Gorenstein categories and Tate cohomology on projective schemes

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GORENSTEIN CATEGORIES AND TATE COHOMOLOGY ONPROJECTIVE SCHEMES

E. ENOCHS, S. ESTRADA AND J.R.GARCIA ROZAS

ABSTRACT. We study Gorenstein categories. We show that such a category hasTate cohomological functors and Avramov-Martsinkovsky exact sequences con-necting the Gorenstein relative, the absolute and the Tate cohomological func-tors. We show that such a category has what Hovey calls an injective modelstructure and also a projective model structure in case the category has enoughprojectives.As examples we show that ifX is a locally Gorenstein projective scheme thenthe categoryQco(X) of quasi-coherent sheaves onX is such a category and sohas these features.

1. INTRODUCTION

Tate homology and cohomology over the group ringZG (with G a finite group)began with Tate’s observation that theZG-moduleZ with the trivial action admitsa complete projective resolution. Apparently motivated byTate’s work, Auslandershowed in [5] that ifA is what Bass in [7] calls a Gorenstein local ring, the finitelygenerated maximal Cohen-Macaulay modules can be characterized as those whichadmit a complete projective resolution of finitely generated projective modules.Auslander calls these modules the modules ofG-dimension0 and goes on to definetheG-dimension of any finitely generated module.

In [16] an easy modification of one of Auslander’s characterizations of thefinitely generated modules ofG-dimension0 was given and so allows one to ex-tend the definition to any module (finitely generated or not).Since this modifieddefinition dualizes it seems appropriate to use the terms Gorenstein projective (cor-responding to modules ofG-dimension0) and Gorenstein injective. Then there is anatural way to define Gorenstein projective and injective dimension of any module.If a moduleM has finite Gorenstein projective dimensionn, then then-th syzygyof M has a complete projective resolution. This complex is a homotopy invariantof M and so can be used to define Tate homological functorsExt

n

R(M,N) and

TorR

n (M,N) for anyn ∈ Z. If on the other handN has finite Gorenstein injectivedimension a similar procedure can be used to define analogousfunctors. A. Iacobin [27] showed that if both conditions hold then the two procedures give us thesame functors, i.e., that we have balance in this situation.

1991Mathematics Subject Classification.Primary: 18E15,16E65. Secondary: 18F20,18G25.Key words and phrases.Gorenstein category, locally Gorenstein scheme.The authors are partially supported by the DGI MTM2005-03227.

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http://arXiv.org/abs/0711.1181v1

2 E. ENOCHS, S. ESTRADA AND J.R.GARCIA ROZAS

In categories of sheaves there are usually not enough projectives but there areenough injectives. So we use the second approach to define Gorenstein injectiveson the category of quasi-coherent sheaves on certain projective schemes. Moreprecisely, we show that if such a schemeX ⊆ P

n(A) (whereA is commutativenoetherian) is what we call a locally Gorenstein scheme thenall objects ofQco(X)have finite Gorenstein injective dimension and that there isa universal bound ofthese dimensions. This allows us to define Tate cohomology inthis situation, to getAvramov-Martsinkovsky sequences and to impose a model structure onQco(X).

The exampleQco(X) mentioned above is our motivation for defining Goren-stein categories. These categories will be Grothendieck categories with propertiesmuch like those of categories of modules over Gorenstein rings. But we would likeour definitions to be such that nice categories of sheaves will be Gorenstein. Sincecategories of sheaves rarely have enough projectives we need a definition whichdoes not involve projective objects. But such categories dohave enough injectivesand so the functors Ext are defined. And so projective dimensions of objects canbe defined in terms of the vanishing of the Ext functors. So we define a Goren-stein category in terms of the global finitistic projective and injective dimensionsof the category. After defining Gorenstein categories and proving some basic re-sults about them, we consider examples. We show that ifX ⊂ P

n(A) (whereAis commutative noetherian) is a projective schemeQco(X) will be a Gorensteincategory whenX is a locally Gorenstein scheme.

The authors give a sincere thanks to Mark Hovey for his interest in our work andfor motivating us by his work on cotorsion pairs and model structures.

For all unexplained terminology see [17].

2. GORENSTEINCATEGORIES

Our object now is to define Gorenstein categories and then exhibit some of theirproperties. In the sectionAwill always be a Grothendieck category with a specifiedgeneratorX. The symbolsY,Z etc. will denote objects ofA. We will use thegeneratorX to assign a cardinal number to every objectY . This cardinal will be|Hom(X,Y )|. We will now give several lemmas with the object of showing that wecan do what is called set-theoretic homological algebra in aGrothendieck category.

We refer the readers to [31] for the definition and basic properties of a Grothendieckcategory.

Lemma 2.1. If I is a set and ifX(I) → Y is an epimorphism, then there is asubsetJ ⊂ I with |J | ≤ |Y | and such that the restrictionX(J) → Y is also anepimorphism.

Proof. For such anX(I) → Y we see that for eachi ∈ I we have an associatedmorphismX → Y . Let J ⊂ I be such that the morphismsX → Y correspondingto thej ∈ J give all these morphisms and such that ifj 6= j′ (j, j′ ∈ J) thenjandj′ correspond to different morphisms. Then|J | ≤ |Hom(X,Y )| = |Y |. Alsothere is a natural factorizationX(I) → X(J) → Y . So X(J) → Y is also anepimorphism. �

GORENSTEIN CATEGORIES AND TATE COHOMOLOGY ON PROJECTIVE SCHEMES 3

Lemma 2.2. For anyY there is an epimorphismX(|Y |) → Y .

Proof. Immediate. �

Corollary 2.3. Given any objectY ofA there is a cardinalκ such that ifU → Yis an epimorphism then there is a subobjectU ′ ⊂ U such thatU ′ → Y is anepimorphism and such that|U ′| ≤ κ.

Proof. Any suchU is a quotient ofX(I) for some setI. ThenX(I) → U → Yis an epimorphism. But from the above we see that there is a subsetJ ⊂ I with|J | ≤ |Y | such thatX(J) → U → Y is an epimorphism. So letU ′ be the image ofX(J) in U . So we see that it is easy to get aκ that bounds allU ′ that we get in thismanner. �

Corollary 2.4. For every cardinalκ there is a set of representatives of objectsYwith |Y | ≤ κ.

Proof. By having such a set we mean that we have a set ofY with |Y | ≤ κ suchthat everyZ with |Z| ≤ κ is isomorphic to someY in our set. Now let|Y | ≤ κ.Then from the aboveY is the quotient of the coproductX(κ). But we can clearlyform a set of representatives of quotients ofX(κ) sinceA is well-powered (see [30,Proposition 10.6.3]). �

Lemma 2.5. If Z ⊂ Y then|Z| ≤ |Y |.


