LOCAL DUALITY FOR REPRESENTATIONS OF FINITE GROUP SCHEMES

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

Abstract. A duality theorem for the stable module category of representations of a finite group scheme is proved. 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 stable category, for each homogeneous prime ideal 𝔭 in the cohomology ring of the group scheme.

1. Introduction

This work concerns the modular representation theory of finite groups and group schemes. A starting point for it is a duality theorem for finite groups due to Tate, that appears already in Cartan and Eilenberg [19]. For our purposes it is useful to recast this theorem in terms of stable module categories. Recall that the stable module category of a finite group scheme G over a field k is the category obtained from the (abelian) category of finite dimensional G-modules by annihilating morphisms that factor through a projective module; we denote it stmod G, and write \underline{Hom}_{G}(−, −) for the morphisms in this category. The category stmod G is triangulated with suspension Ω^{−1}, and Tate duality translates to the statement that for all finite dimensional G-modules M and N there are natural isomorphisms

Hom_{k}(\underline{Hom}_{G}(M, N), k) ≅ \underline{Hom}_{G}(N, Ωδ_{G} ⊗_{k} M) .

Here δ_{G} is the modular character of G, described in Jantzen [28, §I.8.8]; it is isomorphic to the trivial representation k when G is a finite group. This statement can be deduced from a formula of Auslander and Reiten [1] that applies to general associative algebras; see Theorem 4.2.

In the language introduced by Bondal and Kapranov [16] the isomorphism above says that stmod G has Serre duality with Serre functor M ↦ Ωδ_{G} ⊗_{k} M. One of the main results of our work is that such a duality also holds locally.

The precise statement involves a natural action of the cohomology ring H^{∗}(G, k) of G on the graded abelian group

\underline{Hom}_{G}^{∗}(M, N) = ⨁_{n∈ℤ} \underline{Hom}_{G}(M, Ω^{−n}N) .

The ring H^{∗}(G, k) is graded commutative, and also finitely generated as a k-algebra, by a result of Friedlander and Suslin [23]. Fix a homogeneous prime ideal 𝔭 not containing H^{⩾1}(G, k) and consider the triangulated category γ_{𝔭}(stmod G) that is obtained from stmod G by localising the graded morphisms at 𝔭 and then taking the full subcategory of objects such that the graded endomorphisms are 𝔭-torsion; see Section 7 for details. Our interest in the subcategories γ_{𝔭}(stmod G) stems from the fact that they are the building blocks of stmod G and play a key role in the classification of its tensor ideal thick subcategories; see [12]. These subcategories may thus be viewed as analogues of the K(n)-local spectra in stable homotopy theory that give the chromatic filtration of a spectrum; see [35].

Date: 8 July 2017.

2010 Mathematics Subject Classification. 16G10 (primary); 20C20, 20G10 20J06, 18E30.

Key words and phrases. Serre duality, local duality, finite group scheme, stable module category, Auslander-Reiten triangle.

SBI was partly supported by NSF grant DMS-1503044 and JP was partly supported by NSF grants DMS-0953011 and DMS-1501146.

Our local Serre duality statement reads:

Theorem 1.1. Let C := γ_{𝔭}(stmod G) and d the Krull dimension of H^{∗}(G, k)/𝔭. The assignment M ↦ Ω^{d}δ_{G} ⊗_{k} M induces a Serre functor for C. Thus for all M, N in C there are natural isomorphisms

Hom_{H^{∗}(G,k)}(Hom_{C}^{∗}(M, N), I(𝔭)) ≅ Hom_{C}(N, Ω^{d}δ_{G} ⊗_{k} M)

where I(𝔭) is the injective hull of the graded H^{∗}(G, k)-module H^{∗}(G, k)/𝔭.

One of the corollaries is that γ_{𝔭}(stmod G) has Auslander-Reiten triangles. One can thus bring to bear the techniques of AR theory to the study of G-modules. These results are contained in Theorem 7.8.

We deduce Theorem 1.1 from a more general result concerning StMod G, the stable category of all (including infinite dimensional) G-modules. Consider its subcategory Γ_{𝔭}(StMod G) consisting of the 𝔭-local 𝔭-torsion modules; in other words, the G-modules whose support is contained in {𝔭}. This is a compactly generated triangulated category and the full subcategory of compact objects is equivalent, up to direct summands, to γ_{𝔭}(stmod G); this is explained in Remark 7.2. There is an idempotent functor Γ_{𝔭}: StMod G → StMod G with image the 𝔭-local 𝔭-torsion modules; see Section 2 for details. The central result of this work is that Γ_{𝔭}(δ_{G}) is a dualising object for Γ_{𝔭}(StMod G), in the following sense.

Theorem 1.2. For any G-module M and i ∈ ℤ there is a natural isomorphism

Êxt_{G}^{i}(M, Γ_{𝔭}(δ_{G})) ≅ Hom_{H^{∗}(G,k)}(H^{∗−d−i}(G, M), I(𝔭)).

