an additive category FiltModk, in which a morphism φ : (M,F ) → (N,F ) isa k-linear homomorphism φ : M → N such that φ(FiM) ⊂ FiN .

The Rees module of (M,F ) is

Rees (M,F ) = ReesFM :=⊕

i∈Z

FiM · ti ⊂M [t] =M ⊗k k[t]

where t is a central indeterminate of degree 1. We get an additive functor

Rees : FiltModk→ GrModk[t]

where GrMod k[t] is the abelian category of Z-graded k[t]-modules and degree 0homomorphisms.

For a scalar λ ∈ k let us denote by spλ the specialization to λ of a k[t]-module

M , namely

spλ M := M/(t− λ)M.

If λ 6= 0 then

spλ : GrMod k[t]→ Modk

is an exact functor, since spλ M is isomorphic to the degree zero component of the

localization Mt. For λ = 0 we get a functor

sp0 : GrMod k[t]→ GrMod k.

Given any (M,F ) ∈ FiltModk one has

sp0 Rees (M,F ) ∼= gr (M,F ) = grFM.

On the other hand, given a graded k[t]-module M there is a filtration F on M :=

sp1 M defined by

FiM := Im(⊕

j≤iMj →M

).

This is a functor

sp1 : GrMod k[t]→ FiltModk.


If (M,F ) is exhaustive then

sp1 Rees (M,F ) ∼= (M,F ).

For a graded module M ∈ GrMod k[t] we have

Rees sp1 M∼= M/t-torsion.

If A is a filtered k-algebra then A := ReesA and A := grA are graded alge-bras, and we obtain corresponding functors Rees, sp1 and sp0 between FiltModA,

GrMod A and GrMod A.The next lemma says that a filtration can be lifted to the Rees ring.

Lemma 10.4. Let F be an exhaustive nonnegative filtration of the k-algebra A,

and let A := ReesFA ⊂ A[t]. For any integer i define

FiA :=⊕

j∈N(Fmin(i,j)A) · t

j ∈ A[t].

Then:

(1) A =⋃i FiA, F−1A = 0 and FiA · FjA ⊂ Fi+jA. Thus F = FiA is an

exhaustive nonnegative filtration of the algebra A.(2) There is an isomorphism of k-algebras

grF A ∼= (grFA)⊗ k[t]

(not respecting degrees).

Proof. (1) Since F−1A = 0 we get F−1A = 0. Let Aj := (FjA) · tj , the jth graded

component of A. Then for any i, j there is equality

Aj ∩ (FiA) = (Fmin(i,j)A) · tj .

Hence Aj ⊂ FjA. It remains to check the products. For any two pairs of numbers(i, k) and (j, l) one has

min(i, k) + min(j, l) ≤ min(i + j, k + l).

Therefore((Fmin(i,k)A) · t

k)·((Fmin(j,l)A) · t

l)⊂

((Fmin(i+j,k+l)A) · t

k+l).

This says that FiA · FjA ⊂ Fi+jA.

(2) We have isomorphisms

grF A ∼=⊕

0≤i

⊕i≤j

(grFi A) · tj

and

(grFA)⊗ k[t] ∼=⊕

0≤i

⊕0≤j

(grFi A) · tj .

The isomorphism grF A≃−→ (grFA) ⊗ k[t] we want is defined on every summand

(grFi A) · tj by dividing by ti.


11. Differential k-Algebras of Finite Type

In this section we introduce a class of algebras for which Auslander rigid dualizingcomplexes exist.

Let C be a ring. Recall that a C-ring is a ring A together with a ring homomor-phism C → A, called the structural homomorphism. If C → A is centralizing thenA is a C-algebra. Observe that a C-ring is also a C-bimodule.

Definition 11.1. Suppose C is a commutative k-algebra and A is a C-ring. Adifferential C-filtration on A is a filtration F = FiAi∈Z with the following prop-erties:

(i) 1 ∈ F0A and FiA · FjA ⊂ Fi+jA.(ii) F−1A = 0 and A =

⋃FiA.

(iii) Each FiA is a C-sub-bimodule.(iv) The graded ring grFA is a C-algebra.

A is called a differential C-ring if it admits some differential C-filtration.

Note that properties (i) and (iii) imply that the image of the structural homo-morphism C → A lies in F0A, so that (iv) makes sense.

Definition 11.2. Let C be a commutative noetherian k-algebra and A a C-ring.A differential C-filtration of finite type on A is a differential C-filtration F = FiAsuch that the graded C-algebra grFA is a finite module over its center Z(grFA),and Z(grFA) is a finitely generated C-algebra. We say A is a differential C-ring of

finite type if it admits some differential C-filtration of finite type.

If C is a finitely generated commutative k-algebra and A is a differential C-ringof finite type then A is also a differential k-ring of finite type. In this case we alsocall A a differential k-algebra of finite type.

Proposition 11.3. Let C be a finitely generated commutative k-algebra, A a dif-

ferential C-ring of finite type, s ∈ C and S := sii∈N. Then:

(1) The image S of S in A is a denominator set.

(2) Let Cs and As be the Ore localizations w.r.t. S. Then Cs is is a finitely

generated k-algebra and As is a differential Cs-ring of finite type.

Proof. (1) Use Lemma 4.10 and Theorem 3.13.

(2) Suppose F = FiA is a differential C-filtration of finite type. Then setting

FiAs := Cs ⊗C (FiA)⊗C Cs ⊂ As

we obtain a filtration F of As such that grFAs ∼= Cs ⊗C grFA as graded Cs-algebras.

Remark 11.4. The ideas in [KL, Theorem 4.9] can be used to show the following.In the setup of the previous proposition let M be a finite A-module and M :=Im(M → As ⊗AM). Then

GKdimA M = GKdimAs(As ⊗AM).

Example 11.5. Let C be a finitely generated commutative k-algebra and A a finiteC-algebra. Then A is a differential C-ring of finite type. As filtration we can takethe trivial filtration F−1A := 0 and F0A := A.


Example 11.6. Suppose chark = 0, C is a smooth commutative k-algebra andA := D(C) is the ring of k-linear differential operators. Then A is a differentialC-ring of finite type. For filtration we can take the filtration F = FiA by order ofoperator, in which F−1A := 0, F0A := C, F1A := C ⊕ TC and Fi+1A := FiA · F1Afor i ≥ 1. Here TC := Derk(C), the module of derivations.

Example 11.7. A special case of Example 11.6 is when C := k[x1, . . . , xn], apolynomial ring. Then A is called the nth Weyl algebra. Writing yi :=

∂∂xi

thealgebra A is generated by the 2n elements x1, . . . , xn, y1, . . . , yn, with relations[xi, xj ] = [yi, yj ] = [yi, xj ] = 0 for i 6= j and [yi, xi] = 1. In addition to the filtrationF above there is also a differential k-filtration of finite type G = GiA whereG−1A := 0, G0A := k, G1A := k+(

∑i k ·xi)+ (

∑i k ·yi) and Gi+1A := GiA ·G1A

for i ≥ 1.

Lemma 11.8 ([ATV, Theorem 8.2]). Suppose A =⊕

i∈NAi is a graded k-algebra

and t ∈ A is a central homogeneous element of positive degree. The following are

equivalent:

(i) A is left noetherian.

(ii) A/(t) is left noetherian.

The next lemma follows almost directly from the definition.

Lemma 11.9. If A is a differential k-algebra of finite type then it is a noetherian

finitely generated k-algebra.

The class of differential k-algebras of finite type is closed under tensor productsas we now show.

Proposition 11.10. Let C1 and C2 be noetherian commutative k-algebras. Let Aibe a differential Ci-ring of finite type for i = 1, 2. Then A1 ⊗ A2 is a differential

(C1 ⊗ C2)-ring of finite type.

Proof. Choose differential filtrations of finite type FnA1 and FnA2 of A1 andA2. Define a filtration on A1 ⊗A2 as follows:

Fn(A1 ⊗A2) :=∑

l+m=n

FlA1 ⊗ FmA2.

