LOCAL DUALITY FOR THE SINGULARITY CATEGORY OF A FINITE DIMENSIONAL GORENSTEIN ALGEBRA

DAVE BENSON, SRIKANTH B. IYENGAR, HENNING KRAUSE AND JULIA PEVTSOVA

Abstract. A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander-Reiten triangles for the 𝔭-local and 𝔭-torsion subcategories of the derived category, for each homogeneous prime ideal 𝔭 arising from the action of a commutative ring via Hochschild cohomology.

1. Introduction

This work concerns duality phenomena in various triangulated categories of modules over Gorenstein algebras. By a Gorenstein algebra we mean here an algebra A, finite dimensional over a field k, with the property that A has finite injective dimension both as a left module and a right module over itself. For such an A the derived Nakayama functor is an equivalence:

ν: D^{b}(mod A) ⟶^{∼} D^{b}(mod A), where X ↦ Hom_{k}(A, k) ⊗_{A}^{L} X .

Happel [16] proved that for perfect complexes X, Y there is a natural isomorphism

Hom_{k}(Hom_{D}(X, Y), k) ≅ Hom_{D}(Y, νX) .

where D := D^{b}(mod A). In other words, the Nakayama functor on D restricts to a Serre functor, in the sense of Bondal and Kapranov [11], on the full subcategory of perfect complexes.

In this work we discover that Happel’s result is only the tip of an iceberg: It is a special case of a duality on D, analogous to Grothendieck’s local duality for commutative Gorenstein algebras. The duality on D also involves a (graded-)commutative algebra, namely, HH^{∗}(A/k), the Hochschild cohomology, of A over k that acts on D via canonical homomorphisms of k-algebras

HH^{∗}(A/k) ⟶ Ext_{A}^{∗}(X, X)  for each X ∈ D.

In this way D acquires a structure of an HH^{∗}(A/k)-linear category.

In the remainder of the introduction we fix a homogeneous k-subalgebra R of HH^{∗}(A/k) that is finitely generated as a k-algebra. For simplicity of exposition we assume that R^{0} = k. This is not a great loss of generality for given any R as above, we can drop down to the subring k ⊕ R^{⩾1} without sacrificing the finite generation. A natural choice for R is k ⊕ HH^{⩾1}(A/k) but, for example, when A is a Hopf algebra, like the group algebra of a finite group, or group scheme, it is more natural to take R = Ext_{A}^{∗}(k, k), the cohomology ring of A, for this is functorial in the ring argument, whilst the Hochschild cohomology is not.

Date: May 5, 2020.

2010 Mathematics Subject Classification. 16G10 (primary); 16G50, 16E65, 16E35.

Key words and phrases. Gorenstein algebra, local duality, maximal Cohen-Macaulay module, Serre duality, singulartity category.

SBI was partly supported by NSF grants DMS-1503044 and DMS-1700985, and JP was partly supported by NSF grants DMS-0953011 and DMS-1501146. The authors are grateful to the American Institute of Mathematics in San Jose, California for supporting this project by their “Research in Squares” program.

Fix a homogeneous prime ideal 𝔭 of R. Let γ_{𝔭}(D) be the triangulated category obtained from D by localising the graded morphisms at 𝔭 and then taking the full triangulated subcategory of objects such that the graded endomorphisms are 𝔭-torsion. By construction γ_{𝔭}(D) is an R_{𝔭}-linear category, where R_{𝔭} denotes the homogenous localisation of R at 𝔭. The Nakayama functor induces an equivalence

ν_{𝔭}: γ_{𝔭}(D) ⟶^{∼} γ_{𝔭}(D) .

Let Proj R denote the homogenous prime ideals in R not containing R^{⩾1}, the unique maximal homogenous ideal of R. Our local Serre duality statement reads:

Theorem 1.1. Let 𝔭 be in Proj R and let d be the Krull dimension of R/𝔭. Then Σ^{−d} ∘ν_{𝔭} is a Serre functor for γ_{𝔭}(D), in that, for all X, Y in γ_{𝔭}(D) there are natural isomorphisms

Hom_{R_{𝔭}}(Hom_{γ_{𝔭}(D)}^{∗}(X, Y), I(𝔭)) ≅ Hom_{γ_{𝔭}(D)}(Y, Σ^{−d}ν_{𝔭}(X))

where I(𝔭) is the injective hull of the graded R-module R/𝔭.

This result is proved towards the end of Section 5 from a more general statement concerning the category GProj A of (possibly infinite dimensional) Gorenstein projective modules. These are A-modules M with the property that Ext_{A}^{i}(M, P) = 0 for i ≥ 1 and projective A-module P. The connection to D is through its subcategory, Gproj A, consisting of finitely dimensional modules. These are precisely the maximal Cohen-Macaulay A-modules, in the terminology of Buchweitz [13].

The stable category, \underline{GProj} A, of GProj A is a compactly generated triangulated category, with compact objects equivalent to \underline{Gproj} A, the stabilisation of Gproj A. Buchweitz [13] proved that there is a equivalence of triangulated categories

\underline{Gproj} A ⟶^{∼} D_{sg}(A) := D^{b}(mod A)/D^{b}(proj A)

where D_{sg}(A) is the singularity category, also known as the stable derived category, of A. There is a natural R-action on \underline{GProj} A and the equivalence above is compatible with the induced R-actions. For each prime ideal 𝔭 in Proj R the canonical functors induce equivalences of triangulated categories

γ_{𝔭}(\underline{Gproj} A) ⟶^{∼} γ_{𝔭}(D) ⟶^{∼} γ_{𝔭}(D_{sg}(A))

compatible with R_{𝔭}-actions, by Lemma 5.5. Thus to prove Theorem 1.1 it suffices to prove the corresponding statement for \underline{Gproj} A; equivalently, for the singularity category of A. This also explains the title of this paper.

To that end we consider the subcategory Γ_{𝔭}(\underline{GProj} A) of \underline{GProj} A consisting of the 𝔭-local 𝔭-torsion modules. These are the Gorenstein projective A-modules M with the property that for each finite dimensional A-module C, every element of \underline{Hom}_{A}^{∗}(C, M), the graded R-module of morphisms in \underline{GProj} A, is annihilated by some power of 𝔭, and the natural map \underline{Hom}_{A}^{∗}(C, M) → \underline{Hom}^{∗}(C, M)_{𝔭} is bijective. Then Γ_{𝔭}(\underline{GProj} A) is also a compactly generated triangulated category and the full subcategory of compact objects is equivalent, up to direct summands, to γ_{𝔭}(\underline{Gproj} A). There is an idempotent functor Γ_{𝔭}: \underline{GProj} A → \underline{GProj} A with image the 𝔭-local 𝔭-torsion modules; see Section 3 for details. The central result of this work is a local duality theorem for this category:

Theorem 1.2. Let 𝔭 be in Proj R and let d be the Krull dimension of R/𝔭. Let X, Y be Gorenstein projective A-modules and suppose that X is finite dimensional. Then there is a natural isomorphism

Hom_{R}(Ext_{A}^{∗}(X, Y), I(𝔭)) ≅ \underline{Hom}_{A}(Y, Ω^{d}Γ_{𝔭} GP ν(X)) .

Here GP is the Gorenstein projective approximation functor; see Section 2 for details. The theorem above is contained in Theorem 5.1.