This result, proved in Section 5, may be compared with Serre duality for coherent sheaves F on a non-singular projective variety X of dimension n:

Ext_{X}^{i}(F, ω_{X}) ≅ Hom_{k}(H^{n−i}(X, F), k),

see, for example, Hartshorne [27].

When G is a finite group Γ_{𝔭}(k) is the Rickard idempotent module κ_{V}, introduced by Benson, Carlson, and Rickard [6], that is associated to the irreducible subvariety V of Proj H^{∗}(G, k) defined by 𝔭. In this context, Theorem 1.2 was proved by Benson and Greenlees [7]; see the paragraph following Theorem 5.1 below for a detailed comparison with their work, and that of Benson [5].

Concerning Γ_{𝔭}(k), the following consequences of Theorem 1.2 have been anticipated in [4] when G is a finite group.

Theorem 1.3. Suppose that δ_{G} ≅ k. Then the H^{∗}(G, k)-module H^{∗}(G, Γ_{𝔭}(k)) is injective and up to a twist isomorphic to I(𝔭). Also, there is an isomorphism of k-algebras

Êxt_{G}^{∗}(Γ_{𝔭}(k), Γ_{𝔭}(k)) ≅ (H^{∗}(G, k)_{𝔭})^{∧}

where (−)^{∧} denotes completion with respect to the 𝔭-adic topology, and the G-module Γ_{𝔭}(k) is pure injective.

Theorem 1.2 can be interpreted to mean that the category StMod G is Gorenstein, for it is analogous to Grothendieck’s result that a commutative noetherian ring A is Gorenstein if, and only if, Γ_{𝔭}A is the injective hull of A/𝔭, up to suspension, for each 𝔭 in Spec A. In Section 6 we propose a general notion of a Gorenstein triangulated category, in an attempt to place these results in a common framework. There is a plethora of duality results, and categorical frameworks dealing with them. A systematic comparison with those is beyond the scope of this work.

To prove Theorem 1.2 we use a technique from algebraic geometry in the tradition of Zariski and Weil; namely, the construction of generic points for algebraic varieties. Given a point 𝔭 ⊆ H^{∗}(G, k), there is a purely transcendental extension K of k and a closed point 𝔪 of Proj H^{∗}(G_{K}, K) lying above the point 𝔭 in Proj H^{∗}(G, k). Here, G_{K} denotes the group scheme that is obtained from G by extending the field to K. The crux is that one can choose 𝔪 such that the following statement holds.

Theorem 1.4. Restriction of scalars induces a dense exact functor

stmod G_{K} ⊇ γ_{𝔪}(stmod G_{K}) ⟶ γ_{𝔭}(stmod G).

This result is proved in Section 3, building on our work in [14]. It gives a remarkable description of the compact objects in Γ_{𝔭}(StMod G): they are obtained from the finite dimensional objects in Γ_{𝔪}(StMod G_{K}) by restriction of scalars. This allows one to reduce the proof of Theorem 1.2 to the case of a closed point, where it is essentially equivalent to classical Tate duality. The theorem above has other consequences; for example, it implies that the compact objects in Γ_{𝔭}(StMod G) are endofinite G-modules in the sense of Crawley-Boevey [21]; see Section 3.

2. Cohomology and localisation

In this section we recall basic notions concerning certain localisation functors on triangulated categories with ring actions. The material is needed to state and prove the results in this work. The main triangulated category of interest is the stable module category of a finite group scheme, but the general framework is needed in Sections 6 and 7. Primary references for the material presented here are [9, 10]; see [14] for the special case of the stable module category.

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 noetherian ring. In what follows 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.

We say that a 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.

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 [9, 10] for the construction of this functor.

The following observation is clear.

Lemma 2.1. 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. □

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

Lemma 2.2. 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 [10, Proposition 2.9]. □

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

(2.1)  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.

Tensor triangulated categories. Let now T = (T, ⊗, 𝟙) be a tensor triangulated category such that R acts on T via a homomorphism of graded rings R → End_{T}^{∗}(𝟙). It is easy to verify that in this case the functors Γ_{𝔭} and T_{𝔭} are tensor functors:

(2.2)  Γ_{𝔭} ≅ Γ_{𝔭}(𝟙) ⊗ −  and  T_{𝔭} ≅ T_{𝔭}(𝟙) ⊗ −

Now we turn to modules over finite group schemes. We follow the notation and terminology from [14].

The stable module category. Let G be a finite group scheme over a field k of positive characteristic. The coordinate ring and the group algebra of G are denoted k[G] and kG, respectively. These are finite dimensional Hopf algebras over k that are dual to each other. We write Mod G for the category of G-modules and mod G for its full subcategory consisting of finite dimensional G-modules. We often identity Mod G with the category of kG-modules, which is justified by [28, I.8.6].

We write H^{∗}(G, k) for the cohomology algebra, Ext_{G}^{∗}(k, k), of G. This is a graded commutative k-algebra, because kG is a Hopf algebra, and acts on Ext_{G}^{∗}(M, N), for any G-modules M, N. Moreover, the k-algebra H^{∗}(G, k) is finitely generated, and, when M, N are finite dimensional, Ext_{G}^{∗}(M, N) is finitely generated over it; this is by a theorem of Friedlander and Suslin [23].