ThengrF (A1 ⊗A2) ∼= (grFA1)⊗ (grFA2)

as graded rings. Since grFA1 and grFA2 are finite modules over their centers itfollows that (grFA1)⊗ (grFA2) is a finite module over its center.

The next theorem generalizes the case of the nth Weyl algebra and its two filtra-tions (see Examples 11.6 and 11.7 above). McConnell and Stafford also consideredsuch filtrations, and our result extends their [MS, Corollary 1.7]. The basic idea isattributed in [MS] to Bernstein.

We recall that a graded k-algebra A is called connected if A =⊕

i∈NAi, A0 = k

and each Ai is a finite k-module.

Theorem 11.11. Let A be a k-algebra. Assume A has a differential k-filtration

of finite type F = FiAi∈Z. Then there is a nonnegative exhaustive k-filtration

G = GiAi∈Z such that grGA is a commutative, finitely generated, connected

graded k-algebra.


Observe that G is also a differential filtration of finite type on A.The following easy lemma will be used often in the proof of the theorem.

Lemma 11.12. Let F = FiA be a nonnegative exhaustive filtration of A and let

a1, a2 ∈ A be two elements. Define ai := symbF (ai) ∈ grFA and di := degF (ai) ∈N ∪ −∞. Then the commutator [a1, a2] = 0 if and only if

degF ([a1, a2]) ≤ d1 + d2 − 1.

Proof of Theorem 11.11. Step 1. Write A := grFA. Then the center Z(A) is agraded, finitely generated, commutative k-algebra, and A is a finite Z(A)-module.

For any element a ∈ A we write a := symbF (a) ∈ grFA.Let d1 ∈ N be large enough such that Z(A) is generated as Z(A)0-algebra by

finitely many elements of degrees ≤ d1, and A is generated as Z(A)-module byfinitely many elements of degrees ≤ d1.

Choose nonzero elements a1, . . . , am ∈ F0A ∼= A0 that their symbols a1, . . . , amare in Z(A)0, and they generate Z(A)0 as k-algebra. Next choose elements b1, . . . , bn∈ Fd1A−F0A such that the symbols b1, . . . , bn are in Z(A), and they generate Z(A)as Z(A)0-algebra. Finally choose nonzero elements c1, . . . , cp ∈ Fd1A such that the

symbols c1, . . . , cp generate⊕d1

j=0 Aj as Z(A)0-module. This implies that c1, . . . , cpgenerate A as a Z(A)-module.

The symbols a1, . . . , am, b1, . . . , bn, c1, . . . , cp generate A as k-algebra, so byLemma 10.2 the elements a1, . . . , cm, b1, . . . , bn, c1, . . . , cp generate A as k-algebra.Let

k〈x,y, z〉 := 〈x1, . . . , xm, y1, . . . , yn, z1, . . . , zp〉

be the free associative algebra, and define a surjective ring homomorphism φ :k〈x,y, z〉 → A by sending xi 7→ ai, yi 7→ bi and zi 7→ ci. We are now in thesituation of Lemma 10.2. The free algebra k〈x,y, z〉 also has a filtration F , where

degF (xi) := degF (ai), degF (yi) := degF (bi) etc., and φ is a strict surjection, mean-

ing that Fi(A) = φ(Fik〈x,y, z〉).Let us denote substitution by f(a, b, c) := φ(f(x,y, z)). Consider the subrings

k〈a〉 ⊂ k〈a, b〉 ⊂ A = k〈a, b, c〉

with filtrations F induced by the inclusions into A. Warning: these filtrations mightdiffer from the filtrations induced by φ : k〈x〉 ։ k〈a〉 and φ : k〈x,y〉 ։ k〈a, b〉respectively.

We observe that the commutators [ai, aj ] = 0 for all i, j, since ai = ai ∈ Z(A)0.This also says that [ai, bj ] = 0, so according to Lemma 11.12 we get

[ai, bj ] ∈ FdegF (bj)−1A

for all i, j. Therefore by Lemma 10.3, applied to the filtered k-algebra k〈a〉 andthe filtered k〈a〉-module Fd1A, we see that there are noncommutative polynomialsf1i,j,k(x) ∈ k〈x〉 such that

(11.13)

degF (f1i,j,k(x)) + degF (ck) ≤ degF (bj)− 1

and

[ai, bj ] =∑

k

f1i,j,k(a) · ck.


Note that either f1i,j,k(x) 6= 0, in which case degF (f1

i,j,k(x)) = 0; or f1i,j,k(x) = 0

and then degF (f1i,j,k(x)) = −∞. The choice f1

i,j,k(x) = 0 is of course required when

degF (ck) ≥ degF (bj).Likewise [bi, bj] ∈ FdegF (bi)+degF (bj)−1A, so by Lemma 10.3, applied to the fil-

tered k-algebra k〈a, b〉 and the filtered k〈a, b〉-module A, we see that there arenoncommutative polynomials f2

i,j,k(x) and g2i,j,k(y) such that

(11.14)

degF (f2i,j,k(x)) + degF (g2i,j,k(y)) + degF (ck) ≤ degF (bi) + degF (bj)− 1

and

[bi, bj] =∑

k

f2i,j,k(a) · g

2i,j,k(b) · ck.

Similarly there are polynomials f3i,j,k(x) such that

degF (f3i,j,k(x)) + degF (ck) ≤ degF (cj)− 1

and

[ai, cj ] =∑

k

f3i,j,k(a) · ck,

and there are polynomials f4i,j,k(x) and g

4i,j,k(y) such that

degF (f4i,j,k(x)) + degF (g4i,j,k(y)) + degF (ck) ≤ degF (bi) + degF (cj)− 1

and

[bi, cj ] =∑

k

f4i,j,k(a) · g

4i,j,k(b) · ck.

The same idea applies to cicj : there are polynomials f5i,j,k(x) and g

5i,j,k(y) such

that

degF (f5i,j,k(x)) + degF (g5i,j,k(y)) + degF (ck) ≤ degF (ci) + degF (cj)

and

ci · cj =∑

k

f5i,j,k(a) · g

5i,j,k(b) · ck.

Let G be the standard grading on k〈x〉, namely degG(xi) := 1. This induces afiltration G. Define

e0 := max0, degG(f li,j,k(x)),

e1 := e0 + 1 and e2 := e0 + e1 + 1.Put on the free algebra k〈x,y, z〉 a new grading G by declaring

degG(yi) := e2 degF (bi)

degG(zi) := e2 degF (ci) + e1,

and keeping degG(xi) = 1 as above. We get a new filtration G on k〈x,y, z〉. Usingthis we obtain a new filtration G on A with

GiA := φ(Gik〈x,y, z〉).

Step 2. Now we verify that the filtration G has the required properties. Sincethe filtration G on k〈x,y, z〉 is nonnegative exhaustive, and φ is a surjection, itfollows that the filtration G on A is also nonnegative exhaustive. The rest requires


some work, and in order to simplify our notation we are going to “recycle” theexpressions A, ai etc. From here on we define A := grGA. We have a surjectivegraded k-algebra homomorphism

φ := grG(φ) : grGk〈x,y, z〉 → grGA = A.

Let xi := symbG(xi), yi := symbG(yi), etc. Then

grGk〈x,y, z〉 = k〈x, y, z〉 := k〈x1, . . . , xm, y1, . . . , yn, z1, . . . , zp〉

which is also a free algebra. Define ai := φ(xi), bi := φ(yi) and ci := φ(zi). Observe

that either degG(ai) = degG(xi), in which case ai = symbG(ai), and is a nonzero

element of AdegG(ai); or degG(ai) < degG(xi), and then ai = 0. Similar statements

hold for bi and ci.Since φ is surjective we see that A is generated as k-algebra by the elements

a1, . . . , am, b1, . . . , bn, c1, . . . , cp. These elements are either of positive degree or are0, and hence A is connected graded. We claim that A is commutative.

We know already that [ai, aj ] = 0. Let us check that [ai, bj ] = 0. If either

degG(ai) < degG(xi) or degG(bj) < degG(yj) then aibj = bjai = 0. Otherwise

ai = symbG(ai) and bj = symbG(bj). By formula (11.13) we have

degG([ai, bj ]) ≤ maxdegG(f1i,j,k(a)) + degG(ck)

≤ maxdegG(f1i,j,k(x)) + degG(ck).