Theorems 1.1 and 1.2 are formulated in terms of an arbitrary (but fixed) subalgebra R of the Hochschild cohomology of A. It is thus natural to ask how these results are related as we vary R. This point is addressed in Remark 5.8. Another issue is what transpires in Theorem 1.1 if we set 𝔭 = R^{⩾1}. When the ring R is such that, in addition to being noetherian, the R-module Ext_{A}^{∗}(X, X) is finitely generated for each X ∈ D, the subcategory Γ_{𝔪}(D) is precisely the subcategory of perfect complexes and the analogue of Theorem 1.1 is Happel’s duality; see Remark 5.6.

The duality statements above are modeled on, and extensions of, analogous results for representation of modules over finite group schemes established in [10]. We refer to that work for antecedents of these results and for applications, notably, the existence of AR triangles in γ_{𝔭}(D). The proof of the results in op. cit. exploited the tensor structure on the module categories in question, but in fact the arguments can be readily adapted to deal with the general case, as we do here. In doing so, it became clear that local duality is a feature of Gorenstein algebras in general.

2. Gorenstein algebras

Throughout this work k will be a field and A a finite dimensional k-algebra that is Gorenstein (also known as Iwanaga-Gorenstein): the injective dimension of A as a left A-module and as a right A-module is finite. In this case, the injective dimensions are the same; this was proved by Zaks [23]. Evidently when A is Gorenstein so is A^{op}, the opposite algebra of A.

In this section we recall basic notions and results from the homological theory of Gorenstein algebras, mainly pertaining to duality. To begin with we note that the Gorenstein condition on A is equivalent to: An A-module has finite projective dimension if and only if it has finite injective dimension; see, for example, [13, Lemma 5.1.1]. This will be used often in the sequel.

Let Mod A be the category all (left) A-modules and mod A its full subcategory consisting of finitely generated A-modules. The full subcategory of mod A consisting of projective A-modules is denoted proj A. The injective analogue is denoted inj A. In what follows for A-modules M, N we set

\underline{Hom}_{A}(M, N) := Hom_{A}(M, N)/\{ϕ | ϕ factors through a projective\}
\overline{Hom}_{A}(M, N) := Hom_{A}(M, N)/\{ϕ | ϕ factors through an injective\}.

When M and N are finitely generated, it suffices to consider maps ϕ that factor through proj A and inj A, respectively.

In the sequel a duality between categories will mean a contravariant equivalence.

Vector space duality. For any A-module M we set

DM := Hom_{k}(M, k) ,

viewed as an A^{op}-module. The assignment M ↦ DM induces a duality

mod A ⟶^{∼} mod A^{op} .

which restricts to a duality proj A ⟶^{∼} inj A^{op}. The functor D also extends to a duality between the corresponding bounded derived categories:

(2.1)  D: D^{b}(mod A) ⟶^{∼} D^{b}(mod A^{op}) .

Let D^{b}(proj A) denote the bounded derived category of proj A. We identify it with the subcategory of D^{b}(mod A) consisting of perfect complexes. The duality above restricts to D^{b}(proj A) ⟶^{∼} D^{b}(proj A^{op}), since A is Gorenstein.

The singularity category. Buchweitz [13] introduced the stable derived category of A as the Verdier quotient

D_{sg}(A) := D^{b}(mod A)/D^{b}(proj A) .

This category was rediscovered by Orlov [19], who called it the singularity category of A, and our notation reflects this terminology. When the global dimension of A is finite one has D_{sg}(A) = 0, so the singularity category is one measure of the deviation of A from finite global dimension.

The duality (2.1) induces a duality

(2.2)  D: D_{sg}(A) ⟶^{∼} D_{sg}(A^{op}) .

The singularity category can be realized (in more than one way) as a stabilisation of a subcategory of mod A. This is described next.

Gorenstein projective and Gorenstein injective modules. An A-module X is Gorenstein projective if

Ext_{A}^{i}(X, P) = 0 for each projective P and each i ≥ 1.

We write GProj A for the category of Gorenstein projective A-modules and set

Gproj A := GProj A ∩ mod A .

These are the maximal Cohen-Macaulay A-modules, in Buchweitz’s terminology.

Standard arguments (following, for example, [16, Section 9]) yield that GProj A is a Frobenius exact category, with the projective A-modules the projective and injective objects in this category; this requires the hypothesis that A is Gorenstein. We write \underline{GProj} A for the corresponding stable category; it is a triangulated category. Its thick subcategory consisting of the finitely generated modules is denoted \underline{Gproj} A, for it can be realised as a stabilisation of Gproj A.

On Gproj A the syzygy functor Ω has an inverse, denoted Ω^{−1}, that is welldefined up to projective summands. This is the translation on \underline{GProj} A.

An A-module Y is Gorenstein injective if

Ext_{A}^{i}(Q, Y) = 0 for each injective Q and each i ≥ 1.

We write GInj A for the category of Gorenstein injective modules and set Ginj A := GInj A∩mod A. The stabilisation of GInj A is denoted \overline{GInj} A, and the subcategory of finitely generated modules is \overline{Ginj} A. These are triangulated categories, where the translation ΣY of a Y ∈ GInj A is the cokernel of an embedding into an injective A-module:

0 ⟶ Y ⟶ I ⟶ ΣY ⟶ 0

The duality D: mod A ⟶^{∼} mod A^{op} restricts to dualities

(2.3)  Gproj A ⟶^{∼} Ginj A^{op} and proj A ⟶^{∼} inj A^{op} .

and hence induces a duality of triangulated categories

(2.4)  D: \underline{Gproj} A ⟶^{∼} \overline{Ginj} A^{op} .

Here is the result on realising the singularity category as a stabilisation.

Proposition 2.1. The inclusions Gproj A → D^{b}(mod A) and Ginj A → D^{b}(mod A) induce triangle equivalences

p: \underline{Gproj} A ⟶^{∼} D_{sg}(A)  and  q: \overline{Ginj} A ⟶^{∼} D_{sg}(A) .

Proof. The first equivalence is Theorem 4.4.1 from [13], and the second equivalence follows from that statement applied to A^{op} and dualities (2.4). □

Approximations. The following result—see [13, Lemma 5.1.1]—is straightforward to verify.

Lemma 2.2. Let X be a Gorenstein projective module.

(1) Ext_{A}^{i}(X, F) = 0 for any module F of finite projective dimension and i ≥ 1.

(2) If an A-linear map X → N factors through a module of finite projective dimension, then it factors through a projective module.

The analogous statements for Gorenstein injective modules also hold. □

This has the following consequence.

Lemma 2.3. If X is Gorenstein projective and Y is Gorenstein injective, then

\underline{Hom}_{A}(X, Y) = \overline{Hom}_{A}(X, Y) .

Proof. Given Lemma 2.2, one has to verify that a map f: M → N factors through a module of finite projective dimension if and only if it factors through a module of finite injective dimension. This is a tautology as these categories coincide. □

The following result is due to Auslander and Buchweitz, and is the cornerstone of their theory of maximal Cohen-Macaulay approximations. There is an analogous statement involving Gorenstein injectives.

Proposition 2.4. Every finite dimensional A-module M fits into exact sequences

0 → F_{M} → X_{M} → M → 0 and 0 → M → F^{M} → X^{M} → 0

where X_{M}, X^{M} are in Gproj A and F_{M}, F^{M} have finite projective dimension. For X ∈ Gproj A and F of finite projective dimension, these sequences induce bijections

\underline{Hom}_{A}(X, X_{M}) ⟶^{∼} \underline{Hom}_{A}(X, M) and \underline{Hom}_{A}(F^{M}, F) ⟶^{∼} \underline{Hom}_{A}(M, F) .