The stable module category StMod G is obtained from Mod G by identifying two morphisms between G-modules when they factor through a projective G-module. An isomorphism in StMod G will be called a stable isomorphism, to distinguish it from an isomorphism in Mod G. In the same vein, G-modules M and N are said to be stably isomorphic if they are isomorphic in StMod G; this is equivalent to the condition that M are N are isomorphic in Mod G, up to projective summands.

The tensor product over k of G-modules passes to StMod G and yields a tensor triangulated category with unit k and suspension Ω^{−1}, the inverse of the syzygy functor. The category StMod G is compactly generated and the subcategory of compact objects identifies with stmod G, the stable module category of finite dimensional G-modules. See Carlson [18, §5] and Happel [24, Chapter I] for details.

We use the notation \underline{Hom}_{G}(M, N) for the Hom-sets in StMod G. The cohomology algebra H^{∗}(G, k) acts on StMod G via a homomorphism of k-algebras

− ⊗_{k} M: H^{∗}(G, k) = Ext_{G}^{∗}(k, k) ⟶ \underline{Hom}_{G}^{∗}(M, M) .

Thus, the preceding discussion on localisation functors on triangulated categories applies to the H^{∗}(G, k)-linear category StMod G.

Koszul objects. An element b in H^{d}(G, k) corresponds to a morphism k → Ω^{−d}k in StMod G; let k//b denote its mapping cone. This gives a morphism k → Ω^{d}(k//b). For a sequence of elements b := b_{1}, …, b_{n} in H^{∗}(G, k) and a G-module M, we set

k//b := (k//b_{1}) ⊗_{k} ⋯ ⊗_{k} (k//b_{n})  and  M//b := M ⊗_{k} k//b.

It is easy to check that for a G-module N and s = ∑_{i} |b_{i}|, there is an isomorphism

(2.3)  \underline{Hom}_{G}(M, N//b) ≅ \underline{Hom}_{G}(Ω^{n+s}M//b, N).

Let 𝔟 = (b) denote the ideal of H^{∗}(G, k) generated by b. By abuse of notation we set M//𝔟 := M//b. If b′ is a finite set of elements in H^{∗}(G, k) such that \sqrt{(b′)} = \sqrt{(b)}, then, by [12, Proposition 3.10], for any M in StMod G there is an equality

(2.4)  Thick(M//b) = Thick(M//b′) .

Fix 𝔭 in Spec H^{∗}(G, k). We will repeatedly use the fact that Γ_{𝔭}(StMod G)^{c} is generated as a triangulated category by the family of objects (M//𝔭)_{𝔭} with M in stmod G; see [10, Proposition 3.9]. In fact, if S denotes the direct sum of a representative set of simple G-modules, then there is an equality

(2.5)  Γ_{𝔭}(StMod G)^{c} = Thick((S//𝔭)_{𝔭}) .

It turns out that one has Γ_{𝔪}(StMod G) = {0} where 𝔪 denotes H^{⩾1}(G, k), the ideal of elements of positive degree; see Lemma 2.5 below. For this reason, it is customary to focus on Proj H^{∗}(G, k), the set of homogeneous prime ideals not containing 𝔪, when dealing with StMod G.

Tate cohomology. By construction, the action of H^{∗}(G, k) on StMod G factors through \underline{Hom}_{G}^{∗}(k, k), the graded ring of endomorphisms of the identity. The latter ring is not noetherian in general, which is one reason to work with H^{∗}(G, k). In any case, there is little difference, vis a vis their action on StMod G, as the next remarks should make clear.

Remark 2.3. Let M and N be G-modules. The map Hom_{G}(M, N) → \underline{Hom}_{G}(M, N) induces a map Ext_{G}^{∗}(M, N) → \underline{Hom}_{G}^{∗}(M, N) of H^{∗}(G, k)-modules. This map is surjective in degree zero, with kernel PHom_{G}(M, N), the maps from M to N that factor through a projective G-module. It is bijective in positive degrees and hence one gets an exact sequence of graded H^{∗}(G, k)-modules

(2.6) 0 ⟶ PHom_{G}(M, N) ⟶ Ext_{G}^{∗}(M, N) ⟶ \underline{Hom}_{G}^{∗}(M, N) ⟶ X ⟶ 0

with X^{i} = 0 for i ≥ 0. For degree reasons, the H^{∗}(G, k)-modules PHom_{G}(M, N) and X are 𝔪-torsion. Consequently, for 𝔭 in Proj H^{∗}(G, k) the induced localised map is an isomorphism:

(2.7)  Ext_{G}^{∗}(M, N)_{𝔭} ⟶ \underline{Hom}_{G}^{∗}(M, N)_{𝔭} .