For any k such that f1i,j,k(x) 6= 0 we have degF (ck) ≤ degF (bj)− 1, and then

degG(ck) ≤ degG(zk) = e2 degF (ck) + e1 ≤ e2(deg

F (bj)− 1) + e1.

Also degG(fi,j,k(x)) ≤ e0. Because

degG(bj) = degG(yj) = e2 degF (bj)

and

degG(ai) = degG(xi) = 1,

we get

degG([ai, bj]) ≤ e0 + (e2(degF (bj)− 1) + e1)

= degG(ai) + degG(bj)− 2.

Using Lemma 11.12 we conclude that [ai, bj ] = 0.

Next let’s consider the commutator [bi, bj ]. If either degG(bi) < degG(yi) or

degG(bj) < degG(yj) then bibj = bj bi = 0. Otherwise bi = symbG(bi) and bj =

symbG(bj). By formula (11.14), if f2i,j,k(x) 6= 0 then

degF (g2i,j,k(y)) + degF (ck) ≤ degF (bi) + degF (bj)− 1.

Also

degG(g2i,j,k(y)) = e2 degF (g2i,j,k(y))

degG(ck) ≤ degG(zk) = e2 degF (ck) + e1.


Therefore, looking only at indices k s.t. f2i,j,k(x) 6= 0, we obtain

degG([bi, bj]) ≤ maxdegG(f2i,j,k(a)) + degG(g2i,j,k(b)) + degG(ck)

≤ maxdegG(f2i,j,k(x)) + degG(g2i,j,k(y)) + degG(ck)

≤ e0 +maxe2 degF (g2i,j,k(y)) + e2 deg

F (ck) + e1

= e2 maxdegF (g2i,j,k(y)) + degF (ck) + 1+ (e0 + e1 − e2)

≤ e2(degF (bi) + degF (bj))− 1

= degG(bi) + degG(bj)− 1.

So according to Lemma 11.12 we conclude that [bi, bj ] = 0.The calculation for the other commutators is similar.Finally we show that, amusingly, cicj = 0. If either degG(ci) < degG(zi) or

degG(cj) < degG(zj) then automatically cicj = 0. Otherwise ci = symbG(ci) and

cj = symbG(cj). For any k such that f5i,j,k(x) 6= 0 one has

degF (g5i,j,k(y)) + degF (ck) ≤ degF (ci) + degF (cj).

Therefore, looking only at indices k s.t. f5i,j,k(x) 6= 0, we obtain

degG(cicj) ≤ maxdegG(f5i,j,k(a)) + degG(g5i,j,k(b)) + degG(ck)

≤ maxdegG(f5i,j,k(x)) + degG(g5i,j,k(y)) + degG(ck)

≤ e0 +maxe2 degF (g5i,j,k(y)) + e2 deg

F (ck) + e1

= e2maxdegF (g5i,j,k(y)) + degF (ck)+ (e0 + e1)

≤ e2(degF (ci) + degF (cj)) + (e0 + e1)

= degG(ci) + degG(cj)− 1.

So by definition of the product in A we get cicj = 0.

Proposition 11.15. In the situation of Theorem 11.11 assume the k-algebra A is

graded, and also every k-submodule FiA is graded. Then the filtration G can be

chosen such that every k-submodule GiA is graded.

Proof. Simply choose the generators a1, . . . , b1, . . . , c1, . . . , cp ∈ A used in the proofto be homogeneous.

We finish the section with an important example of a differential k-algebra offinite type, which generalizes Example 11.7.

Example 11.16. Let C be a finitely generated commutative k-algebra (not neces-sarily smooth, and chark arbitrary), and let L be a finite C-module (not necessarilyprojective). Suppose L has a k-linear Lie bracket [−,−]. The module of deriva-tions TC := Derk(C) is also a k-Lie algebra. Suppose α : L→ TC is a C-linear Liehomomorphism, namely α(cξ) = cα(ξ) and α([ξ, ζ]) = [α(ξ), α(ζ)] for all c ∈ C andξ, ζ ∈ L. L is then called a Lie algebroid or a Lie-Rinehart algebra (cf. [Ch1] or[Ri]). The ring of generalized differential operators D(C;L), also called the univer-sal enveloping algebra and denoted U(C;L), is defined as follows. Choose k-algebragenerators c1, . . . , cp for C and C-module generators l1, . . . , lq for L. Let

k〈x,y〉 := k〈x1, . . . , xp, y1, . . . , yq〉


be the free associative algebra. We have a ring surjection φ0 : k〈x〉 → C withφ0(xi) := ci. Let I0 := Ker(φ0). Next there is a surjection of k〈x〉-modules

φ1 : k〈x〉q =⊕q

j=1k〈x〉 · yj → L

with φ1(yj) := lj. Define I1 := Ker(φ1) ⊂ k〈x〉q. For any i, j choose polynomialsfi,j(x) and gi,j,k(x) such that [li, lj] =

∑k gi,j,k(c)lk ∈ L and α(li)(cj) = fi,j(c) ∈

C. Now define

U(C;L) :=k〈x,y〉

Iwhere I ⊂ k〈x,y〉 is the two-sided ideal generated by I0, I1 and the polynomials[yi, yj ]−

∑k gi,j,k(x)yk and [yi, xj ]− fi,j(x).

The ring U(C;L) has the following universal property: given any ring D, anyring homomorphism η0 : C → D and any C-linear Lie homomorphism η1 : L→ Dsatisfying [η1(l), η0(c)] = η0(α(l)(c)), there is a unique ring homomorphism η :U(C;L)→ D through which η0 and η1 factor.

Put on k〈x,y〉 the filtration F such that degF (xi) = 0 and degF (yj) = 1. LetF be the filtration induced on U(C;L) by the surjection φ : k〈x,y〉 ։ U(C;L).Then grFU(C;L) is a commutative C-algebra, generated by the elements lj :=grF (φ)(yj), j ∈ 1, . . . , q. We see that U(C;L) is a differential C-ring of finitetype. If C = k[x1, . . . , xn] and L = TC then we are in the situation of Example11.7. If C = k then U(C;L) = U(L) is the usual universal enveloping algebra ofthe Lie algebra L.

12. The Rigid Dualizing Complex of a Differential k-Algebra

We begin this section with the following consequence of previous work.

Theorem 12.1. Let A be a differential k-algebra of finite type. Then A has an

Auslander rigid dualizing complex RA. For a finite A-module M the canonical

dimension CdimM coincides with the Gelfand-Kirillov dimension GKdimM .

Proof. According to Theorem 11.11 A has a nonnegative exhaustive filtration G =GiA such that grGA is a commutative, finitely generated, connected graded k-algebra. Now use [YZ2, Corollary 6.9].

Recall that a ring homomorphism f : A→ B is called finite centralizing if thereexist elements b1, . . . , bn ∈ B that commute with all elements of A and B =

∑iAbi.

Proposition 12.2. Let A be a differential k-algebra of finite type and f : A→ B a

finite centralizing homomorphism. Then B is also a differential k-algebra of finite

type, and the rigid trace TrB/A : RB → RA exists.

Proof. By Theorem 11.11 we can find a differential k-filtration of finite type F =FiA of A such that grFA is connected. By [YZ2, Lemma 6.13] and its proofthere is a filtration F = FiB of B such that grFB is connected, f(FiA) ⊂ FiBand grF (f) : grFA → grFB is a finite centralizing homomorphism. It follows thatgrFB is finite over its center, so B is a differential k-algebra of finite type. By [YZ2,Theorem 6.17] the rigid trace TrB/A : RB → RA exists.

Let A be a differential k-algebra of finite type with rigid dualizing complex RA.The derived category Db

f (ModA) has on it the rigid perverse t-structure induced byRA, whose heart is the category of perverseA-modules pDb

f (ModA)0. See Definition8.4.