Proof. This is part of [13, Theorems 5.1.2, 5.1.4]; see also [3, Theorems 1.8, 2.8]. □

We write GP(M) := X_{M} for the Gorenstein projective approximation of M. It follows from the preceding result that X_{M} is well-defined in \underline{Gproj} A.

Lemma 2.5. The Gorenstein projective approximation GP induces a triangle equivalence \overline{Ginj} A ⟶^{∼} \underline{Gproj} A satisfying q ≅ p ∘ GP, where p, q are the functors in Proposition 2.1.

Proof. Since pF = 0 for any A-module F of finite projective dimension, Proposition 2.4 implies qY ≅ p GP(Y) for any Gorenstein injective module Y. This yields also that GP is a triangle equivalence, since p, q are triangle equivalences. □

Remark 2.6. When A is self-injective the projective and injective A-modules coincide and hence GProj A = GInj A. Conversely, when the (projective) module A is Gorenstein injective, A is self-injective: Consider an exact sequence

0 ⟶ A ⟶ I ⟶ ΣA ⟶ 0

where I is injective. If A is Gorenstein injective, then so is ΣA, and hence the sequence above splits, as the injective dimension of A is finite. Thus A is injective.

Example 6.3 describes a Gorenstein algebra that is not self-injective, and identifies modules that are Gorenstein projective but not Gorenstein injective.

The Nakayama functor. This is the functor ν: mod A → mod A that assigns to each M in mod A the A-module

νM := DA ⊗_{A} M ≅ D Hom_{A}(M, A) .

This functor is an equivalence if A is self-injective but not in general. However the Gorenstein property of A implies that the derived Nakayama functor, for which we use the same notation, is an equivalence:

ν: D^{b}(mod A) ⟶^{∼} D^{b}(mod A) where M ↦ DA ⊗_{A}^{L} M .

From this it is not hard to verify that ν: mod A → mod A restricts to equivalences

proj A  inj A

Gproj A  Ginj A .

ν ∼

ν ∼

Therefore one gets an induced equivalence ν: \underline{Gproj} ⟶^{∼} \overline{Ginj} A.

On the other hand, since the complexes of finite projective dimension are the same as those of finite injective dimension (because A is Gorenstein), DA is in the thick subcategory of D^{b}(mod A^{op}) generated by A, and hence, by duality A is in the thick subcategory of D^{b}(mod A) generated by DA, that is to say, by νA. It follows that the derived Nakayama functor satisfies

ν(D^{b}(proj A)) = D^{b}(proj A) .

This is the essence of Happel duality. It induces an equivalence ν̄ on the singularity category that makes the following square commutative

(2.5)

\underline{Gproj} A  \overline{Ginj} A

D_{sg}(A)  D_{sg}(A) . ∼ p ν ∼ q ∼ ν̄ ∼

The vertical equivalences are from Proposition 2.1.

The Auslander transpose. Let X be a finite dimensional Gorenstein projective A-module. Any projective presentation

P_{1} ⟶ P_{0} ⟶ X ⟶ 0

induces an exact sequence of A^{op}-modules.

0 ⟶ Hom_{A}(X, A) ⟶ Hom_{A}(P_{0}, A) ⟶ Hom_{A}(P_{1}, A) ⟶ Tr X ⟶ 0 .

The A^{op}-module Tr X is the Auslander transpose of X, and it depends only on X up to projective summands. It is a Gorenstein projective A^{op}-module since it identifies with Hom_{A}(Ω^{2}X, A). Applying D yields an exact sequence of A-modules

0 ⟶ D Tr X ⟶ νP_{1} ⟶ νP_{0} ⟶ νX ⟶ 0 .

Since Tr X is Gorenstein projective over A^{op}, the A-module D Tr X is Gorenstein injective and the exact sequence above implies that there is an isomorphism

(2.6)  Σ^{2}(D Tr X) ≅ νX.

in \overline{Ginj} A.

Auslander-Reiten Duality. We are ready to state the main results of this section. For similar statements we refer to [1, 5].

Proposition 2.7. Let A be a Gorenstein algebra, and X, Y Gorenstein projective A-modules. When X is finite dimensional, there is a natural isomorphism

D\underline{Hom}_{A}(X, Y) ≅ \underline{Hom}_{A}(Y, Ω^{−1} GP(D Tr X)) .

Proof. The exact sequence 0 → ΩY → P → Y → 0 with P projective, induces the first of the following isomorphisms

D\underline{Hom}_{A}(X, Y) ≅ D Ext_{A}^{1}(X, ΩY)
≅ \overline{Hom}_{A}(ΩY, D Tr X)
≅ \underline{Hom}_{A}(ΩY, D Tr X)
≅ \underline{Hom}_{A}(ΩY, GP(D Tr X))
≅ \underline{Hom}_{A}(Y, Ω^{−1} GP(D Tr X)) .

The second isomorphism is Auslander-Reiten duality [2, Proposition I.3.4], whilst the third isomorphism is from Lemma 2.3, since ΩY is Gorenstein projective and D Tr X is Gorenstein injective. The fourth isomorphism follows from Proposition 2.4, and the final one is clear since Ω is an equivalence on \underline{Gproj} A. □

Following Bondal and Kapranov [11, §3], a Serre functor on a k-linear, Homfinite, additive category C is an equivalence F: C → C along with natural isomorphisms

D Hom_{C}(X, Y) ≅ Hom_{C}(Y, F X)

for all objects X, Y in C.

Theorem 2.8. Let A be a Gorenstein algebra. Then

Ω ∘ GP ∘ ν: \underline{Gproj} A ⟶^{∼} \underline{Gproj} A and Σ^{−1} ∘ ν̄: D_{sg}(A) ⟶^{∼} D_{sg}(A)

are Serre functors.

Proof. For any Gorenstein projective A-module X, one has a sequence of isomorphisms in \underline{Gproj} A, where the first and the last one are by construction:

Ω^{−1} GP(D Tr X) = Σ GP(D Tr X)
≅ GP(ΣD Tr X)
≅ GP(Σ^{−1}νX)
≅ Σ^{−1}(GP νX)
= Ω(GP νX) .

The second and the fourth isomorphisms hold because GP is a triangle functor, and the third one is by (2.6). Thus Proposition 2.7 yields that Ω ∘ GP ∘ν is a Serre functor on \underline{GProj} A, as claimed. The description of the Serre functor on D_{sg}(A) is then a consequence of Lemma 2.5 and (2.5). □

Compact generation. So far the results have mostly dealt with the categories \underline{Gproj} A and \overline{Ginj} A consisting of finite dimensional modules. To prove the local duality theorem announced in the introduction we need to work in larger categories, \underline{GProj} A and \overline{GInj} A. To this end we recall the following result, which is well-known at least for self-injective algebras.

Proposition 2.9. The stable categories \underline{GProj} A and \overline{GInj} A are compactly generated triangulated categories, and the full subcategories of compact objects identify with \underline{Gproj} A and \overline{Ginj} A, respectively.

Proof. The Nakayama functor induces a triangle equivalence \underline{GProj} A ⟶^{∼} \overline{GInj} A, identifying \underline{Gproj} A with \overline{GInj} A; the quasi-inverse is given by Hom_{A}(D(A), −). It thus suffices to verify the assertions about \overline{GInj} A.

It follows from [17, §5] that \overline{GInj} A is compactly generated; it remains to identify the compact objects. If X is a finite dimensional module, then \overline{Hom}_{A}(X, −) preserves direct sums. Thus every module in \overline{Ginj} A is compact in \overline{GInj} A.