Lemma 2.6. Given a cardinalκ there exists a cardinalλ such that if|Y | ≤ κ andif Z ⊂ Y then|Y/Z| ≤ λ.

Proof. By the Corollary above there is an epimorphismX(|Y |) → Y and so anepimorphismX(κ) → Y . HenceY is isomorphic to a quotient ofX(κ). Sincethere is a set of representatives of subobjectsS of X(κ), we can takeλ to be thesup of all|X(κ)/S| taken over allS ⊂ X(κ). �

Lemma 2.7. For any objectY and setI we have|Y (I)| ≤ |Y I | = |Y ||I|.

Proof. The equality follows from the equality Hom(X,Y I) = Hom(X,Y )I . Sincewe are in a Grothendieck categoryY (I) → Y I is a monomorphism, so we get theinequality. �

Lemma 2.8. For any objectsY andZ we have|Hom(Y,Z)| ≤ |Z||Y |.

Proof. We have an epimorphismX(|Y |) → Y so Hom(Y,Z) ⊂ Hom(X(|Y |), Z) =

Hom(Y,Z)|Y | But |Hom(Y,Z)|Y || = |Z||Y |. �

Lemma 2.9. If γ is an ordinal and if(κα)α<γ is a family of cardinal numbers,then there is a cardinal numberλ such that if(Yα)α<γ is a family of objects withYα ⊂ Yα′ wheneverα ≤ α′, γ and such that|Yα| ≤ κα for eachα < γ then| ∪ Yα| ≤ λ.


Proof. We have an epimorphismX(|Yα|) → Yα for eachα. So since we are ina Grothendieck category we have an epimorphismX(

P

|Yα|) → ∪Yα and so wehave an epimorphismX(

P

κα) → ∪Yα. So∪Yα is a quotient object ofX(P

κα).So now we appeal to Lemma 2.6 and get ourλ. �

Lemma 2.10. Given a cardinalκ there is a cardinalλ such that if|Y | ≤ κ then|E(Y )| ≤ λ (hereE(Y ) is an injective envelope ofY ).

Proof. We only need argue that for a givenκ there is aλ such that if|Y | ≤ κ thenY ⊂ E for an injective objectE where|E| ≤ λ. To show this we use a categoricalversion of Baer’s original proof that every module can be embedded in an injectivemodule. SinceX is a generator Baer’s criterion says an objectE is injective if andonly if it is injective for X, i.e. if for any subobjectS ⊂ X and any morphismS → E there is an extensionX → E. Given the objectY = Y0 Baer firstconstructsY1 with Y0 ⊂ Y1 and such that for anyS ⊂ X and anyS → Y0 thereis an extensionS → Y1. By his constructionY1 is the quotient of the coproductof Y0 and copies ofX where this is a copy ofX for eachS → Y0 (with S ⊂ Xarbitrary). The quotient identifies each suchS ⊂ X for a givenS → Y0 with itsimage inY0. By Lemmas 2.6 and 2.7 we see that ifκ = κ0 is a cardinal we canfind a cardinalκ1 such that if|Y | = |Y0| ≤ κ then theY1 as constructed above issuch that|Y1| ≤ κ1. Then for any ordinalβ Baer constructs a continuous chain(Yα)α≤β of objects (soYα ⊂ Yα′ if α ≤ α′ ≤ β and if γ ≤ β is a limit ordinalthenYγ = ∪Yα (α < γ) whereY0 is a givenY . And then ifα + 1 ≤ β then wegetYα+1 for Yα just asY1 is constructed fromY0 as above and whereYγ = ∪Yα

α < γ. Then for a givenκ we see that we can find a family(κα)α≤β of cardinalnumbers such thatκ0 = κ and such that if(Yα)α≤β is constructed as above where|Y | = |Y0| ≤ κ then|Yα| ≤ κα for eachα ≤ β.We now note that ifS ⊂ X and if S → Yβ is such that there is a factorizationS → Yα → Yβ for someα < β thenS → Yα has an extensionX → Yα+1 sogiving the extensionX → Yα+1 → Yβ of the originalS → Yβ. So we must chooseβ such that every suchS → Yβ has such a factorization. So we want to argue thatwe can find aβ independent ofY so that the correspondingYβ will always beinjective. But this again just uses Baer’s original idea andappeals to the fact thatevery object is small relative to the class of monomorphisms(see [22, pg. 32] forthe terminology and [1, Proposition 2.2] for an argument). The object in questionwould be the coproduct of a representative set of subobjectsS of X. �

Lemma 2.11. Givenn ≥ 0 and a cardinalκ there is a cardinalλ so that ifL isan object of injective dimension at mostn and if S ⊂ L is such that|S| ≤ κ thenthere is anL′ ⊂ L such thatS ⊂ L′, such that|L′| ≤ λ and such that bothL′ andL/L′ have injective dimension at mostn.

Proof. The proof is just the categorical version of [2, Proposition2.5]. There theargument uses Corollary 2.3 and Lemma 2.4. To get our versionof Corollary 2.3we only need use the Baer criterion with respect to a generator X of A. Thecategorical version of Lemma 2.4 is just the preceeding lemma. �


Definition 2.12. A classL of objects ofA is said to be a Kaplansky class (see[19]) of A if for each cardinalκ there is a cardinalλ such that ifS ⊂ L for someL ∈ L where|S| ≤ κ then there is anL′ ⊂ L with S ⊂ L′ where|L′| ≤ λ andwhereL′ andL/L′ are both inL.

Corollary 2.13. If n ≥ 0 and if L is the class of objectsL of A such thatinjdimL ≤ n thenL is a Kaplansky class ofA.

Proof. Immediate from the above. �

In what follows we will no longer need to refer to a fixed generator X of A andso will use the symbolX to stand for an arbitrary object ofA.

Corollary 2.14. If n ≥ 0 and if L is the class of objectsL of A such thatinjdimL ≤ n thenL is preenveloping.