More generally, for each r in 𝔪 localisation induces an isomorphism

Ext_{G}^{∗}(M, N)_{r} ⟶^{≅} \underline{Hom}_{G}^{∗}(M, N)_{r}

of H^{∗}(G, k)_{r}-modules. This means that Proj H^{∗}(G, k) has a finite cover by affine open sets on which ordinary cohomology and stable cohomology agree.

Given the finite generation result due to Friedlander and Suslin mentioned earlier, the next remark can be deduced from the exact sequence (2.6).

Remark 2.4. When M, N are finite dimensional G-modules, \underline{Hom}_{G}^{⩾s}(M, N) is a finitely generated H^{∗}(G, k)-module for any s ∈ ℤ. Moreover the H^{∗}(G, k)_{𝔭}-module

\underline{Hom}_{G}^{∗}(M_{𝔭}, N_{𝔭}) ≅ \underline{Hom}_{G}^{∗}(M, N)_{𝔭}

is finitely generated for each 𝔭 in Proj H^{∗}(G, k).

Lemma 2.5. One has Γ_{𝔪}(StMod G) = {0}, where 𝔪 = H^{⩾1}(G, k).

Proof. Given (2.5) it suffices to check that S//𝔪 = 0 in StMod G, where S is the direct sum of representative set of simple G-modules. For any G-module M, the H^{∗}(G, k)-module \underline{Hom}_{G}^{∗}(M, S//𝔪) is 𝔪-torsion; see [9, Lemma 5.11(1)]. Thus, when M is finite dimensional, the H^{∗}(G, k)-module \underline{Hom}_{G}^{⩾0}(M, S//𝔪) is 𝔪-torsion and finitely generated, by Remark 2.4, so it follows that \underline{Hom}_{G}^{i}(M, S//𝔪) = 0 for i ≫ 0. This implies that S//𝔪 is projective, since kG is self-injective. □

To gain a better understanding of the discussion above, it helps to consider the homotopy category of Inj G, the injective G-modules.

The homotopy category of injectives. Let K(Inj G) and D(Mod G) denote the homotopy category of Inj G and the derived category of Mod G, respectively. These are also H^{∗}(G, k)-linear compactly generated tensor triangulated category, with the tensor product over k. The unit of the tensor product on K(Inj G) is an injective resolution of the trivial G-module k, while that of D(Mod G) is k. The canonical quotient functor K(Inj G) → D(Mod G) induces an equivalence of triangulated category K(Inj G)^{c} ⟶^{∼} D^{b}(mod G), where the target is the bounded derived category of mod G; see [31, Proposition 2.3].

Taking Tate resolutions identifies StMod G with K_{ac}(Inj G), the full subcategory of acyclic complexes in K(Inj G). In detail, let pk and ik be a projective and an injective resolution of the trivial G-module k, respectively, and let tk be the mapping cone of the composed morphism pk → k → ik; this is a Tate resolution of k. Since projective and injective G-modules coincide, one gets the exact triangle

(2.8)  pk ⟶ ik ⟶ tk ⟶

in K(Inj G). For a G-module M, the complex M ⊗_{k} tk is a Tate resolution of M and the assignment M ↦ M ⊗_{k} tk induces an equivalence of categories

StMod G ⟶^{∼} K_{ac}(Inj G),

with quasi-inverse X ↦ Z^{0}(X), the submodule of cycles in degree 0. Assigning X in K(Inj G) to X ⊗_{k} tk is a left adjoint of the inclusion K_{ac}(Inj G) → K(Inj G). These results are contained in [31, Theorem 8.2]. Consider the composed functor

π: K(Inj G) ⟶^{−⊗_{k}tk} K_{ac}(Inj G) ⟶^{∼} StMod G .

A straightforward verification yields that these functors are H^{∗}(G, k)-linear. The result below is the categorical underpinning of Remark 2.3 and Lemma 2.5.

Lemma 2.6. There is a natural isomorphism Γ_{𝔪}X ≅ X ⊗_{k} pk for X ∈ K(Inj G). For each 𝔭 in Proj H^{∗}(G, k), the functor π induces triangle equivalences

K(Inj G)_{𝔭} ⟶^{∼} (StMod G)_{𝔭}  and  Γ_{𝔭}(K(Inj G)) ⟶^{∼} Γ_{𝔭}(StMod G).

Proof. We identify StMod G with K_{ac}(Inj G). This entails Γ_{𝔪}(K_{ac}(Inj G)) = {0}, by Lemma 2.5. It is easy to check that kG is 𝔪-torsion, and hence so is pk, for it is in the localising subcategory generated by kG, and the class of 𝔪-torsion objects in K(Inj G) is a tensor ideal localising subcategory; see, for instance, [9, Section 8]. Thus, applying Γ_{𝔪}(−) to the exact triangle (2.8) yields pk ≅ Γ_{𝔪}(ik). It then follows from (2.2) that X ⊗_{k} pk ≅ Γ_{𝔪}X for any X in K(Inj G).

From the construction of π and (2.8), the kernel of π is the subcategory

{X ∈ K(Inj G) | X ⊗_{k} pk ≅ X}.