On the other hand, for any nonzero module Y in \overline{GInj} A there exists a finite dimensional module M such that \overline{Hom}_{A}(M, Y) ≠ 0; this is because Y is not injective. Choose a Gorenstein injective approximation M → W, using the analogue of Proposition 2.4 for Gorenstein injective modules. Then

\overline{Hom}_{A}(W, Y) ≅ \overline{Hom}_{A}(M, Y) ≠ 0 ,

which implies that \overline{Ginj} A is all the compact objects of \overline{GInj} A. □

3. Cohomology and localisation

In this section we recall basic notions and constructions concerning certain localisation functors on triangulated categories with ring actions. The material is needed to state and prove the results in Section 4 and 5. The main triangulated category of interest is the stable category of Gorenstein projective modules. Primary references for the material presented here are [6, 7].

Triangulated categories with central action. Let T be a triangulated category with suspension Σ. Given objects X and Y in T, consider the graded abelian groups

Hom_{T}^{∗}(X, Y) = ⨁_{i∈ℤ} Hom_{T}(X, Σ^{i}Y) and End_{T}^{∗}(X) = Hom_{T}^{∗}(X, X) .

Composition makes End_{T}^{∗}(X) a graded ring and Hom_{T}^{∗}(X, Y) a left-End_{T}^{∗}(Y) right-End_{T}^{∗}(X) module.

Let R be a graded-commutative ring. We say the triangulated category T is R-linear if for each X in T there is a homomorphism of graded rings ϕ_{X}: R → End_{T}^{∗}(X) such that the induced left and right actions of R on Hom_{T}^{∗}(X, Y) are compatible in the following sense: For any r ∈ R and α ∈ Hom_{T}^{∗}(X, Y), one has

ϕ_{Y} (r)α = (−1)^{|r||α|}αϕ_{X}(r) .

An exact functor F: T → U between R-linear triangulated categories is R-linear if the induced map

Hom_{T}^{∗}(X, Y) ⟶ Hom_{U}^{∗}(F X, F Y)

of graded abelian groups is R-linear for all objects X, Y in T.

In what follows, we fix a compactly generated R-linear triangulated category T and write T^{c} for its full subcategory of compact objects.

Graded modules. In the remainder of this section R will be a graded commutative noetherian ring. We will only be concerned with homogeneous elements and ideals in R. In this spirit, ‘localisation’ will mean homogeneous localisation, and Spec R will denote the set of homogeneous prime ideals in R.

Given graded R-modules M and N, we denote by Hom_{R}(M, N) the R-linear maps ϕ: M → N such that ϕ(M^{i}) ⊆ N^{i} for all i ∈ ℤ, and

Hom_{R}^{∗}(M, N) = ⨁_{i∈ℤ} Hom_{R}(M, N[i])

where N[i]^{p} = N^{i+p} for all i, p ∈ ℤ.

Localisation. Fix an ideal 𝔞 in R. An R-module M is 𝔞-torsion if M_{𝔮} = 0 for all 𝔮 in Spec R with 𝔞 ⊈ 𝔮. Analogously, an object X in T is 𝔞-torsion if the R-module Hom_{T}^{∗}(C, X) is 𝔞-torsion for all C ∈ T^{c}. The full subcategory of 𝔞-torsion objects

Γ_{𝒱(𝔞)} T := {X ∈ T | X is 𝔞-torsion}

is localising and the inclusion Γ_{𝒱(𝔞)} T ⊆ T admits a right adjoint, denoted Γ_{𝒱(𝔞)} .

Fix a 𝔭 in Spec R. An R-module M is 𝔭-local if the localisation map M → M_{𝔭} is invertible, and an object X in T is 𝔭-local if the R-module Hom_{T}^{∗}(C, X) is 𝔭-local for all C ∈ T^{c}. Consider the full subcategory of T of 𝔭-local objects

T_{𝔭} := {X ∈ T | X is 𝔭-local}

and the full subcategory of 𝔭-local and 𝔭-torsion objects

Γ_{𝔭}T := {X ∈ T | X is 𝔭-local and 𝔭-torsion}.

Note that Γ_{𝔭}T ⊆ T_{𝔭} ⊆ T are localising subcategories. The inclusion T_{𝔭} → T admits a left adjoint X ↦ X_{𝔭} while the inclusion Γ_{𝔭}T → T_{𝔭} admits a right adjoint. We denote by Γ_{𝔭}: T → Γ_{𝔭}T the composition of those adjoints; it is the local cohomology functor with respect to 𝔭; see [6, 7] for the construction of this functor.

The functor Γ_{𝒱(𝔞)} commutes with exact functors preserving coproducts.

Lemma 3.1. Let F: T → U be an exact functor between R-linear compactly generated triangulated categories such that F is R-linear and preserves coproducts. Suppose that the action of R on U factors through a homomorphism f: R → S of graded commutative rings. For any ideal 𝔞 of R there is a natural isomorphism

F ∘ Γ_{𝒱(𝔞)} ≅ Γ_{𝒱(𝔞S)} ∘ F

of functors T → U, where 𝔞S denotes the ideal of S that is generated by f(𝔞).

Proof. The statement follows from an explicit description of Γ_{𝒱(𝔞)} in terms of homotopy colimits; see [7, Proposition 2.9]. □

The following observation is clear.

Lemma 3.2. For any element r in R \ 𝔭, say of degree n, and 𝔭-local object X, the natural map X ⟶^{r} Σ^{n}X is an isomorphism. □

Koszul objects. Fix objects X, Y in T. Any element b in R^{d} induces a morphism X → Σ^{d}X and let X//b denote its mapping cone. This gives a morphism X → Σ^{−d}(X//b). For a sequence of elements b := b_{1}, …, b_{n} in R set X//b := X_{n} where

X_{0} := X and X_{i} := X_{i−1}//b_{i} for 1 ≤ i ≤ n .

It is easy to check that for s = ∑_{i} |b_{i}| there is an isomorphism

(3.1)  Hom_{T}(X, Y//b) ≅ Hom_{T}(X//b, Σ^{s+n}Y).

Injective cohomology objects. Given an object C in T^{c} and an injective Rmodule I, Brown representability yields an object T (C, I) in T such that

(3.2)  Hom_{R}^{∗}(Hom_{T}^{∗}(C, −), I) ≅ Hom_{T}^{∗}(−, T (C, I)) .

This yields a functor

T: T^{c} × Inj R ⟶ T.

For each 𝔭 in Spec R, we write I(𝔭) for the injective hull of R/𝔭 and set

T_{𝔭} := T (−, I(𝔭)) ,

viewed as a functor T^{c} → T. For objects C and D in T^{c}, we have a natural R-linear isomorphism

(3.3)  Hom_{T}^{∗}(T (C, I), T (D, I)) ≅ Hom_{R}^{∗}(Hom_{R}^{∗}(Hom_{T}^{∗}(C, D), I), I) .

4. Hochschild and Tate cohomology

Let k be a field and A a finite dimensional Gorenstein k-algebra. The enveloping algebra of A is the k-algebra A^{e} := A ⊗_{k} A^{op}; it is also Gorenstein, by Proposition 6.1, but this observation does not play a role in the sequel. The Hochschild cohomology of the k-algebra A is

HH^{∗}(A/k) := Ext_{A^{e}}^{∗}(A, A) .

This is a graded-commutative k-algebra.

When X and Y are Gorenstein projective A-modules, we set

\underline{Hom}_{A}^{∗}(X, Y) = ⨁_{i∈ℤ} \underline{Hom}_{A}^{∗}(X, Ω^{−i}Y) .