Proof. Given the objectX of A we consider morphismsX → L whereL ∈ L.Using Lemmas 2.5,2.6 and the Corollaries 2.4 and 2.13 we see that we can find acardinalλ such that for anyX → L with L ∈ L there is a factorizationX → L′ ⊆L with L′ ∈ L and|L′| ≤ λ. Then using Corollary 2.4 we see that there is a setL∗

of objects inL such that ifL ∈ L and if |L| ≤ λ thenL ∼= L∗ for someL∗ ∈ L∗.So noting that for eachL∗ ∈ L∗, Hom(X,L∗) is a set we see that we can find afamily (σi)i∈I of morphismsσi : X → Li with Li ∈ L

∗ so that ifL∗ ∈ L∗ andif σ : X → L∗ is a morphism, thenσ = σi for somei ∈ I. Then the morphismX →

∏i∈I Li given by the family(σi)i∈I is the desired preenvelope. �

We remark that for anyX of A we haveX ⊆ E for an injective objectE of C.SinceE ∈ L we have the factorizationX → L → E for any suchL-preenvelopeX → L. HenceX → L is necessarily a monomorphism.

We will eventually want another property of a classL of A. In the next Lemmawe will use the notion of transfinite extensions. For a definition see ([23, Section6]).

We recall that sinceA is a Grothendieck category (and so has enough injectives)we can define the functors Extn(X,Y ) for all n ≥ 0.

Definition 2.15. If X is an object ofA we say thatprojdim X ≤ n if

Exti(X,−) = 0 for i ≥ n + 1.

Analogously if an objectX has finite injective dimension at mostn we writeinjdim Y ≤ n. So then we defineprojdim X andinjdim Y as usual. We defineFPD(A) as the supremum ofprojdim X over allX such thatprojdim X <∞.We defineFID(A) in a similar manner.

Definition 2.16. A class of objectsL of A is said to be closed under transfiniteextensions if whenever(Lα)α≤λ is a continuous chain of objects ofA such thatL0 = 0 and such thatLα+1/Lα ∈ L wheneverα + 1 ≤ λ we also haveLλ in L.

Lemma 2.17.Givenn ≥ 0, if L is the class of objectsL ofA such thatprojdim L ≤n thenL is closed under transfinite extensions.


Proof. This result (for modules) is due to Auslander. Our result follows fromHovey’s proof of [23, Lemma 6.2]. In that proof he shows that for any objectY of A the class ofX such that Ext1(X,Y ) = 0 is closed under transfinite exten-sions (this is a categorical version of a theorem of Eklof [9]). If we let Y rangethrough the class ofn-th cosyzygies of objects ofA we get the result by using thefact that Extn+1(X,Z) = Ext1(X,Y ) if Y is such a cosyzygy forZ. �

Definition 2.18. We will say thatA is a Gorenstein category if the following hold:1) For any objectL of A, projdim L <∞ if and only if injdim L <∞.2) FPD(A) <∞ andFID(A) <∞.3)A has a generatorL such thatprojdim L <∞.

So now when we say that(A,L) is a Gorenstein category we mean thatA is sucha category and thatL is the class of objectsL of A such thatprojdim L <∞.

If furthermoreFPD(A) ≤ n andFID(A) ≤ n we will say that(A,L) hasdimension at mostn and then define the dimension of(A,L) to be the least suchn.

Remark. Generators with finite projective dimension in Grothendieck cate-gories without enough projectives were also considered by Hovey in [24, Section2].

Definition 2.19. By a complete projective resolution inA we mean a complexP = (Pn) for n ∈ Z (so the complex is infinite in both directions) of projectiveobjects such thatP is exact and such that the complex Hom(P, Q) is also exactfor any projective objectQ of A. If C is the kernel ofP 0 → P 1 then we say thatP is a complete projective resolution ofC. An objectC is said to be Gorensteinprojective if it admits such a complete projective resolution. Complete injectiveresolutions and Gorenstein injective objects are defined dually (see [16]).

Definition 2.20. Given a classC of objects ofA then the class of objectsY of Asuch that Ext1(C, Y ) = 0 for all C ∈ C is denotedC⊥. Similarly ⊥C denotes theclass ofX such that Ext1(X,C) = 0 for all C ∈ C.

Proposition 2.21. LetA be a Grothendieck category and letC (J ) be the class ofGorenstein projective (injective) objects ofA (there may not be any other than 0).ThenC⊥ ( ⊥J ) is a thick subcategory ofA containing all injective and all projec-tive objects ofA. SoC⊥ ( ⊥J ) contains all objects of finite injective dimensionand all objects having a finite projective resolution.

Proof. We prove the result forC andC⊥. A dual argument gives the result forJand⊥J .

We note thatY ∈ C⊥ if and only if for every complete projective resolutionPthe complex Hom(P, Y ) is exact. So easilyC⊥ is closed under retracts.

If 0 → Y ′ → Y → Y ′′ → 0 is an exact sequence inA and if P = (Pn) is acomplete projective resolution, then since eachPn is projective we get that

0→ Hom(P, Y ′)→ Hom(P, Y )→ Hom(P, Y ′′)→ 0

is an exact sequence of complexes. Hence if any two of these complexes is exact,so is the third. HenceC⊥ is a thick subcategory ofA.


If E is an injective object ofA, then since any complete projective resolutionP is exact, Hom(P, E) is also exact. SoE ∈ C⊥. If Q is a projective objectof A, then by the definition of a complete projective resolutionP the complexHom(P, Q) is exact. Hence we get all the claims aboutC⊥. �

Definition 2.22. If A is Grothendieck with enough projectives defineGpd(X) (theGorenstein projective dimension ofX) in the usual way. That is,Gpd(X) = n ifthe first syzygy ofX that is Gorenstein projective is then-th one andGpd(X) =∞if there is no such syzygy. Then defineglGpd(A) (the global Gorenstein projectivedimension ofA). Then also defineGid(Y ) andglGid(A) (without assumingAhas enough projectives).

Definition 2.23. A pair (F , C) of classes of objects ofA is said to be cotorsion pairif F⊥ = C and if⊥C = F . It is said to be complete if for eachX andY ofA thereexist exact sequences0→ C → F → X → 0 and0→ Y → C ′ → F ′ → 0 whereF,F ′ ∈ F and whereC,C ′ ∈ C.We furthermore say that(F , C) is functoriallycomplete if these sequences can be chosen in a functorial manner (depending onX and onY ) (see Definition 2.3 of [23]).

We now give our main result.

Theorem 2.24. If (A,L) is a Gorenstein category then(L,L⊥) is a complete andhereditary cotorsion pair onA andL⊥ is the class of Gorenstein injective objectsofA. If (A,L) has dimension at mostn thenGid(Y ) ≤ n for all objectsY ofA.