These are precisely the 𝔪-torsion objects in K(Inj G), by the already established part of the result. Said otherwise, X ∈ K(Inj G) is acyclic if and only if Γ_{𝔪}X = 0. It follows that K_{ac}(Inj G) contains the subcategory K(Inj G)_{𝔭} of 𝔭-local objects, for each 𝔭 in Proj H^{∗}(G, k). On the other hand, the inclusion K_{ac}(Inj G) ⊆ K(Inj G) preserves coproducts, so its left adjoint π preserves compactness of objects and all compacts of K_{ac}(Inj G) are in the image of π. Given this a simple calculation shows that K(Inj G)_{𝔭} contains K_{ac}(Inj G)_{𝔭}. Thus K_{ac}(Inj G)_{𝔭} = K(Inj G)_{𝔭}. □

3. Passage to closed points

Let G be a finite group scheme over a field k of positive characteristic. In this section we describe a technique that relates the 𝔭-local 𝔭-torsion objects in StMod G, for a point 𝔭 in Proj H^{∗}(G, k), to the corresponding modules at a closed point defined over a field extension of k. Recall that a point 𝔪 is closed when it is maximal with respect to inclusion: 𝔪 ⊆ 𝔮 implies 𝔪 = 𝔮 for all 𝔮 in Proj H^{∗}(G, k). In what follows, k(𝔭) denotes the graded residue field of H^{∗}(G, k) at 𝔭.

For a field extension K/k extension of scalars and restriction give exact functors

K ⊗_{k} (−): StMod G ⟶ StMod G_{K} and (−)↓_{G}: StMod G_{K} ⟶ StMod G .

Moreover, since H^{∗}(G_{K}, K) ≅ K ⊗_{k} H^{∗}(G, k) as K-algebras, K ⊗_{k} (−) yields a homomorphism H^{∗}(G, k) → H^{∗}(G_{K}, K) of rings. It induces a map

Proj H^{∗}(G_{K}, K) ⟶ Proj H^{∗}(G, k) ,

with 𝔮 mapping to 𝔭 := 𝔮 ∩ H^{∗}(G, k). We say that 𝔮 lies over 𝔭 to indicate this. The main objective of this section is the proof of the following result.

Theorem 3.1. Fix 𝔭 in Proj H^{∗}(G, k) and K/k a purely transcendental extension of degree dim(H^{∗}(G, k)/𝔭) − 1. There exists a closed point 𝔪 in Proj H^{∗}(G_{K}, K) lying over 𝔭 with k(𝔪) ≅ k(𝔭) such that the functor (−)↓_{G} restricts to functors

Γ_{𝔪}(StMod G_{K}) → Γ_{𝔭}(StMod G) and Γ_{𝔪}(StMod G_{K})^{c} → Γ_{𝔭}(StMod G)^{c}

that are dense.

The proof of the theorem yields more: There is a subcategory of Γ_{𝔪}(StMod G_{K}) on which (−)↓_{G} is full and dense; ditto for the category of compact objects. However, the functor need not be full on all of Γ_{𝔪}(StMod G_{K})^{c}; see Example 3.7.

Here is one consequence of Theorem 3.1.

Corollary 3.2. The compact objects in Γ_{𝔭}(StMod G) are precisely the restrictions of finite dimensional G_{K}-modules in Γ_{𝔪}(StMod G_{K}).

Proof. By [9, Theorem 6.4], for any ideal 𝔞 in H^{∗}(G, k), we have

Γ_{V(𝔞)}(StMod G)^{c} = Γ_{V(𝔞)}(StMod G) ∩ stmod G .

Applying this observation to the ideal 𝔪 of H^{∗}(G_{K}, K) and noting that Γ_{𝔪} = Γ_{V(𝔪)}, since 𝔪 is a closed point, the desired result follows from Theorem 3.1. □

The closed point in Theorem 3.1 depends on the choice of a Noether normalisation of H^{∗}(G, k)/𝔭 as is explained in the construction below, from [14, §7].

Construction 3.3. Fix 𝔭 in Proj H^{∗}(G, k); the following construction is relevant only when 𝔭 is not a closed point. Choose elements a := a_{0}, …, a_{d−1} in H^{∗}(G, k) of the same degree such that their image in H^{∗}(G, k)/𝔭 is algebraically independent over k and H^{∗}(G, k)/𝔭 is finitely generated as a module over the subalgebra k[a]. Thus the Krull dimension of H^{∗}(G, k)/𝔭 is d. 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 H^{∗}(G_{K}, K). Let 𝔭′ denote the extension of 𝔭 to H^{∗}(G_{K}, K), and set

𝔮 := 𝔭′ + (b)  and  𝔪 := \sqrt{𝔮} .

It is proved as part of [14, Theorem 7.7] that the ideal 𝔪 is a closed point in Proj H^{∗}(G_{K}, K) with the property that 𝔪∩H^{∗}(G, k) = 𝔭. What is more, it follows from the construction (see in particular [14, Lemma 7.6, and (7.2)]) that the induced extension of fields is an isomorphism

k(𝔭) ⟶^{≅} k(𝔪) .