This is the Tate cohomology of X, Y. There is a canonical homomorphism

(4.1)  Ext_{A}^{∗}(X, Y) ⟶ \underline{Hom}_{A}^{∗}(X, Y),

of graded abelian groups, induced from the canonical morphism tX → pX from a complete projective resolution to a projective resolution of X. In particular, this map is surjective in degree 0 and bijective in positive degrees.

Action on GProj A. For any A-module M there is a canonical map

HH^{∗}(A/k) ⟶^{−⊗_{A}M} Ext_{A}^{∗}(M, M)

that is a morphism of graded k-algebras. When X is Gorenstein projective, composing the map above with the one in (4.1) one gets a homomorphism of k-algebras

ϕ_{X}: HH^{∗}(A/k) ⟶ \underline{Hom}_{A}^{∗}(X, X)

and this induces a linear action of HH^{∗}(A/k) on \underline{GProj} A, in the sense of Section 3.

Assumption 4.1. We fix a homogenous k-subalgebra R of HH^{∗}(A/k) such that (1) R is connected, that is to say, R^{0} = k;

(2) R is finitely generated as a k-algebra. Connectedness implies that R^{≥1} is the unique maximal ideal of R, which allows us to import the results from [10]. This is not a serious restriction. Indeed A decomposes as a direct product of connected algebras, and for a connected algebra A the ring HH^{0}(A/k), being the center of A, is a finite dimensional local ring, so the inclusion R^{0} ⊆ HH^{0}(A/k) induces a bijection on the spectra.

As usual Proj R denotes the set of prime ideals that do not contain R^{⩾1}.

Condition (2) is equivalent to the condition that the ring R is noetherian. Since the HH^{∗}(A/K)-action on \underline{GProj} A restricts to an R-action, the noetherian property of R allows one to invoke the constructions and results presented in Section 3.

Base change. Let K/k be an extension of fields and set

A_{K} := K ⊗_{k} A .

This is a finite dimensional Gorenstein K-algebra, and extension of scalars

Mod A ⟶ Mod A_{K}, X ↦ X_{K} = K ⊗_{k} X

and restriction

Mod A_{K} ⟶ Mod A, X ↦ X↓_{A} = Hom_{K}(K, X)

form an adjoint pair of exact functors.

Lemma 4.2. For A-modules X and Y, the canonical K-linear map

K ⊗_{k} Hom_{A}(X, Y) ⟶ Hom_{A_{K}} (X_{K}, Y_{K})

is a isomorphism when X is finite dimensional over k. □

Lemma 4.3. Extension and restriction preserve projectivity and injectivity. Thus A is Gorenstein if and only if A_{K} is Gorenstein. In that case extension and restriction preserve Gorenstein projectivity. □

There are isomorphisms of K-algebras

(A_{K})^{e} ≅ (A^{e})_{K} and HH^{∗}(A_{K}/K) ≅ K ⊗_{k} HH^{∗}(A/k) .

Thus extension of scalars yields a ring homomorphism HH^{∗}(A/k) → HH^{∗}(A_{K}/K), and setting R_{K} := K ⊗_{k} R one gets a ring homomorphism

R ⟶ R_{K} .

Observe that R_{K} is connected and a noetherian K-subalgebra of HH^{∗}(A_{K}/K).

Next we recall a construction from [9] of a field extension K/k and a closed point in Proj R_{K} lying over a given a point in Proj R.

Construction 4.4. Let R be a finitely generated, graded, connected, k-algebra. Fix a point 𝔭 in Proj R, and let d be the Krull dimension of R/𝔭.

Choose elements a := a_{0}, …, a_{d−1} in R of the same degree such that their image in R/𝔭 is algebraically independent over k and R/𝔭 is finitely generated as a module over the subalgebra k[a]. Set K := k(t_{1}, …, t_{d−1}), the field of rational functions in indeterminates t_{1}, …, t_{d−1} and

b_{i} := a_{i} − a_{0}t_{i} for i = 1, …, d − 1

viewed as elements in R_{K}. Let 𝔭′ denote the extension of 𝔭 to R_{K}, and set

𝔮 := 𝔭′ + (b)  and  𝔪 := \sqrt{𝔮} .

The following statements hold:

(1) 𝔪 is a closed point in Proj R_{K} with the property that 𝔪 ∩ R = 𝔭;

(2) the induced extension of fields k(𝔭) ⟶^{≅} k(𝔪) is an isomorphism.

The first part is contained in [9, Theorem 7.7]. The second one holds by construction; see [9, Lemma 7.6, and (7.2)].

Fix an object X in \underline{GProj} A. The sequence of elements b in R_{K} yields a morphism X_{K} → Ω^{s}(X_{K}//b), where s = ∑_{i} |b_{i}|, and composing its restriction to A with the canonical morphism X → (X_{K})↓_{G} gives in \underline{GProj} A a morphism

f_{X}: X ⟶ Ω^{s}(X_{K}//b)↓_{A} .

Since the a_{i} are not in 𝔭, when X is 𝔭-local Lemma 3.2 yields a natural isomorphism

(4.2)  Ω^{s}X ≅ X

in \underline{GProj} A.

The result below extends [10, Theorem 3.4] that concerns modules over finite group schemes, but the argument is essentially the same.

Theorem 4.5. For any Gorenstein projective A-module X, the morphism f_{X} induces a natural isomorphism

Γ_{𝔭}X ≅ Γ_{𝔪}(X_{K}//b)↓_{A} .

in \underline{GProj} A. When X is 𝔭-torsion, this induces a natural isomorphism

Γ_{𝔭}X ≅ (X_{K}//b)↓_{A} .

Proof. Given the second isomorphism, the first one can be checked as follows: Let X be a Gorenstein projective A-module and 𝔭′ the ideal in Construction 4.4. Since Γ_{𝒱(𝔭)} X is 𝔭-torsion, one gets the second isomorphism below:

Γ_{𝔭}X ≅ (Γ_{𝒱(𝔭)} X)_{𝔭}
≅ ((Γ_{𝒱(𝔭)} X)_{K}//b)↓_{A}
≅ (Γ_{𝒱(𝔭′)} (X_{K})//b)↓_{A}
≅ (Γ_{𝒱(𝔭′)} (X_{K}//b))↓_{A}
≅ (Γ_{𝒱(𝔭′+(b))} (X_{K}//b))↓_{A}
≅ (Γ_{𝔪}(X_{K}//b))↓_{A} .

The third one is by Lemma 3.1, applied to the functor K ⊗_{k} (−) from \underline{GProj} A to \underline{GProj} A_{K}. The next one is standard while the penultimate one holds because X_{K}//b is (b)-torsion.

It remains to verify the second isomorphism in the statement. The modules satisfying this isomorphism form a localising subcategory of \underline{GProj} A. Moreover, by [7, Proposition 2.7], the 𝔭-torsion modules form a localising subcategory of \underline{GProj} A generated by the modules X//𝔭, for X ∈ Gproj A. It thus suffices to verify the desired isomorphism for such modules. Since X//𝔭 is 𝔭-torsion, the natural map

Γ_{𝔭}(X//𝔭) = Γ_{𝒱(𝔭)} (X//𝔭)_{𝔭} → (X//𝔭)_{𝔭}

is an isomorphism. Thus the task reduces to verifying that f_{(X//𝔭)_{𝔭}} is an isomorphism for each X ∈ Gproj A. This is proved in [9, Theorem 8.8] for the case of finite group schemes and X = k, the trivial representation. However the argument only uses [9, Proposition 6.2(2)] which in turn is a formal consequence of Lemma 3.1, and thus carries over to the present context. □

5. The Gorenstein property

Let F: \underline{Gproj} A → \underline{Gproj} A be the Serre functor from Theorem 2.8, given by

F (X) = (Ω ∘ GP ∘ ν)(X).