Proposition 12.3. Let A→ B be a finite centralizing homomorphism between two

differential k-algebras of finite type. Denote by restB/A : D(ModB) → D(ModA)the restriction of scalars functor.

(1) Let M ∈ Dbf (ModB). Then M ∈ pDb

f (ModB)0 if and only if restB/AM ∈pDb

f (ModA)0.(2) If A→ B is surjective then the functor

restB/A : pDbf (ModB)0 → pDb

f (ModA)0

is fully faithful.

Proof. (1) Define the duality functors DA := RHomA(−, RA) and DB :=RHomB(−, RB). According to [YZ2, Proposition 3.9(1)] the trace TrB/A : RB →RA gives rise to a commutative diagram

Dbf (ModB)

restB/A−−−−−→ Db

f (ModA)yDB

yDA

Dbf (ModBop)op

restBop/Aop

−−−−−−−−→ Dbf (ModAop)op

in which the vertical arrows are equivalences. By definition M ∈ pDbf (ModB)0 if

and only if HiDBM = 0 for all i 6= 0. Likewise restB/AM ∈ pDbf (ModA)0 if and

only if HiDA restB/AM = 0 for all i 6= 0. But

HiDA restB/AM ∼= restBop/Aop HiDBM.

(2) In view of (1) we have a commutative diagram

pDbf (ModB)0

restB/A−−−−−→ pDb

f (ModA)0yDB

yDA

(Modf Bop)op

restBop/Aop

−−−−−−−−→ (Modf Aop)op

where the vertical arrows are equivalences. The lower horizontal arrow is a fullembedding, since it identifies Modf B

op with the full subcategory of Modf Aop con-

sisting of modules annihilated by Ker(Aop → Bop). Hence the top horizontal arrowis fully faithful.

Lemma 12.4. Suppose A and B are k-algebras, M,M ′ ∈ Db(ModAe) and N,N ′ ∈Db(ModBe). Then there is a functorial morphism

µ : RHomA(M,M ′)⊗ RHomB(N,N′)→ RHomA⊗B(M ⊗N,M

′ ⊗N ′)

in D(Mod (A⊗B)e). If A and B are left noetherian and all the modules HpM and

HpN are finite then µ is an isomorphism.

Proof. Choose projective resolutions P → M and Q → N over Ae and Be respec-tively. So P ⊗Q →M ⊗N is a projective resolution over (A ⊗ B)e, and we get amap of complexes

µ : HomA(P,M′)⊗HomB(Q,N

′)→ HomA⊗B(P ⊗Q,M′ ⊗N ′)

over (A⊗B)e.Now assume the finiteness of the cohomologies. To prove µ is a quasi-isomorphism

we might as well forget the right module structures. Choose resolutions Pf → M


and Qf → N by complexes of finite projective modules over A and B respectively.We obtain a commutative diagram

HomA(P,M′)⊗HomB(Q,N

′)µ

−−−−→ HomA⊗B(P ⊗Q,M ′ ⊗N ′)y

y

HomA(Pf ,M′)⊗HomB(Qf , N

′)µ

−−−−→ HomA⊗B(Pf ⊗Qf ,M′ ⊗N ′)

in which the vertical arrows are quasi-isomorphism and the bottom arrow is anisomorphism of complexes.

Recall that by Proposition 11.10 the tensor product of two differential k-algebrasof finite type is also a differential k-algebra of finite type.

Theorem 12.5. Suppose A and B are differential k-algebras of finite type. Then

the rigid dualizing complexes satisfy

RA⊗B∼= RA ⊗RB

in D(Mod (A⊗B)e).

Proof. We will prove that RA ⊗RB is a rigid dualizing complex over A⊗B.Consider the Kunneth spectral sequence

(HpRA)⊗ (HqRB)⇒ Hp+q(RA ⊗RB).

Since A⊗B is noetherian it follows that Hp+q(RA⊗RB) is a finite (A⊗B)-moduleon both sides.

From Lemma 12.4 we see that the canonical morphism

A⊗B → RHomA⊗B(RA ⊗RB, RA ⊗RB)

in D(Mod (A⊗B)e) is an isomorphism. Likewise for RHom(A⊗B)e .Next using this lemma with Ae and Be, and by the rigidity of RA and RB, we

get isomorphisms

RHom(A⊗B)e(A⊗B, (RA ⊗RB)⊗ (RA ⊗RB)

)

∼= RHomAe⊗Be

(A⊗B, (RA ⊗RA)⊗ (RB ⊗RB)

)

∼= RHomAe(A,RA ⊗RA)⊗ RHomBe(B,RB ⊗RB)∼= RA ⊗RB

in D(Mod (A⊗B)e).It remains to prove that the complex RA⊗RB has finite injective dimension over

A⊗ B and over (A ⊗B)op. This turns out to be quite difficult (cf. Corollary 12.6below). By Theorem 11.11 there is a filtration F of A such that grFA is connected,

finitely generated and commutative. Let A := ReesFA ⊂ A[s], which is a noetherian

connected graded k-algebra. By [YZ2, Theorem 5.13], A has a balanced dualizing

complex RA ∈ Db(GrMod (A)e). The same holds for B: there’s a filtration G, a

Rees ring B := ReesGB ⊂ B[t] and a balanced dualizing complex RB. Accordingto [VdB1, Theorem 7.1] the complex RA⊗RB is a balanced dualizing complex over

A⊗ B. In particular RA ⊗RB has finite graded-injective dimension over A⊗ B.

Now A ∼= A/(s− 1), so by [YZ2, Lemma 6.3] the complex

Q := (A⊗ B)⊗A⊗B (RA ⊗RB)


has finite injective dimension over A ⊗ B. But the algebra A ⊗ B is graded (the

element 1⊗t has degree 1), andQ is a complex of graded (A⊗B)-modules. ThereforeQ has finite graded-injective dimension over this graded ring. Applying [YZ2,Lemma 6.3] again – it works for any graded ring, connected or not – we see that

(A⊗B)⊗A⊗B Q∼= (A⊗ B)⊗A⊗B (RA ⊗RB)

has finite injective dimension over A⊗B.According to [YZ2, Theorem 6.2] there is an isomorphism RA ∼= A ⊗A RA[−1]

in D(ModA). Likewise RB ∼= B ⊗B RB[−1]. Hence

RA ⊗RB ∼= (A⊗B)⊗A⊗B (RA ⊗RB)[−2]

has finite injective dimension over A⊗B.By symmetry RA ⊗RB has finite injective dimension also over (A⊗B)op.

Corollary 12.6. Suppose A and B are differential k-algebras of finite type, and

the complexes M ∈ Dbf (ModA) and N ∈ Db

f (ModB) have finite injective dimension

over A and B respectively. Then M ⊗N has finite injective dimension over A⊗B.

Proof. Let M∨ := RHomA(M,RA) and N∨ := RHomB(N,RB). The complexesM∨ ∈ Db

f (ModAop) and N∨ ∈ Dbf (ModBop) have finite projective dimension (i.e.

they are perfect); see [Ye4, Theorem 4.5]. Since the tensor product of projectivemodules is projective it follows that M∨ ⊗ N∨ ∈ Db

f (Mod (Aop ⊗ Bop)) has finiteprojective dimension. Using Theorem 12.5 and Lemma 12.4 we see that

RHomA⊗B(M ⊗N,RA⊗B) ∼= RHomA⊗B(M ⊗N,RA ⊗RB) ∼=M∨ ⊗N∨.

Applying RHomA⊗B(−, RA⊗B) to these isomorphisms we get

M ⊗N ∼= RHomA⊗B(M∨ ⊗N∨, RA⊗B),

so this complex has finite injective dimension.

Question 12.7. Is there a direct proof of the corollary? Is it true in greatergenerality, e.g. for any two noetherian k-algebras A and B?

Remark 12.8. We take this opportunity to correct a slight error in [YZ2]. In

[YZ2, Theorem 6.2(1)] the complex R should be defined as R := (Rt)0, namely thedegree 0 component of the localization with respect to the element t. The rest ofthat theorem (including the proof!) is correct.