The sequence of elements b in H^{∗}(G_{K}, K) yields a morphism K → Ω^{s}(K//b), where s = ∑_{i} |b_{i}|, and composing its restriction to G with the canonical morphism k → K↓_{G} gives in StMod G a morphism

f: k ⟶ Ω^{s}(K//b)↓_{G} .

Since the a_{i} are not in 𝔭, Lemma 2.1 yields a natural stable isomorphism

(3.1)  Ω^{s}M ≅ M

for any 𝔭-local G-module M. This remark will be used often in the sequel.

By [13, Lemma 2.1], for any G_{K}-module N there is a natural isomorphism

(3.2)  M ⊗_{k} N↓_{G} ≅ (M_{K} ⊗_{K} N)↓_{G}.

The result below extends [14, Theorem 8.8]; the latter is the case M = k//𝔭.

Theorem 3.4. For any G-module M, the morphism M ⊗_{k} f induces a natural stable isomorphism of G-modules

Γ_{𝔭}M ≅ M ⊗_{k} Γ_{𝔪}(K//b)↓_{G} ≅ (M_{K} ⊗_{K} Γ_{𝔪}(K//b))↓_{G} .

When M is 𝔭-torsion, these induce natural stable isomorphisms

Γ_{𝔭}M ≅ M_{𝔭} ≅ M ⊗_{k} (K//b)↓_{G} ≅ (M_{K} ⊗_{K} K//b)↓_{G} .

Proof. We begin by verifying the second set of isomorphisms. As M is 𝔭-torsion so is M_{𝔭} and then it is clear that the natural map Γ_{𝔭}M = Γ_{V(𝔭)}M_{𝔭} → M_{𝔭} is an isomorphism. The third of the desired isomorphisms follows from (3.2). It thus remains to check that M ⊗_{k} f induces an isomorphism

M_{𝔭} ≅ M ⊗_{k} (K//b)↓_{G} .

It is easy to verify that the modules M having this property form a tensor ideal localising subcategory of StMod G. Keeping in mind (3.1), from [14, Theorem 8.8] one obtains that this subcategory contains k//𝔭. The desired assertion follows since the 𝔭-torsion modules form a tensor ideal localising subcategory of StMod G that is generated by k//𝔭; see [10, Proposition 2.7].

Now we turn to the first set of isomorphisms. There the second one is by (3.2), so we focus on the first. Let M be an arbitrary G-module, and let 𝔭′ be as in Construction 3.3. Since Γ_{V(𝔭)}M is 𝔭-torsion, the already established isomorphism yields the second one below.

Γ_{𝔭}M ≅ (Γ_{V(𝔭)}M)_{𝔭}
≅ ((Γ_{V(𝔭)}M)_{K} ⊗_{K} K//b)↓_{G}
≅ (Γ_{V(𝔭′)}(M_{K}) ⊗_{K} K//b)↓_{G}
≅ (M_{K} ⊗_{K} Γ_{V(𝔭′)}(K//b))↓_{G}
≅ (M_{K} ⊗_{K} Γ_{V(𝔭′+(b))}(K//b))↓_{G}
≅ (M_{K} ⊗_{K} Γ_{𝔪}(K//b))↓_{G}

The third one is by Lemma 2.2, applied to the functor K ⊗_{k} (−) from StMod G to StMod G_{K}. The next one is standard while the penultimate one holds because K//b is (b)-torsion. This completes the proof. □

In the next remark we recast part of Theorem 3.4.

Remark 3.5. Fix a point 𝔭 in Proj H^{∗}(G, k), and let K, b and 𝔪 be as in Construction 3.3. Consider the following adjoint pair of functors.

λ: StMod G ⟶ StMod G_{K} and ρ: StMod G_{K} ⟶ StMod G
λ(M) = M_{K} ⊗_{K} K//b  ρ(N) = Hom_{K}(K//b, N)↓_{G}

It is easy to check that this induces an adjoint pair

Γ_{𝔭}(StMod G)  Γ_{𝔪}(StMod G_{K}) .

λ

ρ

Theorem 3.4 implies that (λM)↓_{G} ≅ M for any M in Γ_{𝔭}(StMod G).

Proof of Theorem 3.1. Let 𝔪, 𝔮, and b be as in Construction 3.3. As noted there, 𝔪 is a closed point in Proj H^{∗}(G_{K}, K) lying over 𝔭 and k(𝔪) ≅ k(𝔭). The modules in Γ_{𝔭}(StMod G) are precisely those with support contained in {𝔭}. It then follows from [14, Proposition 6.2] that (−)↓_{G} restricts to a functor

Γ_{𝔪}(StMod G_{K}) ⟶ Γ_{𝔭}(StMod G) .

This functor is dense because for any G-module M that is 𝔭-local and 𝔭-torsion one has M ≅ (λM)↓_{G} where λ is the functor from Remark 3.5.