Given the description of F, the result below contains Theorem 1.2. It extends [10, Theorem 5.1], which deals with the case A is the group algebra of a finite group scheme and R = Ext_{A}^{∗}(k, k), its cohomology ring.

Theorem 5.1. Let k be a field, A a finite dimensional, Gorenstein, k-algebra and R ⊆ HH^{∗}(R/k) as in 4.1. Fix 𝔭 ∈ Proj R, and let d be the Krull dimension of R/𝔭. On \underline{Gproj} A there is a natural isomorphism of functors

Γ_{𝔭} ∘ F ≅ Ω^{−d+1} ∘ T_{𝔭} .

Thus for any object X in Gproj A there is a natural isomorphism

\underline{Hom}_{A}(−, Ω^{d−1}Γ_{𝔭}F (X)) ≅ Hom_{R}(Ext_{A}^{∗}(X, −), I(𝔭)) .

This result is proved further below, following some preparatory remarks.

Remark 5.2. Let X, Y be Gorenstein projective A-modules. The natural map

Ext_{A}^{∗}(X, Y) → \underline{Hom}_{A}^{∗}(X, Y)

is compatible with action of HH^{∗}(A/k), and hence of R. The map is surjective in degree zero, with kernel PHom_{A}(X, Y), the maps from X to Y that factor through a projective A-module. Since it is bijective in positive degrees one gets an exact sequence of graded R-modules

(5.1)  0 ⟶ PHom_{A}(X, Y) ⟶ Ext_{A}^{∗}(X, Y) ⟶ \underline{Hom}_{A}^{∗}(X, Y) ⟶ C ⟶ 0

with C^{i} = 0 for i ≥ 0. For degree reasons, the R-modules PHom_{A}(X, Y) and C are R^{⩾1}-torsion so for 𝔭 in Proj R the induced localised map is an isomorphism:

(5.2)  Ext_{A}^{∗}(X, Y)_{𝔭} ⟶^{∼} \underline{Hom}_{A}^{∗}(X, Y)_{𝔭} .

It follows from Lemma 4.3 that the functor F is compatible with base change.

Lemma 5.3. Let K/k be a field extension and F_{K}: \underline{Gproj} A_{K} → \underline{Gproj} A_{K} the corresponding Serre functor. For X in Gproj A there is a natural isomorphism F_{K}(X_{K}) ≅ F (X)_{K}. □

The argument below is direct adaptation of the one for [10, Theorem 5.1].

Proof of Theorem 5.1. The proof uses the following observation: For any A-modules X, Y that are 𝔭-local and 𝔭-torsion, there is an isomorphism X ≅ Y in \underline{GProj} A if and only if there is a natural isomorphism

\underline{Hom}_{A}(M, X) ≅ \underline{Hom}_{A}(M, Y)

for 𝔭-local and 𝔭-torsion A-modules M. This follows from Yoneda’s lemma.

In anticipation of using the preceding remark, we note that Γ_{𝔭}(X) and T_{𝔭}(X) are 𝔭-local and 𝔭-torsion. This is clear for Γ_{𝔭}(X) and follows for T_{𝔭}(X) from the fact that I(𝔭) is a 𝔭-local and 𝔭-torsion R-module. Another observation is that, by (5.2), for any 𝔭-local R-module I, there is an isomorphism

Hom_{R}(Ext_{A}^{∗}(X, −), I) ≅ Hom_{R}(\underline{Hom}_{A}^{∗}(X, −), I).

Consequently, one can rephrase the defining isomorphism (3.2) for the object T_{𝔭}(X) as a natural isomorphism

\underline{Hom}_{A}(−, T_{𝔭}(X)) ≅ Hom_{R}(Ext_{A}^{∗}(X, −), I(𝔭)) .

Our task is to verify that, on \underline{Gproj} A, there is an isomorphism of functors

Γ_{𝔭}F ≅ Ω^{−d+1}T_{𝔭} .

We verify this when 𝔭 is closed and then use a reduction to closed points.

Claim. The desired isomorphism holds when 𝔪 is a closed point in Proj R.

The injective hull, I(𝔪), of the R-module R/𝔪 is the same as that of the R_{𝔪}- module k(𝔪), viewed as an R-module via restriction of scalars along the localisation map R → R_{𝔪}. Let Y be a Gorenstein projective A-module that is 𝔪-local and 𝔪torsion. The claim is a consequence of the following computation:

\underline{Hom}_{A}(Y, Γ_{𝔪}(F X)) ≅ \underline{Hom}_{A}(Y, F X)
≅ \underline{Hom}_{A}(X, Y)^{∨}
≅ Hom_{R_{𝔪}}(\underline{Hom}_{A}^{∗}(X, Y), I(𝔪))
≅ Hom_{R}(\underline{Hom}_{A}^{∗}(X, Y), I(𝔪))
≅ \underline{Hom}_{A}(Y, T_{𝔪}(X)) .

The first isomorphism holds because Y is 𝔪-torsion; the second is Serre duality, Proposition 2.7, and the next one is by [10, Lemma A.2], which applies because \underline{Hom}_{A}^{∗}(X, Y) is 𝔪-local and 𝔪-torsion as an A-module.

Let 𝔭 be a point in Proj R that is not closed, and let K, b, and 𝔪 be as in Construction 4.4. Recall that 𝔪 is a closed point in R_{K} lying over 𝔭.

Claim. In \underline{GProj} A there is an isomorphism of A-modules

(5.3)  (T_{𝔪}(X_{K})//b)↓_{A} ≅ Ω^{−d+1}T_{𝔭}(X)

where d is the Krull dimension of R/𝔭.

Let Y be an A-module that is 𝔭-local and 𝔭-torsion. Then we have the following:

\underline{Hom}_{A}(Y, Ω^{d−1}(T_{𝔪}(X_{K})//b)↓_{A}) ≅ \underline{Hom}_{A_{K}} (Y_{K}, Ω^{d−1}(T_{𝔪}(X_{K})//b))
≅ \underline{Hom}_{A_{K}} (Y_{K}//b, T_{𝔪}(X_{K}))
≅ Hom_{R_{K}} (\underline{Hom}_{A_{K}}^{∗} (X_{K}, Y_{K}//b), I(𝔪))
≅ Hom_{R}(\underline{Hom}_{A_{K}}^{∗} (X_{K}, Y_{K}//b), I(𝔭))
≅ Hom_{R}(\underline{Hom}_{A}^{∗}(X, (Y_{K}//b)↓_{A}), I(𝔭))
≅ Hom_{R}(\underline{Hom}_{A}^{∗}(X, Y), I(𝔭))
≅ \underline{Hom}_{A}(Y, T_{𝔭}(X)) .

The first and fifth isomorphisms are by adjunction. The second one is a direct computation using (3.1) and (4.2). The next one is by definition and the fourth isomorphism is by [10, Lemma A.3], applied to the homomorphism R → R_{K}; it applies as the R_{K}-module \underline{Hom}_{A_{K}}^{∗} (X_{K}, Y_{K}//b) is 𝔪-torsion. The sixth isomorphism is by Theorem 4.5, and the last one by definition. This justifies the claim.

Consider now the chain of isomorphisms:

Γ_{𝔭}F (X) ≅ Γ_{𝔪}(F (X)_{K}//b)↓_{A}
≅ Γ_{𝔪}(F_{K}(X_{K})//b)↓_{A}
≅ (Γ_{𝔪}(F_{K}(X_{K}))//b)↓_{A}
≅ (T_{𝔪}(X_{K})//b)↓_{A}
≅ Ω^{−d+1}T_{𝔭}(X).