Proof. To get that(L,L⊥) is a cotorsion pair we only need argue that⊥(L⊥) = L.ClearlyL ⊂ ⊥(L⊥). But sinceprojdim L ≤ n for all L ∈ L, L⊥ contains allthe n-th cosyzygiesY of objects ofA. If Ext1(L, Y ) = 0 for all suchY thenprojdim L ≤ n and so⊥(L⊥) ⊂ L and so we getL = ⊥(L⊥).

As a first step toward arguing that this cotorsion pair is complete, we want toargue that it is cogenerated by a set, i.e. there is a setS with S ⊆ L such thatS⊥ =L⊥. But this follows from the fact thatL is closed under transfinite extensions andthat it is a Kaplansky class (see Lemma 2.17 and Corollary 2.13 respectively). Forlet κ besup|T | whereT is the image of an arbitrary morphismX → L for L ∈ L(every suchT ∼= X/Z for someZ ⊆ X, so we are using Corollary 2.3). Now letλ be as in Definition 2.12 for thisκ andL. Now letS be a set of representatives ofS ∈ L such that|S| ≤ λ (here we are using Corollary 2.4 withκ replaced byλ).Then we see that everyL ∈ L can be written as the union of a continuous chain(Lα)α≤β (for some ordinalβ) of subobjects such thatα + 1 ≤ β, Lα+1/Lα isisomorphic to anS ∈ S. Then we appeal to [10, Lemma 1] to see thatS⊥ = L⊥.

So now to get functorial completeness we want to appeal to [23, Theorem 6.5].To do so we need to show that(L,L⊥) is small according to [23, Definition 6.4]. Inthis definition, Hovey gives three conditions on a cotorsionpair in a Grothendieckcategory that are required for it to be small. In our situation the cotorsion pair is(L,L⊥). Applied to this pair (and, again, in our situation) Hovey’sconditions are:i) L contains a set of generators of the category,ii) (L,L⊥) is cogenerated by a setS ⊂ L, iii) for eachL ∈ S there is a given exact sequence0→ K → U → L→ 0


such that for anyY , Y ∈ L⊥ if and only if Hom(U, Y ) → Hom(K,Y ) → 0 isexact for all such exact sequence.

We have conditioni) by our definition of a Gorenstein category. We haveii)from the above.

We now argue that(L,L⊥) satisfies a slightly weaker version ofiii). We arguethat for eachL ∈ S we have a set of exact sequences0→ K → U → L→ 0 (oneset for eachL ∈ S) so thatY ∈ L⊥ if and only if Hom(U, Y ) → Hom(K,Y ) →0 is exact for all such exact sequences. Our set for a givenL will be a set ofrepresentatives of all short exact sequences0 → K → U → L → 0 where|U | ≤ κ and where we get theκ from Corollary 2.3 when we let theY of thatlemma be ourL.

So now supposeG is an object such that Hom(U,G) → Hom(K,G) → 0is exact for all of the exact sequences in our set. We want to argue then thatExt1(L,G) = 0 for all L ∈ L. BecauseS cogenerates(L,L⊥) it suffices toargue that Ext1(L,G) = 0 for all L ∈ S. So let0 → G → V → L → 0 be exactwith L ∈ S. We want to argue that this sequence splits. But we know that there isaU ⊂ V such that|U | ≤ κ and such thatU → L is an epimorphism. Then, up toisomorphism, we can suppose that withK = G ∩ U , 0 → K → U → L → 0 isone of our sequences.

So consider the commutative diagram

0 - K - U - L - 0

? ?

0 - G - V - L - 0

Since by hypothesisK → G can be extended toU → G we get that the bottomsequence splits.

If Z is a Gorenstein injective object then using a complete injective resolutionof Z it is easy to argue that Ext1(L,Z) = 0 whenprojdim L ≤ n. For such aZ isann-th cosyzygy of some objectW of A and Ext1(L,Z) = Extn+1(L,W ) = 0.HenceL⊥ contains all the Gorenstein injective objects ofA.

We now argue that ifG ∈ L⊥ thenG is Gorenstein injective. If0 → G →E0 → E1 → · · · is an injective resolution ofG, then sinceE ∈ L for any injectiveobjectE we get that Hom(E,−) applied to0 → G → E0 → E1 → · · · givesan exact sequence. So we have the right half of a complete injective resolutionof G. We must now show that we can construct the left half. Since(L,L⊥) iscomplete there is an exact sequence0 → K → L → G → 0 with K ∈ L⊥ andwith L ∈ L. SoL → G is a specialL-precover ofG. Let L ⊂ E whereE isinjective. Theninjdim E/L < ∞ and soE/L ∈ L and Ext1(E/L,G) = 0.This means thatL → G can be extended to (necessarily epimorphic)E → G.Let 0 → H → E → G → 0 be exact. ThenE → G is also anL-precover.So Hom(E,E) → Hom(E,G) → 0 is exact for any injectiveE. We now see thatExt1(E,H) = 0. This follows from the exact0→ Hom(E,H)→ Hom(E,E)→


Hom(E,G) → Ext1(E,H) → Ext1(E,E) = 0 and the fact that Hom(E,E) →Hom(E,G) is surjective. So now replaceG with H in the argument above and wefind an analogousE′ → H with E′ injective and such that this morphism is anL-precover ofH. Since we can continue this procedure we see that we can constructa complete injective resolution

· · · → E′ → E → E0 → E1 → · · ·

of G. Finally we want to argue thatGid(Y ) ≤ n for any objectY of A. But ifZ is ann-th cosyzygy ofY we have Ext1(L,Z) = 0 for all L ∈ L. HenceZ isGorenstein injective.

We now argue that our cotorsion pair(L,L⊥) is hereditary. In this situationthis means that we need argue that if0 → G′ → G → G′′ → 0 is exact withG′, G ∈ L⊥, thenG′′ ∈ L⊥. So we need to argue thatG′′ is Gorenstein injective.We letL→ G be a specialL-precover ofG. This means we have an exact sequence

0→ H → L→ G→ 0

with H Gorenstein injective. So we get a commutative diagram

0 0

? ?

H H

? ?

0 - G - L - G′′ - 0

? ?

0 - G′ - G - G′′ - 0

? ?