If A is a differential k-algebra of finite type then so is the enveloping algebra Ae.Hence the rigid dualizing complex RAe exists, as does the rigid perverse t-structureon Db

f (ModAe), whose heart is the category pDbf (ModAe)0 of perverse Ae-modules.

Theorem 12.9. Let A be a differential k-algebra of finite type with rigid dualizing

complex RA. Then RA ∈ pDbf (ModAe)0.

Proof. Consider the k-algebra isomorphism

τ : (Aop)e = Aop ⊗A≃−→ A⊗Aop = Ae

with formula τ(a1 ⊗ a2) := a2 ⊗ a1. Given an Ae-module M let τM be the (Aop)e-module with action via τ , i.e.

(a1 ⊗ a2) ·τ m := τ(a1 ⊗ a2) ·m = a2ma1

for m ∈ M and a1 ⊗ a2 ∈ (Aop)e. Doing this operation to the complex RA ∈D(ModAe) we obtain a complex τRA ∈ D(Mod (Aop)e). Each of the conditions in


Definitions 2.1 and 2.10 is automatically verified, and hence RAop := τRA is a rigiddualizing complex over Aop.

According to Theorem 12.5 we get an isomorphism

RAe ∼= RA ⊗RAop = RA ⊗ (τRA)

in D(Mod (Ae)e). But the left (resp. right) Ae action on RA ⊗ (τRA) is exactly theoutside (resp. inside) action on RA ⊗RA. By rigidity (cf. Definition 2.10) we have

RA ∼= RHomAe(A,RA ⊗RA) ∼= RHomAe(A,RAe)

in D(Mod Ae).Finally since Ae ∼= (Ae)op, via the involution τ , we may view RHomAe(−, RAe)

as an auto-duality of Dbf (Mod Ae). By definition of the rigid t-structure this duality

exchanges Modf Ae and pDb

f (Mod Ae)0. Since A ∈ Modf Ae it follows that RA ∈

pDbf (Mod Ae)0.

We know that the cohomology bimodules HiRA are central Z(A)-modules. Thenext lemma shall be used a few times.

Lemma 12.10. Let A be a differential k-algebra of finite type and a ∈ A a non-

invertible central regular element. Define B := A/(a). Let RA and RB denote

the rigid dualizing complexes of A and B respectively. Then there is a long exact

sequence

· · · → HiRB → HiRAa−→ HiRA → Hi+1RB → · · ·

of A-bimodules.

Proof. Trivially A→ B is a finite centralizing homomorphism. By Proposition 12.2the trace morphism TrB/A : RB → RA exists. In particular, RB ∼= RHomA(B,RA).There is an exact sequence of bimodules

0→ Aa−→ A→ B → 0.

Applying the functor RHomA(−, RA) to it and taking cohomologies we obtain thelong exact sequence we want.

Here are a couple of examples of differential k-algebras of finite type and theirrigid dualizing complexes.

Example 12.11. Let C be a smooth n-dimensional k-algebra in characteristic 0and A := D(C) the ring of differential operators. Then the rigid dualizing complexis RA = A[2n]; see [Ye3].

Example 12.12. Let g be an n-dimensional Lie algebra over k and A := U(g)its universal enveloping algebra. By [Ye3] the rigid dualizing complex is RA =A⊗(

∧ng)[n], where

∧ng has the adjoint A action on the left and the trivial action

on the right.

Suppose A is a ring with nonnegative exhaustive filtration F such that the Rees

ring A := ReesFA is left noetherian. We remind that a filtered (A,F )-module

(M,F ) is called good if it is bounded below, exhaustive, and ReesFM is a finite

A-module.In the two previous examples the cohomology bimodules HiRA all came equipped

with filtrations that were both differential and good on both sides. These propertiesturn out to hold in general, as Theorems 12.13 and 12.14 show.


Theorem 12.13. Let A be a differential k-algebra of finite type, and let RA be the

rigid dualizing complex of A. Let F be some differential k-filtration of finite type of

A. Then for every i there is an induced filtration F of HiRA, such that (HiRA, F )is a good filtered (A,F )-module on both sides.

Proof. Define A := ReesFA ⊂ A[t]. Let F = FiA be the filtration from Lemma10.4. Then

grF A ∼= (grFA)⊗ k[t]

as k-algebras. The center is

Z(grF A) ∼= Z(grFA)⊗ k[t],

which is a finitely generated commutative k-algebra. Also grF A is a finite Z(grF A)-

module. We conclude that F is a differential k-filtration of finite type on A. More-

over each k-submodule FiA is graded, where A ⊂ A[t] has the grading F in which

degF (t) = 1.

Applying Theorem 11.11 and Proposition 11.15 to the filtered k-algebra (A, F )

we obtain another filtration G on A. This new filtration is also differential k-filtration of finite type, and each k-submodule GiA ⊂ A is graded (for the grading

F ). Furthermore grGA is a connected graded k-algebra (when considered as Z-

graded ring with the grading G).Define

B := ReesGA ⊂ A[s] ⊂ A[s, t].

This is a Z2-graded ring with grading (G, F ), in which deg(G,F )(s) = (1, 0) and

deg(G,F )(t) = (0, 1).

Consider the k-algebra B with its grading G. This is a connected graded k-

algebra. The quotient B/(s) ∼= grGA is a finitely generated commutative k-algebra.Therefore by [YZ2, Theorem 5.13], B has a balanced dualizing complex RB ∈Db(GrModBe). By [VdB1, Theorem 6.3] we get

RB ∼= (RΓmB)∗

in D(GrModBe). Here Γm is the torsion functor with respect to the augmentationideal m of B, and

(M)∗ := Homgrk(M, k) =

⊕iHomk(M−i, k),

the graded dual of the graded k-module M . In particular, for every p there is anisomorphism of B-bimodules H−pRB ∼=

(HpmB

)∗where

HpmB := HpRΓmB ∼= limk→

ExtpB(B/mk, B).

Now for each k

ExtpB(B/mk, B) =

⊕(i,j)∈Z2

ExtpB(B/mk, B)(i,j)

where (i, j) is the (G, F ) degree. Therefore in the direct limit we get a doublegrading

HpmB =⊕

(i,j)∈Z2(HpmB)(i,j).


Since for every i the k-module (HpmB)i =⊕

j∈Z(HpmB)(i,j) is finite it follows that

the graded dual (HpmB)∗, which is computed with respect to the G grading, is alsoZ2-graded. We see that H−pRB is in fact a Z2-graded B-bimodule.

By [YZ2, Theorem 6.2] the complex RA := ((RB[−1])s)0 is a rigid dualizing

complex over the ring A ∼= B/(s− 1); cf. Remark 12.8. Hence each cohomology

HpRA∼=

Hp−1RB(s− 1) ·Hp−1RB

= spG1 Hp−1RB

is a graded A-bimodule (with the Z-grading F in which degF (t) = 1), finite on bothsides.

Next we have A ∼= A/(t − 1). Because t − 1 is a central regular non-invertibleelement, Lemma 12.10 says there is an exact sequence

· · · → HpRA → HpRAt−1−−→ HpRA → Hp+1RA → Hp+1RA → · · ·

of A-bimodules. Since HpRA is graded the element t − 1 is a non-zero-divisor onit, and therefore we get an exact sequence

0→ Hp−1RAt−1−−→ Hp−1RA → HpRA → 0.

Thus the bimodule

HpRA ∼=Hp−1RA

(t− 1) · Hp−1RA= spF1 Hp−1RA

inherits a bounded below exhaustive filtration F , and Rees (HpRA, F ), being a

quotient of Hp−1RA, is a finite A-module on both sides. By definition (HpRA, F )is then is a good filtered (A,F )-module on both sides.

The next theorem will have the utmost significance once we pass from rings toringed spaces (cf. Theorem 9.13).

Theorem 12.14. Let C be a finitely generated commutative k-algebra, let A be

differential C-ring of finite type, and let RA be the rigid dualizing complex of A.Then for every i the C-bimodule HiRA is differential.

Proof. Using the same setup as in the proof of Theorem 12.13, define

A := grFA = spF0 A = A/(t).