The first isomorphism is by Theorem 4.5, the second by Lemma 5.3, the third is clear, the fourth follows from the first claim, since 𝔪 is a closed point for A_{K}, and the last one follows from the second claim.

This completes the proof that the functors Γ_{𝔭} ∘F and Ω^{−d+1} ∘T_{𝔭} are isomorphic. Given this and the alternative description of T_{𝔭} above, the last isomorphism in the statement follows. □

Next we record a corollary of Theorem 5.1 concerning γ_{𝔭}(\underline{Gproj} A), the 𝔭-torsion objects in the 𝔭-localisation of \underline{Gproj} A; see [10, §7] and the references therein for details of this construction. The R-linear triangle equivalences

ν: \underline{Gproj} A → \overline{Ginj} A and GP: \overline{Ginj} A → \underline{Gproj} A

induce R_{𝔭}-linear triangle equivalences

(5.4)  γ_{𝔭}(\underline{Gproj} A)  γ_{𝔭}(\overline{Ginj} A) (\underline{Gproj} A)_{𝔭}  (\overline{Ginj} A)_{𝔭} \begin{matrix}ν_{𝔭} \\ ∼ \end{matrix} \begin{matrix}ν_{𝔭} \\ ∼ \end{matrix} and  γ_{𝔭}(\overline{Ginj} A)  γ_{𝔭}(\underline{Gproj} A) (\overline{Ginj} A)_{𝔭}  (\underline{Gproj} A)_{𝔭} \begin{matrix}GP & 𝔭 \\ ∼ \end{matrix} \begin{matrix}GP & 𝔭 \\ ∼ \end{matrix}

compatible with the localisation functor; see [10, Remark 7.1]. The result below can be interpreted as the statement that the category γ_{𝔭}(\underline{Gproj} A), and hence also γ_{𝔭}(\overline{Ginj} A), has a Serre functor.

Corollary 5.4. For X, Y in C := γ_{𝔭}(\underline{Gproj} A), there is a natural isomorphism

Hom_{R_{𝔭}}(\underline{Hom}_{C}^{∗}(X, Y), I(𝔭)) ≅ \underline{Hom}_{C}(Y, Ω^{d} GP_{𝔭} ν_{𝔭}(X)) .

Proof. Up to direct summands, the categories C and Γ_{𝔭}(\underline{GProj} A)^{c} are equivalent, and the compact objects in Γ_{𝔭}(\underline{GProj} A) are of the form M_{𝔭}, for some M ∈ \underline{Gproj} A that is 𝔭-torsion; see [10, Remark 7.2]. We obtain the desired isomorphism by reinterpreting the isomorphisms in Theorem 5.1, as follows.

Fix 𝔭-torsion objects M, N in \underline{Gproj} A. One has the first isomorphism below because M is finite dimensional:

Hom_{R_{𝔭}}(\underline{Hom}_{A}^{∗}(M_{𝔭}, N_{𝔭}), I(𝔭)) ≅ Hom_{R_{𝔭}}(\underline{Hom}_{A}^{∗}(M, N)_{𝔭}, I(𝔭))
≅ Hom_{R_{𝔭}}(Ext_{A}^{∗}(M, N)_{𝔭}, I(𝔭))
≅ Hom_{R}(Ext_{A}^{∗}(M, N), I(𝔭)) .

The second one is by (5.2), and the last one is by adjunction.

On the other hand, since M is 𝔭-torsion, so is F (M) and hence one has the first isomorphism below:

Ω^{d−1}Γ_{𝔭}F (M) ≅ Ω^{d−1}F (M)_{𝔭} ≅ Ω^{d−1}(Ω GP ν(M))_{𝔭} ≅ Ω^{d} GP_{𝔭} ν_{𝔭}(M_{𝔭})

The second one is by the definition of F and the third one follows by the discussion around (5.4). Applying \underline{Hom}_{A}(N, −) to the composition yields the first isomorphism below, whilst the second one holds as the covariant argument is 𝔭-local:

\underline{Hom}_{A}(N_{𝔭}, Ω^{d} GP_{𝔭} ν_{𝔭}(M_{𝔭})) ≅ \underline{Hom}_{A}(N_{𝔭}, Ω^{d−1}Γ_{𝔭}F (M))
≅ \underline{Hom}_{A}(N, Ω^{d−1}Γ_{𝔭}F (M)) .

The isomorphisms above and Theorem 5.1, applied with X = M and Y = N, yield the desired result. □

Lemma 5.5. For any 𝔭 ∈ Proj R the quotient functor D^{b}(mod A) → D_{sg}(A) induces equivalences

D^{b}(mod A)_{𝔭} ⟶^{∼} D_{sg}(A)_{𝔭} and γ_{𝔭}(D^{b}(mod A)) ⟶^{∼} γ_{𝔭}(D_{sg}(A))

compatible with the R_{𝔭} actions.

Proof. The quotient functor is essentially surjective and hence so is its 𝔭-localisation D^{b}(mod A)_{𝔭} → D_{sg}(A)_{𝔭}. The latter is also fully faithful, by (5.2), and so is an equivalence of categories. It is also R_{𝔭}-linear, by construction. Given this, it is immediate from the definition that the 𝔭-torsion subcategories are equivalent. □

Proof of Theorem 1.1. The hypotheses is that A is a finite dimensional Gorenstein algebra and R is a finitely generated, homogenous, k-subalgebra of HH^{∗}(A/k) with R^{0} = k. Also, D := D^{b}(mod A). We fix a homogeneous prime ideal 𝔭 in Proj R. We want to verify that for all X, Y in γ_{𝔭}(D) there are natural isomorphisms

Hom_{R_{𝔭}}(Hom_{γ_{𝔭}(D)}^{∗}(X, Y), I(𝔭)) ≅ Hom_{γ_{𝔭}(D)}(Y, Σ^{−d}ν_{𝔭}(X))

with d the Krull dimension of R/𝔭. Lemma 5.5 gives the second equivalence below:

γ_{𝔭}(\underline{Gproj} A) ⟶^{∼} γ_{𝔭}(D) ⟶^{∼} γ_{𝔭}(D_{sg}(A)) ,

whereas the first one is induced by Proposition 2.1, and both these are compatible with the R_{𝔭} actions. It remains to recall Corollary 5.4. □

The (Fg) condition. As in (4.1), let R ⊆ HH^{∗}(A/k) be a connected, noetherian, k-subalgebra. Assume in addition that for each M ∈ mod A the R-module Ext_{A}^{∗}(M, M) is finitely generated. Said otherwise, the algebra A satisfies the (Fg) condition with respect to R, introduced in [15]. As noted in [22, Proposition 5.7], this condition implies that the Hochschild cohomology algebra HH^{∗}(A/k) is finitely generated. The (Fg) condition holds for several interesting classes of algebras, including group algebras of finite groups (or group schemes) and finite dimensional commutative complete intersection rings; see [15, 22].

When A satisfies the (Fg) condition, it follows from Corollary 5.4 that the triangulated category γ_{𝔭}(\underline{Gproj} A), and hence also anything equivalent to it, has ARtriangles. This is explained in [10, Section 7], for which we refer the reader also for other consequences of Serre duality.

Remark 5.6. Assume that the algebra A satisfies the (Fg) condition with respect to R and set 𝔯 := R^{⩾1}, the homogenous maximal ideal of R. Set D := D^{b}(mod A).

Claim. The triangulated category γ_{𝔯}(D) is equivalent to D^{b}(proj A).