0 0

with exact rows and columns. Then sinceL⊥ is closed under extensions we getthatL ∈ L⊥. SoL ∈ L⊥ ∩ L. Considering an exact sequence

0→ L→ E →M → 0

with E injective we get by Proposition 2.21 thatM ∈ L. So the sequence splitsand we haveL injective. We have the exact0 → G → L → G′′ → 0 with GGorenstein injective and where Hom(E,L) → Hom(E,G′′) → 0 is exact ifE isinjective, since Ext1(E,G) = 0 by Proposition 2.21. So using the left half of acomplete injective resolution ofG along with the exact0 → G → L → G′′ → 0and an injective resolution ofG′′ we get a complete injective resolution ofG′′. �


Theorem 2.25. If (A,L) be a Gorenstein category of dimension at mostn havingenough projectives. Then for an objectC ofA the following are equivalent:

1) C is ann-th syzygy.2) C ∈ ⊥L.3) C is Gorenstein projective.As a consequence we get thatglGpd(A) ≤ n and that(⊥L,L) is a complete an

hereditary cotorsion pair.

Proof. 1) ⇒ 2) If C is ann-th syzygy ofX then Ext1(C,L) = Extn+1(X,L) forany objectL of A. If L ∈ L theninjdimL ≤ n so Extn+1(X,L) = 0. HenceExt1(C,L) = 0 and soC ∈ ⊥L.

2)⇒ 3) By Corollary 2.14 we knowC has anL-preenvelopeC → L. As notedafter the proof of that Corollary,C → L is a monomorphism. Let0 → L′ →P → L → 0 be exact whereP is projective. Then sinceprojdimL < ∞ we getprojdimL′ < ∞ and soL′ ∈ L. Since Ext1(C,L′) = 0, C → L can be factoredC → P → L. ThenC → P is a monomorphism and is also andL-preenvelope ofC. So continuing this procedure we get that we get an exact sequence

0→ C → P 0 → P 1 → P 2 → · · ·

where eachPn is projective and such that ifQ is projective then the functorHom(−, Q) leaves the sequence exact.

Now let· · · → P2 → P1 → C → 0

be a projective resolution ofC. Since Exti(C,Q) = 0 for i ≥ 1 andQ projectivewe see that Hom(−, Q) leaves this sequence exact. Consequently pasting we seethat we get a complete projective resolution ofC.

3)⇒ 1) is trivial.The equivalent1)⇔ 3) gives thatglGpd(A) ≤ n.We know argue that(⊥L,L) is a complete cotorsion pair. The fact thatglGpd(A ≤

n gives that for each objectX of A there is an exact sequence

0→ L→ C → X → 0

with C Gorenstein projective andprojdimL ≤ n − 1 in casen ≥ 1 (and withL = 0 if n = 0). The argument is essentially the dual to the proof of [17, Theorem11.2.1].

Then using what is called the Salce trick (see [17, Proposition 7.1.7]) we getthat for every objectX of A there is an exact sequence0 → X → L → C → 0with L ∈ L andC ∈ ⊥L. Hence ifX ∈ (⊥L)⊥ the sequence splits. SoX asa summand ofL ∈ L is also inL. So we get that(⊥L)⊥ = L and that(⊥L,L)is a cotorsion pair. then completeness follows from the above. Since an exact0 → L′ → L → L′′ → 0 with L,L′′ ∈ L givesL′ ∈ L and sinceA has enoughprojectives we get that(⊥L,L) is hereditary. �

Lemma 2.26. If A is Grothendieck with enough projectives then

glGpd(A) ≤ m ⇒ FID(A) ≤ m


and the converse holds ifA is Gorenstein (soglGpd(A) = FID(A) in this case).Dually we have thatglGid(A) ≤ k ⇒ FPD(A) ≤ k and the converse holds ifAis Gorenstein (soglGid(A) = FPD(A) in this case).

Proof. Let L have finite injective dimension. Now, given any objectX of A,let 0 → C → Pm−1 → · · · → P0 → X → 0 be a partial projective resolu-tion of X. ThenC is Gorenstein projective and so Ext1(C,L) = 0. This givesthat Extm+1(X,L) = 0. SinceX was arbitrary,injdim L ≤ m. Now assumeFID(A) ≤ m and thatA is Gorenstein. Let0 → C → Pm−1 → · · · → P0 →X → 0 be a partial projective resolution of anyX. Then if L ∈ L we haveinjdim L < ∞ soinjdim L ≤ m. So Extm+1(X,L) = 0, i.e., Ext1(C,L) = 0.So C ∈ ⊥L. But this meansC is Gorenstein projective by Theorem 2.25. Theargument for the rest of the proof is dual to this argument. �

Theorem 2.27. LetA be a Grothendieck category with enough projectives. Thenthe following are equivalent:

1)A is Gorenstein.2) glGpd(A) <∞ andglGid(A) <∞.Moreover, if (1) (or (2)) holds we have

FID(A) = FPD(A) = glGpd(A) = glGid(A).

Proof. We have1) implies2) by Theorem 2.24 and Theorem 2.25.So now assume2). LetL consist of all objectL with projdimL < ∞ and let

L′ consist of all objectsL′ with injdimL′ < ∞. Now assumeglGpd(A) ≤ nand thatglGid(A) ≤ n. If C is ann-th syzygy of an objectX of A thenC isGorenstein projective. So by Proposition 2.21, Ext1(C,L) = 0 for anyL ∈ L. SoExtn+1(X,L) = 0 for anyX and anyL ∈ L. SoinjdimL ≤ n and soL ∈ L′. Adual argument gives thatL′ ⊆ L and that ifL′ ∈ L′ thenprojdimL′ ≤ n. So weget thatL′ = L and so that(A,L) is a Gorenstein category.

To get the desired equality we note we haveFPD(A) = glGid(A) andFID(A) =glGpd(A) by Lemma 2.26. So we argue thatglGid(A) = glGpd(A). We use thefact that(⊥L,L) and(L,L⊥) are complete cotorsion pairs with⊥L = C the classof Gorenstein projectives andL⊥ = J the class of Gorenstein injectives.

Furthermore we know Hom(−,−) is right balanced byC×J (see [20, Theorem1.2.19]). So we can define relative derived functors of Hom(−,−). These aredenoted byGextn(X,Y ) (n ≥ 0 for any X, Y objects inA). Once we havethis machinery, then we use the usual argument that the following are equivalentfor an abelian categoryA with enough injectives and projectives and ann with0 ≤ n <∞:

1) projdim X ≤ n for all X,2) injdim Y ≤ n for all Y ,3) Gexti(−,−) = 0 for i ≥ n + 1. �

We get examples of Gorenstein categories with enough projectives by consid-ering RMod whereR is an (Iwanaga) Gorenstein ring (see 9.1 of [17]). Theserings are noetherian. But there are many nontrivial nonnoetherian R such that


RMod is Gorenstein (see [13] and [14]). By a trivial example we mean one withl.gldim R <∞).