So A is a C-algebra. Let RA be the rigid dualizing complex of A. By [YZ2, Corollary3.6] the A-bimodule HiRA is Z(A)-central, and hence it is a central C-bimodule.

According to Lemma 12.10 there is an exact sequence of A-bimodules

Hi−1RAt−→ Hi−1RA → HiRA.

Thereforesp0 (H

i−1RA) = (Hi−1RA)/t · (Hi−1RA) → HiRA

is a central C-bimodule.To conclude the proof, consider the filtration F of HiRA from Theorem 12.13.

Because (HiRA, F ) ∼= sp1 (Hi−1RA) is a good filtered (A,F )-module, say on the

left, we see that (HiRA, F ) is exhaustive and bounded below. Now

Rees (HiRA, F ) ∼= (Hi−1RA)/t-torsion,

so gr (HiRA, F ) is a quotient of sp0 (Hi−1RA). It follows that gr (HiRA, F ) is a

central C-bimodule. Thus F is a differential C-filtration of HiRA.


13. Differential Quasi-Coherent Ringed Schemes of Finite Type

In this section all the pieces of our puzzle come together. We consider a quasi-coherent ringed scheme (X,A) in which for any affine open set U ⊂ X the ringΓ(U,A) is a differential Γ(U,OX)-ring of finite type. The rigid dualizing complexesobtained locally are perverse bimodules, and therefore can be glued to yield a globalrigid dualizing complex.

Definition 13.1. Let X be a finite type k-scheme and let A be a quasi-coherentOX -ring. A differential quasi-coherent OX-filtration of finite type on A is an as-cending filtration F = FiA by subsheaves with the following properties:

(i) Each FiA is an OX -sub-bimodule of A, quasi-coherent on both sides.(ii) F−1A = 0 and A =

⋃FiA.

(iii) 1 ∈ F0A and (FiA) · (FjA) ⊂ Fi+jA.(iv) The graded sheaf of rings grFA is an OX -algebra.(v) The center Z(grFA) is a finite type quasi-coherent OX -algebra.(vi) grFA is a coherent Z(grFA)-module.

By properties (i) and (iii) we get a ring homomorphism OX → grFA. Property(iv) tells us that the image of OX is inside Z(grFA).

Observe that the definition is left-right symmetric: if A is a differential quasi-coherent OX -ring of finite type then so is Aop.

Definition 13.2. Let X be a finite type k-scheme. A differential quasi-coherent

OX-ring of finite type is an OX -ring A that admits some differential quasi-coherentOX -filtration of finite type. The pair (X,A) is then called a differential quasi-

coherent ringed k-scheme finite type.

Proposition 13.3. Let C be a noetherian k-algebra and U := SpecC.

(1) Given a differential quasi-coherent OU -ring of finite type A the ring A :=Γ(U,A) is a differential C-ring of finite type.

(2) Given a differential C-ring of finite type A there is a differential quasi-

coherent OU -ring of finite type A, unique up to isomorphism, such that

Γ(U,A) ∼= A as C-rings.

Proof. The proof of (1) is straightforward. For (2) use Proposition 11.3 and Corol-lary 3.15. As filtration we may take FiA := OU ⊗C FiA where FiA is anydifferential C-filtration of finite type on A.

Corollary 13.4. Let (X,A) be a differential quasi-coherent ringed scheme of finite

type over k. Then (X,A) a noetherian quasi-coherent ringed scheme in the sense

of definition 3.2.

Proof. This follows from Proposition 13.3(1) and Lemma 11.9.

Proposition 13.5. Suppose (X,A) and (Y,B) are two differential quasi-coherent

ringed schemes of finite type over k. Then the product (X × Y,A ⊠ B) exists (cf.Definition 4.1), and it too is a differential quasi-coherent ringed scheme of finite

type over k.

Proof. By Proposition 11.10, Lemma 4.10 and Theorem 4.4. A differential filtrationon A ⊠ B can be constructed out of given filtrations of A and B as in the proof ofProposition 11.10, plus gluing.


Here is the main result of the paper.

Theorem 13.6. Let (X,A) be a separated differential quasi-coherent ringed

scheme of finite type over k. Then A has a rigid dualizing complex (RA,ρ). It

is unique up to a unique isomorphism in Dbc (ModAe).

Proof. First we note that the hypotheses of Theorem 9.13 are satisfied for the ringedschemes (X,A) and (X2,Ae). Indeed by Corollary 13.4 (X,A) is a noetherianquasi-coherent ringed scheme. By Proposition 13.3 for every affine open set U ⊂ Xthe ring A := Γ(U,A) is a differential C-ring of finite type, where C := Γ(U,OX).Theorem 12.1 asserts that the rigid dualizing complex RA exists, and by Theorem12.14 the C-bimodules HiRA are differential. So by Proposition 4.11 the HiRAhave supports in ∆(U) ⊂ U2. According to Proposition 11.10 Ae is noetherian. Wesee that the hypotheses of Theorem 9.13 are satisfied for (X,A). Since (X2,Ae)is also a separated differential quasi-coherent ringed scheme over k, it too satisfiesthe hypotheses of Theorem 9.13. By Theorem 9.9 we have the rigid perverse t-structure on Db

c (ModAe), and by Theorem 9.11 there is a stack of abelian categoriesW 7→ pDb

c (ModAe|W )0 on X2.Let us choose, for ease of notation, an indexing Ui of the set AffX of affine

open sets of X . For any index i let Ai := Γ(Ui,A), which is a differential k-algebraof finite type. Then Ai has a rigid dualizing complex (Ri, ρi). By Theorem 12.9 thecomplex Ri is a perverse bimodule, i.e. Ri ∈ pDb

f (ModAei )

0. And by Theorem 12.14the cohomology bimodules HqRi are supported on the diagonal ∆(Ui) ⊂ Ui × Ui.

For a pair of indices i, j let A(i,j) := Γ(Ui ∩ Uj ,A). According to Theorem 6.2the dualizing complex

Ri→(i,j) := A(i,j) ⊗Ai Ri ⊗Ai A(i,j)

has a unique rigidifying isomorphism ρi→(i,j) such that the homomorphismqA(i,j)/Ai

: RAi → Ri→(i,j) with formula r 7→ 1 ⊗ r ⊗ 1 is a rigid localization

morphism. Likewise we obtain a rigid dualizing complex (Rj→(i,j) , ρj→(i,j)). Andthere is a unique isomorphism

φ(i,j) : Ri→(i,j)≃−→ Rj→(i,j)

in D(ModAe(i,j)) that’s a rigid trace. The isomorphisms φ(i,j) will then satisfy the

cocycle condition in D(ModAe(i,j,k)) for a triple intersection A(i,j,k) :=

Γ(Ui ∩ Uj ∩ Uk,A).Now consider the affine ringed scheme (Ui × Ui,Ae|Ui×Ui). Denote by Ri ∈

Dbc (ModAe|Ui×Ui) the sheafification of Ri. By definition of the rigid perverse t-

structure on Dbc (ModAe|Ui×Ui) we have Ri ∈

pDbc (ModAe|Ui×Ui)

0. By Lemma 5.6we obtain induced isomorphisms

φ(i,j) : Ri|(Ui×Ui)∩(Uj×Uj)≃−→ Rj |(Ui×Ui)∩(Uj×Uj)

in pDbc (ModAe|(Ui×Ui)∩(Uj×Uj

)0. Let V := (X ×X)−∆(X). Since

Ri|(Ui×Ui)∩V = 0

we have sufficient gluing data corresponding to the open covering

X ×X =(⋃

iUi × Ui

)∪ V,


and by Theorem 9.11 we deduce the existence and uniqueness of a global complex

RA ∈pDb

c (ModAe)0 together with isomorphisms φi : RA|Ui

≃−→ Ri. By Proposi-

tion 7.3 RA is a local dualizing complex over A. And by construction RA comesequipped with a collection ρ = ρi of rigid localization morphisms that is com-patible with the sheaf structure.

Finite morphisms of ringed schemes were defined in Definition 7.8.