Consider the restriction of (−)↓_{G} to compact objects in Γ_{𝔪}(StMod G_{K}). First we verify that its image is contained in the compact objects of Γ_{𝔭}(StMod G). To this end, it suffices to check that there exists a generator of Γ_{𝔪}(StMod G_{K})^{c}, as a thick subcategory, whose restriction is in Γ_{𝔭}(StMod G)^{c}.

Let S be the direct sum of a representative set of simple G-modules. Each simple G_{K}-module is (isomorphic to) a direct summand of S_{K}, so from (2.5) one gets the first equality below:

Γ_{𝔪}(StMod G_{K})^{c} = Thick(S_{K}//𝔪) = Thick(S_{K}//𝔮) .

The second one is by (2.4). From Theorem 3.4 one gets isomorphisms of G-modules

(S_{K}//𝔮)↓_{G} ≅ ((S//𝔭)_{K} ⊗_{K} K//b)↓_{G} ≅ (S//𝔭)_{𝔭} .

It remains to note that (S//𝔭)_{𝔭} is in Γ_{𝔭}(StMod G)^{c}, again by (2.5).

The last item to verify is that restriction is dense also on compacts. Since K//b is compact, the functor ρ from Remark 3.5 preserves coproducts, and hence its left adjoint λ preserves compactness. Thus Theorem 3.4 gives the desired result. □

Theorem 3.4 yields that f ≅ (f_{K})↓_{G} for any morphism f in Γ_{𝔭}(StMod G); in particular, the restriction functor is full and dense on the subcategory of Γ_{𝔪}(StMod G_{K}) consisting of objects of the form λM, where M is a 𝔭-local 𝔭-torsion G-module. It need not be full on the entire category, or even on its subcategory of compact objects; see Example 3.7, modeled on the following one from commutative algebra.

Example 3.6. Let k be a field and k[a] the polynomial ring in an indeterminate a. Let D(k[a]) denote its derived category; it is k[a]-linear in an obvious way. For the prime 𝔭 := (0) of k[a] the 𝔭-local 𝔭-torsion subcategory Γ_{𝔭}(D(k[a])) is naturally identified with the derived category of k(a), the field of rational functions in a.

With k(t) denoting the field of rational functions in an indeterminate t, the maximal ideal 𝔪 := (a−t) of k(t)[a] lies over the prime ideal 𝔭 of k[a]. The inclusion k[a] ⊂ k(t)[a] induces an isomorphism k(a) ≅ k(t)[a]/𝔪 ≅ k(t). The analogue of Theorem 3.4 is that restriction of scalars along the inclusion k[a] ⊂ k(t)[a] induces a dense functor

Γ_{𝔪}(D(k(t)[a])) ⟶ Γ_{𝔭}(D(k[a])) ≃ D(k(a)) .

This property can be checked directly: The 𝔪-torsion module k(t)[a]/(a−t) restricts to k(a), and each object in D(k(a)) is a direct sum of shifts of k(a). This functor is however not full: For n ≥ 1, the k(t)[a]-module L := k(t)[a]/(a − t)^{n} is 𝔪-torsion, and satisfies

rank_{k(a)} End_{D}(L) = n and rank_{k(a)} End_{D}(L↓_{k[a]}) = n^{2}

where D stands for the appropriate derived category. In particular, if n ≥ 2, the canonical map End_{D}(L) → End_{D}(L↓_{k[a]}) is not surjective.

Indeed, the module of endomorphisms of L in D(k(t)[a]) is

End_{D}(L) = Hom_{k(t)[a]}(L, L) ≅ L .

In particular, it has rank n as an k(a)-vector space. On the other hand, restricted to k[a], the k(t)[a]-module k(t)/(a − t) is isomorphic to k(a). It then follows from the exact sequences

0 ⟶ \frac{k(t)[a]}{(a − t)} ⟶^{1↦(a−t)^{i}} \frac{k(t)[a]}{(a − t)^{i+1}} ⟶ \frac{k(t)[a]}{(a − t)^{i}} ⟶ 0

of k(t)[a]-modules that L restricts to a direct sum of n copies of k(a), so that

End_{D}(L↓_{k[a]}) = Hom_{k(a)}(k(a)^{n}, k(a)^{n}) ≅ k(a)^{n^{2}} .

In particular, this has rank n^{2} as a k(a)-vector space.

Example 3.7. Let V = ℤ/2×ℤ/2 and k a field of characteristic two. As k-algebras, one has H^{∗}(V, k) ≅ k[a, b], where a and b are indeterminates of degree one. For the prime ideal 𝔭 = (0) of k[a, b], Construction 3.3 leads to the field extension K := k(t) of k, and the closed point 𝔪 = (b − at) of Proj H^{∗}(V_{K}, K).

Set F := \underline{End}_{V} (k_{𝔭}); this is the component in degree 0 of the graded field k[a, b]_{𝔭} and can be identified with K; see Construction 3.3.