Proposition 2.28. If A is a Gorenstein category of dimension at most n, then forevery objectY there is an exact sequence

0→ Y → G→ L→ 0

whereG is Gorenstein injective and whereinjdim L ≤ n− 1.

Proof. We can mimic the proof for modules given in [17, Theorem 11.2.1]. �

We note that if0 → Y → G → L → 0 is as above, thenY → G is a specialGorenstein injective preenvelope ofY . Using these preenvelopes we get a versionof relative homological algebra that is called Gorenstein homological algebra. Wesee that for each objectY we get a Gorenstein injective resolution ofY . This justmeans an exact sequence

0→ Y → G0 → G1 → · · ·

where all theGn are Gorenstein injective and such that Hom(−, G) leaves thesequence exact wheneverG is Gorenstein injective. Such a complex is unique upto homotopy and can be used to give right derived functorsGexti(X,Y ) of Hom(these functors were introduced in [15] and were later studied in [6] with differentnotation). There are obvious natural mapsGexti(X,Y ) → Exti(X,Y ) for alli ≥ 0.

The Tate cohomology functorsExti

A(X,Y ) (for any i ∈ Z) are defined as fol-lows.Let 0 → Y → E0 → E1 → · · · → En−1 → G → 0 be a partial injective reso-lution of Y . ThenG is Gorenstein injective so has a complete injective resolutionwhich we can take to be

E = · · · → En−1 → En → En+1 → · · ·

with G = Ker(En → En+1). This complex is unique up to homotopy and so

we define the groupsExti(X,Y ) to be thei-th cohomology groups of the complex

Hom(X,E).If 0 → Y → E′0 → E′1 → · · · is an injective resolution ofY then there is a

commutative diagram

... - 0 - 0 -E′0 -E′1 -E′2 - ...

? ? ? ? ?

... -E−2 -E−1 -E0 -E1 -E2 - ...

The associated map of complexes is unique up to homotopy and gives rise tonatural maps

Exti(X,Y )→ Exti(X,Y )

for i ≥ 0. In [6], Avramov and Martsinkovsky gave a beautiful result relatingthese two collections of natural maps. In their situation they considered finitely


generated modulesM with Gpd(M) < ∞ over a noetherian ring. And so used acomplete projective resolution as above to get the Tate cohomology functors. A.Iacob in [26] removed the finitely generated assumption and also showed how touse the hypothesisGid(Y ) < ∞ instead of the Gorenstein projective dimensionhypothesis. Then she showed that if bothGpd(X) < ∞ andGid(Y ) < ∞ then

the two procedures give the same groupsExti(X,Y ). So due to Iacob’s extension

of the Avramov and Martsinkovsky results we get:

Proposition 2.29. If A is a Gorenstein category of dimension at most n then forall objectsX andY ofA there exist natural exact sequences

0→ Gext1(X,Y )→ Ext1(X,Y )→ Ext1(X,Y )→ Gext2(X,Y )→ · · · →

Gextn(X,Y )→ Extn(X,Y )→ Extn(X,Y )→ 0.

The following is immediate:

Proposition 2.30. In the situation above, the following are equivalent:1) Gext1(X,Y )→ Ext1(X,Y ) is an isomorphism for allX, Y .2) Gexti(X,Y )→ Exti(X,Y ) is an isomorphism for allX, Y and all i ≥ 1.

3) Exti(X,Y ) = 0 for all X, Y andi ≥ 1.

4)L = A.5) J = I (I is the class of injective objects).6) C = P (P is the class of projective objects, only in caseA has enough

projectives).

7) Exti(X,Y ) = 0 for all X, Y andi ∈ Z.

8) For a fixedi, Exti(X,Y ) = 0 for all X, Y .

Proof. This is just an application of [23, Theorem 2.2], (also see [23, Theorem8.6]). For the first claim we use the notation of that paper andlet C = A,W = LandF = L⊥. For the second part letC = ⊥L,W = L andF = A. �

3. THE CATEGORY OF QUASI-COHERENT SHEAVES OVERPn(A)

Let A be a commutative noetherian ring. This section deals with the categoryof quasi-coherent sheaves overP

n(A) and then over closed subschemesX ⊂P

n(A). We want to prove that for certain suchX this category is Gorenstein. Inthis section we will use the fact thatQco(X) over any schemeX is equivalent tothe category of certain representations of some quiverQ, which may be chosen invarious ways, (see [11, Section 2] or [25, 3.1] for an explanation of this viewpoint).In caseX ⊂ P

n(A) is a closed subscheme, this quiver can be taken to have anespecially nice form. First,Pn(A) andQco(Pn(A)) can be associated with thequiverQ whose vertices are the subsets

v ⊆ {0, 1, 2, · · · , n}, v 6= ∅,

where there is a unique arrowv → w whenv ⊆ w. The associated ringR issuch thatR(v) is the ring of polynomials with coefficients inA in the variablesxi/xj , where0 ≤ i ≤ n andj ∈ v. Then, whenv ⊆ w, R(v) → R(w) is just alocalization (by the multiplicative set generated by thexi/xj , i ∈ w, j ∈ v).


Example. If we consider the projective schemeP1(A), the previous quiver is

{0} → {0, 1} ← {1}

Let us consider the projective schemeP2(A). Then the corresponding quiver

has the form

{1,2}

{0,1,2}

{0,1}

{0} {2}{1}

{0,2}

Notice that, for example, we may delete the arrow from{0} to {0, 1, 2} becausethis map is the obvious composition{0} → {0, 1} → {0, 1, 2}

A closed subschemeX ⊆ Pn(A) is given by a quasi-coherent sheaf of ideals,

i.e. we have an idealI(v) ⊆ R(v) for eachv with

R(w) ⊗R(v) I(v) ∼= I(w),

whenv ⊆ w. This just meansI(v)→ I(w) is the localization ofI(v) by the samemultiplicative set as above. But then

R

I(v) =

R(v)

I(v)→

R(w)

I(w)

(whenv ⊆ w) is also a localization. So now, to simplify the notation, wewill useR(v) in place ofR(v)

I(v) to give theR associated withX.The next result is standard in algebraic geometry. Those whowork in this area

will recognize ourDv as essentially thei∗ of [21, Proposition II.5.8].

Proposition 3.1. For a given vertexv, the functor

Hv : Qco(X)→ R(v)Mod

given byHv(M) = M(v) has a right adjoint.