Definition 13.7. Let f : (Y,B) → (X,A) be a finite morphism between twonoetherian quasi-coherent ringed schemes over k. We say f is finite centralizing if forany affine open set U ⊂ X the finite ring homomorphism f∗ : Γ(U,A)→ Γ(U, f∗B)is centralizing.

Proposition 13.8. Let f : (Y,B) → (X,A) be a finite centralizing morphism

between two separated differential quasi-coherent ringed schemes of finite type over

k. Consider the the rigid perverse t-structures on these ringed schemes.

(1) Let M ∈ Dbc (ModB). Then M ∈ pDb

c (ModB)0 if and only if Rf∗M ∈pDb

c (ModA)0.(2) Assume f∗ : A → f∗B is surjective. Then the functor

Rf∗ : pDbc (ModB)0 → pDb

c (ModA)0

is fully faithful.

Proof. (1) Let U ⊂ X be an affine open set and V := f−1(U). Define A := Γ(U,A),B := Γ(V,B) and M := RΓ(V,M). By Lemma 9.6 it is enough to show thatM ∈ pDb

c (ModB)0 iff restB/AM ∈ pDbc (ModA)0. This is done in Proposition

12.3(1).

(2) This follows from Theorem 9.11 and Proposition 12.3(2).

Rigid trace morphisms between rigid dualizing complexes were defined in Defi-nition 7.9.

Theorem 13.9. Let (X,A) and (Y,B) be two separated differential quasi-coherent

ringed schemes of finite type over k, and let f : (Y,B)→ (X,A) be a finite central-

izing morphism. Then there exists a unique rigid trace

Trf : Rf e∗RB →RA.

Proof. The morphism f e : (Y 2,Be) → (X2,Ae) is finite centralizing, so by Propo-sition 13.8 we get Rf e

∗RB ∈pDb

c (ModAe)0. Hence by Theorem 9.11 the morphismsRf e

∗RB →RA are determined locally.Choose an affine open covering X =

⋃ni=1 Ui and let Ai := Γ(Ui,A) and Bi :=

Γ(f−1(Ui),B) ∼= Γ(Ui, f∗B). Since Ai → Bi is a finite centralizing homomor-phism, Proposition 12.2 asserts the existence of the trace TrAi/Bi

: RBi → RAi in

Dbf (ModAe). The uniqueness of TrAi/Bi

is always true. Now

RBi∼= RΓ(Ui × Ui,Rf

e∗RB),

so by Corollary 5.5 we get a morphism

Tri : (Rfe∗RB)|Ui×Ui →RA|Ui×Ui

in pDbc (Mod (Ae)|Ui×Ui)

0.By Proposition 6.3 a localization of the rigid trace is the rigid trace. Therefore

the morphisms Tri coincide on intersections (Ui × Ui) ∩ (Uj × Uj). Both Rf e∗RB


and RA are supported on the diagonal ∆(X). Therefore we have gluing data for aglobal morphism

Trf : Rf e∗RB →RA

in pDbc (ModAe)0 as required.

The uniqueness of Trf is a consequence of the fact that pDbc (ModAe)0 is a stack

and the uniqueness of the traces Tri.

Here are several examples of differential quasi-coherent ringed schemes of finitetype over k and their rigid dualizing complexes.

Example 13.10. Let X be any separated k-scheme of finite type and A a coherentOX -algebra. Then A is a differential quasi-coherent OX -ring of finite type. Asfiltration F we may take the trivial filtration where F−1A := 0 and F0A := A. Inthis case the rigid dualizing complex RA can be chosen to be RHomOX (A,RX)where RX is the central rigid dualizing complex of X ; cf. Proposition 13.15. If Ahappens to be an Azumaya OX -algebra then RA

∼= A⊗OXRX . See [YZ3, Theorem6.2].

Example 13.11. Suppose chark = 0 and X is a smooth separated scheme. Thenthe ring DX of differential operators on X is a differential quasi-coherentOX -ring offinite type. As filtration F we may take the order filtration, in which F0DX := OX ,F1DX := OX ⊕TX , and Fi+1DX := FiDX ·F1DX . Here TX := HomOX (Ω1

X/k,OX)

is the tangent sheaf. The product turns out to be

(X2,DeX) ∼= (X2,DX2(OX ⊠ ωX))

where n := dimX , ωX := ΩnX/k, OX ⊠ωX := OX2 ⊗p−12 OX

p−12 ωX and DX2(−) is

the ring of twisted differential operators. By [Ye3] the rigid dualizing complex ofDX is DX [2n] ∈ Db

c (ModDeX).

Example 13.12. Let X be any finite type separated k-scheme,M a coherent OX -module and DX(M) := DiffOX/k(M,M) the ring of differential operators fromM to itself (cf. [EGA IV]). The order filtration F makes DX(M) into a differentialquasi-coherent OX -ring, but unless chark = 0, X is smooth andM is locally free,this is usually not of finite type. However if we take a coherent OX -sub-bimoduleN ⊂ DX(M), e.g.N := FdDX(M) for some d, then the subring OX〈N〉 ⊂ DX(M)is a differential quasi-coherent OX -ring of finite type.

Example 13.13. Generalizing Examples 12.12 and 13.11, supposeX is a separatedfinite type k-scheme and L is a coherent OX -module endowed with a k-linear Liebracket [−,−] and an OX -linear Lie homomorphism α : L → TX , satisfying theconditions stated in Example 11.16 on affine open sets. Such L is called a Liealgebroid on X . By Proposition 11.3 and Corollary 3.15 the universal envelopingalgebras U(C;L) sheafify to a differential quasi-coherent OX -ring of finite typeU(OX ;L). Perverse U(OX ;L)-modules were considered in the recent paper [AB].

Now suppose chark = 0 and X is a smooth scheme of dimension n. WhenL = TX then U(OX ;L) = DX as in Example 13.11. More generally when L is alocally free OX -module of rank r the rigid dualizing complex RU(OX ;L) of U(OX ;L)was computed by S. Chemla [Ch2] for X affine. Since the rigidifying isomorphismused there was canonical it glues and we obtain

RU(OX ;L) = U(OX ;L)⊗OX ΩnX/k ⊗OX

(∧r

OX

L)[n+ r].


Example 13.14. Assume chark = 0. Let C := k[x] with x an indeterminate,and A := D(C) the first Weyl algebra. Writing y := ∂

∂x ∈ A we have A ∼=

k〈x, y〉/([y, x]−1). Consider the filtration F on A in which degF (x) = degF (y) = 1,

and let A := ReesFA. Then A is the homogenized Weyl algebra, with linear

generators u := xt, v := yt and t, such that t is central and [v, u] = t2. Inside A we

have the commutative subring C := ReesFC ∼= k[u, t].

Since A is a differential C-ring the localizations At and Au exist. Their degree 0

components glue to a quasi-coherent sheaf of rings A on Proj C ∼= P1k. We obtain

a differential quasi-coherent ringed k-scheme of finite type (P1,A). The restrictionto the open set t 6= 0 recovers the ringed scheme (A1,DA1), and in particularΓ(t 6= 0,A) = A. Thus (P1,A) can be viewed as a partial compactification of(A1,DA1). Note that the ringed scheme (P1,A) is not isomorphic to (P1,DP1),since the function t/u ∈ Γ(u 6= 0,A) is in the center, whereas the center of DP1

consists only of constant functions.Consider another geometric object associated to this situation: the projective

spectrum Proj A in the sense of Artin-Zhang [AZ]. We claim that (P1,A) is an

“open subscheme” of the “complete surface” Proj A, whose complement consistsof one point. To make this statement precise we have to pass to abelian cate-

gories. Recall that QGr A is the quotient category GrMod A/m-torsion, where

m is the augmentation ideal of A; and Proj A is the geometric object such that

QCohProj A = QGr A. Because QCohA is the gluing of the categoriesQCohA|t6=0

and QCohA|u6=0 we see that QCohA is equivalent to the quotient of QGr A by

the localizing subcategory a-torsion, where a is the two-sided ideal (t, u) ⊂ A.