Indeed, by definition, γ_{𝔯}(D) consists of complexes M ∈ D for which Ext_{R}^{∗}(M, M) is 𝔯-torsion as an R-module; equivalently, Ext_{R}^{∗}(M, N) is 𝔯-torsion for each N ∈ D. In particular, Ext_{R}^{∗}(M, A/J) is 𝔯-torsion, where J is the Jacobson radical of A. Since the R-module Ext_{R}^{∗}(M, A/J) is finitely generated, by the (Fg) condition, this last property is equivalent to Ext_{R}^{∗}(M, A/J) = 0 for i ≫ 0, that is to say, M is perfect.

Given this claim, one can extend Theorem 1.1 even to 𝔯: For perfect complexes X, Y from [16] we get the second isomorphism below:

Hom_{R}(Hom_{D}^{∗}(X, Y), I(𝔯)) ≅ Hom_{k}(Hom_{D}^{∗}(X, Y), k)
≅ Hom_{D}(Y, ν(X)) .

The first one is by adjunction as I(𝔯) = Hom_{k}(R, k); see [10, Lemma A.2]. This is the stated Serre duality on γ_{𝔯}(D), for the Krull dimension of R/𝔯 is 0 and R_{𝔯} = R.

Remark 5.7. Concerning the claim in Remark 5.6: Even when A does not satisfy the (Fg) condition with respect to R, the subcategory γ_{𝔯}(D) contains the perfect complexes, but it is possible that it contains more. To see what is at stake, consider the special case that A is self-injective. Let M ∈ mod A be a module containing A as a direct summand and satisfying Ext_{A}^{i}(M, M) = 0 for all i > 0. It is easy to verify that M is in γ_{𝔯}(D). However, it is still unknown, and a conjecture of Auslander and Reiten [4], whether such an M has finite projective dimension, equivalently that M is projective.

Remark 5.8. With k and A as before, let R, S be connected, finitely generated, k-subalgebras of the Hochschild cohomology algebra HH^{∗}(A/k). Theorem 5.1, and so also its corollaries, applies to the action of R, and also of S, on \underline{GProj} A. In reconciling the two, one can replace S by the k-subalgebra of HH^{∗}(A/k) generated by R and S and assume R ⊆ S. Then one has an induced map φ: Proj S → Proj R defined by the assignment 𝔮 ↦ 𝔮 ∩ R.

Given 𝔮 in Proj S, it is clear that there is an inclusion

Γ_{𝔮}(\underline{GProj} A) ⊆ Γ_{𝔭}(\underline{GProj} A) where 𝔭 = 𝔮 ∩ R.

So the version of Theorem 5.1 for the action of S may be seen as a refinement of the one for the action of R. Indeed, in the extremal case R = k, one has 𝔭 = 0 for any 𝔮 in Spec S and Γ_{𝔭}(\underline{GProj} A) = \underline{Gproj} A.

A more interesting situation occurs when S is finite as an R-module. Then there are only finitely many primes 𝔮 in Proj S lying over a given 𝔭 ∈ Proj R, and [8, Corollary 7.10] yields a direct sum decomposition

Γ_{𝔭}(\underline{GProj} A) ≅ ⨁_{𝔮∈φ^{−1}(𝔭)} Γ_{𝔮}(\underline{GProj} A) .

This decomposition thus reflects the ramification, in the sense of commutative algebra, of the inclusion R ⊆ S.

6. Examples

In this section we describe some examples of Gorenstein algebras. Throughout k will be a field. The injective dimension of a finite dimensional k-algebra A is the injective dimension of A viewed as a (left) module over itself; we denote it inj.dim A. This is also the projective dimension of the A-module Hom_{k}(A, k).

Proposition 6.1. Let Γ and Λ be finite dimensional k-algebras. The finite dimensional k-algebra Γ ⊗_{k} Λ satisfies

inj.dim(Γ ⊗_{k} Λ) = inj.dim Γ + inj.dim Λ .

In particular Γ ⊗_{k} Λ is Gorenstein if, and only if, both Γ and Λ are Gorenstein.

Proof. Set M := Hom_{k}(Γ, k) and N := Hom_{k}(Λ, k). Let P, Q be minimal projective resolutions of M, N respectively. Then P ⊗_{k} Q is a minimal projective resolution of M ⊗_{k} N by the lemma below. It remains to observe that the latter is isomorphic to Hom_{k}(Γ ⊗_{k} Λ, k), as modules over Γ ⊗_{k} Λ. □

Lemma 6.2. Let P, Q be minimal projective resolutions of modules M, N over finite dimensional algebras Γ, Λ respectively. Then P ⊗_{k} Q is a minimal projective resolution of the Γ ⊗_{k} Λ-module M ⊗_{k} N.

Proof. With J(−) denoting the Jacobson radical, J(Γ)⊗_{k}Λ+Γ⊗_{k}J(Λ) is a nilpotent two-sided ideal in Γ ⊗_{k} Λ, and therefore it is contained in J(Γ ⊗_{k} Λ). So for any projective Γ-module U and projective Λ-module V, we have

Rad(U) ⊗_{k} V + U ⊗_{k} Rad(V) ⊆ Rad(U ⊗_{k} V)

as Γ⊗_{k}Λ-modules. The lemma now follows from the fact that a projective resolution is minimal if and only if the image of each differential lands in the radical of the next module. □

Using the result above, one can construct Gorenstein algebras of any given injective dimension, as long as we find one whose injective dimension is one. The next example describes such an algebra. In particular its class of Gorenstein projective modules is not the same as the class of Gorenstein injective modules; confer Remark 2.6. Another noteworthy feature of the algebra is that it is not of finite global dimension.

Example 6.3. Let k be a field, k[ε] the k-algebra of dual number, and set

Λ := k[ε] ⊗_{k} Γ where Γ = [\begin{matrix}k & k \\ 0 & k \end{matrix}].

The k-algebra Λ has injective dimension one over itself, and can be realised as the path algebra of the quiver

\begin{matrix}1 & 2 \\ ε_{1} \\ α \\ ε_{2} \end{matrix}

modulo the relations

ε_{1}^{2} = 0 = ε_{2}^{2}  and  αε_{1} = ε_{2}α.

The algebra Λ is representation finite and has precisely nine indecomposable modules. There are two simple modules corresponding to the vertices 1 and 2. The following diagram shows the Auslander-Reiten quiver. The vertices represent the indecomposables via their composition series. There is a solid arrow X → Y if

18 BENSON, IYENGAR, KRAUSE, AND PEVTSOVA

there is an irreducible morphism, and a dotted arrow X −−→ Y when Y = D Tr X.

{1 2 \atop 1 2}

1 1  2 2

1  2 1 2 1  2

{1 2 \atop 1}  {2 \atop 1 2}

2 1

The Gorenstein projectives have a bold frame, the Gorenstein injectives are shaded, and the modules of finite projective and injective dimension have rectangular shape. A module belongs to all three classes if and only if it is projective and injective; there is a unique indecomposable with this property.

One way to justify these computations is via [21, Theorem 2] due to Ringel and Zhang that sets up a bijection between indecomposable non-projective Gorenstein projective modules over k[ε] ⊗_{k} kQ and the indecomposable kQ-modules, for any quiver Q.

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Dave Benson, Institute of Mathematics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland U.K.

Srikanth B. Iyengar, Department of Mathematics, University of Utah, Salt Lake City, UT 84112, U.S.A.

Henning Krause, Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany.

Julia Pevtsova, Department of Mathematics, University of Washington, Seattle, WA 98195, U.S.A.