Proof. We considerv as fixed. LetN be anR(v)-module. We construct a quasi-coherentR-moduleDv(N) as follows: for anyw letDv(N)(w) = R(v∪w)⊗R(v)

N (soDv(N)(w) is a localization ofN ). If w1 ⊆ w2 we have the obvious map

Dv(N)(w1) = R(v ∪ w1)⊗R(v) N → R(v ∪ w2)⊗R(v) N

given byR(v ∪ w1) → R(v ∪ w2). The quasi-coherence ofDv(N) follows fromthe definition ofDv(N). Given a quasi-coherentR-moduleM we have

HomQco(X)(M,Dv(N))→ Hom(Hv(M), N).

On the other hand, givenHv(M) = M(v)→ N , then for anyw we get

R(v ∪w)⊗R(v) M(v)→ R(v ∪ w)⊗R(v) N = Dv(N)(w).


But we haveM(w)→M(v∪w), so composing we getM(w)→ Dv(N)(w) withthe required compatibility. Then it is not hard to check thatwe have the requiredadjoint functor. �

Example. We considerQco(P2(A)). Then, ifN is anR({0, 1})-module, thequasi-coherentR-moduleD{0,1}(N) is given by

N

0

0 0

0

N

N

The next result is also standard in algebraic geometry (cf. the comment preced-ing Proposition 3.1).

Corollary 3.2. With the previous notation, we have:

• If N is injective thenDv(N) is an injectiveR-module.• If N ′ → N → N ′′ is an exact sequence ofR(v)-modules thenDv(N ′)→

Dv(N)→ Dv(N ′′) is also exact.

In particular an injective resolution

0→ N → E0 → E1 → · · ·

gives an injective resolution

0→ Dv(N)→ Dv(E0)→ Dv(E1)→ · · ·


From the previous Corollary we have

Corollary 3.3. GivenN andv as above

ExtiQco(X)(M,Dv(N)) ∼= Exti

R(v)(Hv(M), N).

The next Corollary says that ifM has finite projective dimension then “locallyM has finite projective dimension”.

Corollary 3.4. If projdim M <∞ thenprojdim M(v) <∞ for everyv.

Now we focus our attention in proving the converse of Corollary 3.4. To do thiswe will use a special case of theN , v, Dv(N) construction above. Given anR-moduleM , M 6= 0, choosev ∈ Q maximal (with respect to⊆) so thatM(v) 6= 0.Then if v w we haveR(w) ⊗R(v) M(v) = 0. Let N = M(v) and we usethis v to constructDv(N). Now also note that starting withM andR(v)N , anyM(v) → N givesM → Dv(N). We use this procedure in the special case abovewhere we chosev maximal such thatM(v) 6= 0 and letN = M(v). So letting


M(v) → N = M(v) be id we get a morphismM → Dv(N) and so an exactsequence

0→ Kv →M → Dv(N)→ Cv → 0.

Moreover, if the previous maximalv is not unique and we denote byB the set ofmaximal elements we have the exact sequence

0→ K →M → ⊕v∈BDv(N)→ C → 0.

Definition 3.5. By Supp(M) for any quasi-coherentR-moduleM we mean theset ofv ∈ Q such thatM(v) 6= 0.

By the construction ofDv(N) fromM as above it is easy to see thatSupp(Dv(N)) ⊆Supp(M), so alsoSupp(K), Supp(C) ⊆ Supp(M). But also we see (by the con-struction) thatv /∈ Supp(K), Supp(C) since(M(v) → Dv(N)(v)) = idN forsuchv. SoSupp(K), Supp(C) ( Supp(M) whenM 6= 0.

Remark. It follows from the previous definition thatSupp(M) = ∅ if, and onlyif, M = 0. If |Supp(M)| = 1 then Supp(M) is of the form{i} for somei,0 ≤ i ≤ n. If w ∈ Supp(M) andw′ ⊆ w thenw′ ∈ Supp(M). If |Supp(M)| = 1andSupp(M) = {i} then choosingv as above (maximal such thatM(v) 6= 0) wesee thatv = {i} and that withN = M({i}) we have in factM = Dv(N).

Lemma 3.6. For anyM , projdim M <∞ if, and only if,Exti(M,Dv(N)) = 0for i >> 0 and for anyN, v.

Proof. The condition is clearly necessary. Now let us assume the condition. Wewant to prove Exti(M,U) = 0 for i >> 0 and any quasi-coherent moduleU . IfSupp(U) = ∅ there is nothing to prove. If|Supp(U)| = 1, thenU = Dv(N)for someN, v and so we have Exti(M,U) = 0 for i >> 0 by hypothesis. So weproceed by induction on|Supp(U)|. But givenU 6= 0 we construct

0→ K → U → ⊕v∈BDv(U(v))→ C → 0

as above (lettingB be the set of maximal elements withU(v) 6= 0, for all v ∈ B).Then sinceSupp(K), Supp(C) ( Supp(U) we use our induction hypothesis andeasily get Exti(M,U) = 0 for i >> 0. �

Corollary 3.7. LetM be a quasi-coherentR-module. Thenprojdim M <∞ if,and only if,projdim M(v) < ∞ for all v ∈ Q. Moreover, ifprojdim M(v) < sfor all v ∈ Q, thenprojdim M < s + n.

Proof. The first statement follows by the isomorphism

Exti(M,Dv(N)) ∼= Exti(M(v), N).

To see the second one, letN be any quasi-coherentR-module and letwN be thefollowing non negative integer:wN = max{j | Supp(N) contains a subset ofcardinalityj}.

We consider the exact sequence given in the proof of the Lemma3.6:

0→ K → N → ⊕v∈BDv(N)→ C → 0,


which splits into two short exact sequences:

0→ K → N → L→ 0,

0→ L→ ⊕v∈BDv(N)→ C → 0.

From the second short exact sequence we get the long exact sequence:

· · · → Exti−1(M,C)→ Exti(M,L)→ Exti(M,⊕v∈BDv(N))→ · · · .

Then, we know, by hypothesis, that

Exti(M,⊕v∈BDv(N)) = ⊕v∈BExti(M(v), N(v)) = 0

for all i > s. Hence ifwN = 1 we have Exti(M,N) = 0 for all i > s becauseNis a direct sum ofDv(T ) for several objectsT and verticesv. Then we can proveby induction onwN that Exti(M,N) = 0 for all i > s + wN − 1. If wN = 1the result is proved above. So letN such thatwN = t. Then, by the constructionof the exact sequence above, we deduce thatwC < t andwK < t. Therefore,by induction, Exti−1(M,C) = 0 for all i − 1 > s + wC − 1. This implies thatExti(M,L) = 0 for all i > s + wL − 1 (becausewL > wC ). So, from the firstshort exact sequence, we get the long exact sequence of Ext:· · · → Exti(M,K)→Exti(M,N) → Exti(M,L) → · · · and, again by induction applied toK, weconclude that Exti(M,N) = 0 for all i > s + wN − 1 (note thatwL = wN ).