In the final results we look at commutative schemes. Let X be a separated finitetype k-scheme with structural morphism π. We are going to compare the rigiddualizing complex of X to Grothendieck’s dualizing complex π!k. First we needto know that the rigid dualizing complex of X is really a complex of OX -modules.Consider the diagonal embedding ∆ : X → X2. A central rigid dualizing complex

over X is a pair (R,ρ), where R ∈ Dbc (ModOX) and (R∆∗R,ρ) is a rigid dualizing

complex in the sense of Definition 7.6. To say it plainly, for every affine open setU = SpecC ⊂ X the complex RΓ(U,R) is a rigid dualizing complex over C withrigidifying isomorphism ρU (Definition 2.10), and the collection of isomorphismsρ = ρU is compatible with localization.

Proposition 13.15. Let X be a separated finite type k-scheme. Then X has a cen-

tral rigid dualizing complex (RX ,ρX), and it is unique up to a unique isomorphism

in D(ModOX).

Proof. By [Ye4, Proposition 5.7] for each affine open set U ⊂ X the ring C :=Γ(U,OX) has a central rigid dualizing complex RC ∈ Db

f (ModC); cf. Example2.12. Now proceed like in the proof of Theorem 13.6, but replacing (X2,Ae) with(X,OX) everywhere; or alternatively use the fully faithful functor

R∆∗ : pDbc (ModOX)0 → pDb

c (ModOX2)0

(see Proposition 13.8).

As in [RD] we say X is an embeddable k-scheme if the structural morphismπ : X → Spec k factors into π = g f with f finite and g smooth. Under thisassumption the twisted inverse image functor π! exists.


Theorem 13.16. Let X be a separated finite type embeddable k-scheme with struc-

tural morphism π : X → Spec k. Let (RX ,ρX) be the central rigid dualizing complex

of X, and let π!k ∈ Dbc (ModOX) be the dualizing complex from [RD]. Then there

is a canonical isomorphism

RX ∼= π!k

in Dbc (ModOX).

Proof. By the uniqueness in Proposition 13.15 it suffices to prove that the complexR′ := π!k has a canonical collection of rigidifying isomorphisms ρ′ = ρ′UU∈AffX .

Let us denote by f : X2 → Spec k the structural morphism of X2; so π = f ∆and f = π p1 = π p2 (see diagram).

X∆

// X ×Xp1

wwww

wwww

wp2

##GGGG

GGGG

G

f

X

π

##GGGG

GGGG

G X

π

wwww

wwww

w

Spec k

Since p2 is flat, by [RD, Proposition III.8.8] we get

f !k ∼= p!2π!k ∼= (p∗2π

!k)⊗LOX2

(p!2OX).

By flat base change [RD, Theorem III.8.7] we have

p!2OX∼= p!2π

∗k ∼= p∗1π!k.

Combining these two formulas we obtain

f !k ∼= (p∗2R′)⊗L

OX2(p∗1R

′)

in Dbc (ModOX). By the variance for finite morphisms in [RD, Theorem III.8.7]

there is an isomorphism

(13.17)R′ = π!k

≃−→ ∆!f !k ∼= ∆−1RHomOX2 (∆∗OX , f

!k)

∼= ∆−1RHomOX2 (∆∗OX , (p∗2R

′)⊗LOX2

(p∗1R′)).

Now take any affine open set U ⊂ X , and let C := Γ(U,OX) and R′ :=RΓ(U,R′). Applying the functor RΓ(U,−) to the isomorphism (13.17) we obtainan isomorphism

ρ′U : R′ ≃−→ RHomCe(C,R′ ⊗R′)

in Dbf (ModCe). By Lemma 5.6 the collection of isomorphisms ρ′U is compatible

with localizations.

Remark 13.18. If A is a finite k-algebra of finite global dimension then the rigiddualizing complex of A is A∗ := Homk(A, k). It is known thatM 7→ A∗⊗L

AM is theSerre functor of Db

f (ModA). Namely A∗⊗LA− is an auto-equivalence of Db

f (ModA),and there is a bifunctorial nondegenerate pairing

HomD(ModA)(M,N)×HomD(ModA)(N,A∗ ⊗L

AM)→ k

for M,N ∈ Dbf (ModA). Likewise, for a smooth n-dimensional projective scheme X

the rigid dualizing complex is ΩnX/k[n], and the Serre functor isM 7→ ΩnX/k[n]⊗OX

M. Kontsevich has asked recently whether this is true in greater generality. We


believe we can prove the following statement: let X be a projective variety and Aa regular coherent OX -algebra (i.e. for any affine open set U the ring Γ(U,A) hasfinite global dimension). Let RA ∈ D(Mod (A ⊗OX A

op)) be the rigid dualizingcomplex of A. ThenM 7→ RA ⊗L

AM is the Serre functor of Dbc (ModA).

The next result shows that in the commutative case the perverse coherent sheavesare precisely the Cohen-Macaulay complexes inside Db

c (ModOX). Recall that acomplex M ∈ D+(ModOX) is called Cohen-Macaulay if there is an isomorphismM ∼= EM in D(ModOX), where EM is the Cousin complex. See [RD, SectionIV.3], and also [YZ3, Theorem 2.11].

Theorem 13.19. Let X be a finite type scheme over k, let RX be the central rigid

dualizing complex of X, and let D be the duality functor RHomOX (−,RX). Then

the following conditions are equivalent for M ∈ Dbc (ModOX).

(i) M is a perverse coherent sheaf (for the rigid perverse t-structure).(ii) DM is a coherent sheaf, i.e. HiDM = 0 for all i 6= 0.(iii) M is a Cohen-Macaulay complex.

Proof. All three conditions can be checked locally, so we may assume X is affine.Then (i) and (ii) are equivalent by definition.

Since the dualizing complex RX is adjusted to Krull dimension it follows thatthe Cousin complex KX := ERX is a residual complex. Thus for every i we haveK−iX =

⊕KX(x), where x runs over the points of dimension i. Each KX(x) is a

quasi-coherent OX -module with support the closed set x, and as module it isan injective hull of the residue field k(x) as OX,x-module. Since each K−i

X is aninjective OX -module it follows that DM∼= HomOX (M,KX).

Let us prove that (ii) implies (iii). Let N be the coherent sheaf H0DM; soM∼= DN in D(ModOX). For every i the sheaf

HomOX (N ,K−iX ) ∼=

⊕

dim x= i

HomOX (N ,KX(x))

is flasque and pure of dimension i (or it is 0). Therefore

EHomOX (N ,KX) = HomOX (N ,KX).

We conclude that EM∼= EDN ∼= DN ∼=M in D(ModOX).Now let us prove that (iii) implies (i). We are assuming that EM ∼= M in

D(ModOX). It suffices to prove that M ∈ pDbf (ModC)0, where C := Γ(X,OX)

and M := RΓ(X,M). By Lemma 8.9 it is enough to show that dimH−iM ≤ i

and HjMiM = 0 for all i and all j < −i. Because RΓ(X,−) can be computed

using flasque resolutions we get M ∼= Γ(X,EM). But dimΓ(X,EM)−i ≤ i, andhence dimH−iM ≤ i. For the vanishing we use the fact that RΓMi can be com-puted by flasque resolutions (cf. [YZ3, Definition 1.13]), and that each C-moduleΓ(X,EM)−i is a flasque C-module and is pure of dim = i (unless it is zero). Thisgives

RΓMiM∼= ΓMiΓ(X,EM) = Γ(X,EM)≥−i.

Thus HjMiM = 0 for all j < −i.

Remark 13.20. P. Sastry has communicated to us another proof of the implication(iii) ⇒ (ii) in Theorem 13.19, using local duality at closed points of X .


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A. Yekutieli: Department of Mathematics Ben Gurion University, Be’er Sheva 84105,

Israel

E-mail address: [email protected]

J.J. Zhang: Department of Mathematics, Box 354350, University of Washington,

Seattle, Washington 98195, USA

E-mail address: [email protected]


http://arXiv.org


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arxiv:math/0211309v1 [math.ag] 20 nov 2002coherent ox-algebra with a big center. this class of...

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