If we takeN such thatwN = n + 1, we immediately get thatprojdimM ≤s + n. �

Note. We also knowinjdim M < ∞ if, and only if, injdim M(v) < ∞ for allv. In fact injdim M = supv injdim M(v). As a result of the previous Corollary,the corresponding statement forprojdim M is not true.

Now we shall find a family of generators forQco(X) with finite projective di-mension. We have the family ofO(k), k ∈ Z for P

n(A). These give the family{i∗(O(k)) : k ∈ Z}, wherei : X → P

n(A) (see [21, pg. 120] for notation andterminology) we will letO(k) denotei∗(O(k)). SinceprojdimR(v) O(k)(v) = 0for all k ∈ Z and all vertexv, we get, by Corollary 3.7, thatprojdimO(k) ≤ n forall k ∈ Z (so for an example, by [21, Theorem 5.1(c)], Extn(O(0),O(−n−1)) 6= 0in theX = P

n(A) case, so we getprojdim O(0) = n).It was proved by Serre (see for example [21]) that every coherent sheaf onX ⊆

Pn(A) is the quotient of a finite sum of elements of the family{O(k) : k ∈ Z}.

But every quasi-coherent sheaf onX is the filtered union of coherent subsheafs. Sowe get thatL = ⊔l∈ZO(k) is a generator forQco(X). Furthermore we know thatprojdim L ≤ n < ∞. (For a different, and more general, way to get this familyof generators with finite projective dimension see [24, Proposition 2.3]).

Now recall that for a Gorenstein ringB (here commutative noetherian andinjdim B <∞) we have

projdim L <∞ ⇔ injdim L <∞


for anyB-moduleL (see [29]). Now suppose thatX ⊆ Pn(A) is such thatR(v)

is Gorenstein, for any vertexv (X will be called, as usual, a locally Gorensteinscheme). Then putting all of the above together we get.

Theorem 3.8. If A = Qco(X) for X ⊆ Pn(A) a locally Gorenstein scheme, then

A is a Gorenstein category.

Lemma 3.9. LetM be a quasi-coherentR-module, and let

0→M → E0 → E1 → E2 → · · ·

be an injective resolution ofM . Then

0→M(v)→ Hv(E0)→ Hv(E1)→ Hv(E2)→ · · ·

is an injective resolution ofM(v).

Proof. It is immediate, because the functorHv(−) is exact and preserves injectiveobjects. �

As a consequence we get that for a quasi-coherent sheaf beingGorenstein injec-tive is a local property.

Corollary 3.10. Let X ⊆ Pn(A) be a locally Gorenstein scheme andM be a

quasi-coherentR-module overX. ThenM is Gorenstein injective if and only ifM(v) is a Gorenstein injectiveR(v)-module, for all vertexv.

Proof. We shall use the pair of adjoint functors(Hv,Dv) (with v ∈ Q) obtainedin Proposition 3.1.

⇒) Let· · · → E−2 → E−1 → E0 → E1 → E2 → · · ·

be an exact sequence of injective quasi-coherentR-modules such thatM = ker(E0 →E1). Then, for a fixed vertexv, we have an exact sequence of injectiveR(v)-modules

· · · → Hv(E−2)→ Hv(E−1)→ Hv(E0)→ Hv(E1)→ Hv(E2)→ · · ·

Then if we take an integer sufficiently large in absolute value, and apply Lemma3.9 and [17, Theorem 9.1.11(7)] we have thatHv(E−m)→ Hv(E−m+1)→ · · · →Hv(E1)→ Hv(E0)→ · · · remains exact when we apply the functor Hom(E,−),for all m ≥ 0 and for all injectiveR(v)-moduleE (in fact the previous is a leftInj-resolution, see page 167 of [17] ). SoM(v) is Gorenstein injective.

⇐) Let M be a quasi-coherentR-module such thatM(v) is Gorenstein injectiveR(v)-module. SinceQco(X) is a Gorenstein category we may find an exact se-quence0 → M → G → L → 0 with G Gorenstein injective andL ∈ L. SinceM(v) andG(v) are Gorenstein injective (the last by the previous implication) it fol-lows thatL(v) is also Gorenstein injective (L⊥ is a coresolving class). ThenL(v)is Gorenstein injective with finite projective dimension, hence an injective module,for all v. Now we take the injective cover ofG (which is an epimorphism with aGorenstein injective kernel) so we get the exact sequence0→ K → E → G→ 0


with E injective andK Gorenstein injective. Now we make the pull-back squareof E → G andM → G,

0 0

? ?

K K

? ?

0 - U - E - L - 0

? ?

0 - M - G - L - 0

? ?

0 0

ThenU is a quasi-coherent module with finite projective dimension, because it ispart of the exact sequence0 → U → E → L → 0, and the classL is a resolvingclass. FurthermoreU(v) is a Gorenstein injectiveR(v)-module, for allv, becauseit is in the middle of the exact sequence0 → K → U → M → 0. It follows thatU(v) is an injectiveR(v)-module, for allv, so U is an injective quasi-coherentR-module. So again, sinceL⊥ is coresolving, we conclude thatM is a Gorensteininjective quasi-coherentR-module. �

Remark. It is easy to see that the methods of this section apply to other schemes.One of the main properties we require of such a scheme is that the associatedquiverQ can be chosen so that (with the obvious notation) eachR(v) → R(w) isa localization. This is the case, for example, of toric varieties. We also point outthat the category of quasi-coherent sheaves over these schemes is equivalent to aquotient categoryS − gr/T , for a suitable graded ringS and torsion classT (see[8]). This fact could be useful in order to give a new focus in the topic treatedin this paper, taking into account that the Gorenstein property in graded rings iswell-behaved (see [3, 4]).

Acknowledgement.The authors are grateful to the referee. He not only suggested improvements in

the paper but also supplied us with a proof that was lacking.The final version of this paper was completed during Sergio Estrada’s stay at

department of Mathematics of the University of Kentucky with the support of aMEC/Fulbright grant from the Spanish Secretarıa de Estadode Universidades eInvestigacion del Ministerio de Educacion y Ciencia.


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Gorenstein categories and Tate cohomology on projective schemes

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