Fix an integer n ≥ 1 and set N := K//(b − at)^{n}. This is a finite-dimensional 𝔪-torsion V_{K}-module and hence compact in Γ_{𝔪}(StMod V_{K}). We claim that

rank_{F} \underline{End}_{V_{K}} (N) = 2n and rank_{F} \underline{End}_{V} (N↓_{V}) = n^{2} ,

and hence that the map \underline{End}_{V_{K}} (N) → \underline{End}_{V} (N↓_{V}) is not surjective when n ≥ 3.

The claim can be checked as follows: Set S := \underline{End}_{V_{K}}^{∗} (K)_{𝔪} ≅ K[a, b]_{𝔪}. Since (b − at)^{n} is not a zerodivisor on S, applying \underline{Hom}_{V_{K}} (K, −) to the exact triangle

K ⟶^{(b−at)^{n}} Ω^{−n}K ⟶ N ⟶

one gets that \underline{Hom}_{V_{K}}^{∗} (K, N) is isomorphic to S/(b − at)^{n}, as an S-module; in particular (b − at)^{n} annihilates it. Given this, applying \underline{Hom}_{V_{K}} (−, N) to the exact triangle above yields that the rank of \underline{End}_{V_{K}} (N), as an F -vector space, is n.

As to the claim about N↓_{V}: the category Γ_{𝔭}(StMod V) is semisimple for its generator k_{𝔭} has the property that \underline{End}_{V}^{∗} (k_{𝔭}) is a graded field. It thus suffices to verify that N↓_{V} ≅ k_{𝔭}^{n}; equivalently, that rank_{F} \underline{Hom}_{V} (k_{𝔭}, N↓_{V}) = n. This follows from the isomorphisms

\underline{Hom}_{V} (k_{𝔭}, N↓_{V}) ≅ \underline{Hom}_{V} (k, N↓_{V}) ≅ \underline{Hom}_{V_{K}} (K, N) ≅ F^{n} .

The first one holds because N↓_{V} is 𝔭-local, the second one is by adjunction.

There is a close connection between this example and Example 3.6. Namely, the Bernstein-Gelfand-Gelfand correspondence sets up an equivalence between StMod V and the derived category of dg modules over R := k[a, b], viewed as a dg algebra with zero differential, modulo the subcategory of (a, b)-torsion dg modules; see, for example, [11, §5.2.2]. The BGG correspondence induces the equivalences in the following commutative diagram of categories.

Γ_{𝔪}(StMod V_{K})  Γ_{𝔪}(D(S))

Γ_{𝔭}(StMod V)  D(R_{𝔭}) (−)↓_{V} ≃ ≃

where D(−) denotes the derived category of dg modules. The functor on the right is restriction of scalars along the homomorphism of rings R_{𝔭} → S, which is induced by the inclusion R = k[a, b] ⊂ K[a, b]. Under the BGG equivalence, the V_{K}-module N corresponds to S/(b − at)^{n}, viewed as dg S-module with zero differential. Since R_{𝔭} is a graded field, isomorphic to K[a^{±1}], each dg R_{𝔭}-module is isomorphic to a direct sum of copies of R_{𝔭}. Arguing as in Example 3.6 one can verify that the dg S-module S/(b − at)^{n} restricts to a direct sum of n copies of R_{𝔭}. This is another way to compute the endomorphism rings in question.

The remainder of this section is devoted to a further discussion of the compact objects in Γ_{𝔭}(StMod G). This is not needed in the sequel.

Endofiniteness. Following Crawley-Boevey [20, 21], a module X over an associative ring A is endofinite if X has finite length as a module over End_{A}(X).

An object X of a compactly generated triangulated category T is endofinite if the End_{T}(X)-module Hom_{T}(C, X) has finite length for all C ∈ T^{c}; see [33].

Let A be a self-injective algebra, finite dimensional over some field. Then an A-module is endofinite if and only if it is endofinite as an object of StMod A. This follows from the fact X is an endofinite A-module if and only if the End_{A}(X)- module Hom_{A}(C, X) has finite length for every finite dimensional A-module C.

Lemma 3.8. Let F: T → U be a functor between compactly generated triangulated categories that preserves products and coproducts. Let X be an object in T. If X is endofinite, then so is F X and the converse holds when F is fully faithful.

Proof. By Brown representability, F has a left adjoint, say F′. It preserves compactness, as F preserves coproducts. For X ∈ T and C ∈ U^{c}, there is an isomorphism

Hom_{U}(C, F X) ≅ Hom_{T}(F′C, X)

of End_{T}(X)-modules. Thus if X is endofinite, then Hom_{U}(C, F X) is a module of finite length over End_{T}(X), and therefore also over End_{U}(F X). For the converse, observe that each compact object in T is isomorphic to a direct summand of an object of the form F′C for some C ∈ U^{c}. □

Proposition 3.9. Let 𝔭 be a point in Proj H^{∗}(G, k) and M a G-module that is compact in Γ_{𝔭}(StMod G). Then M is endofinite both in StMod G and in Γ_{𝔭}(StMod G).

Proof. By Corollary 3.2, the module M is of the form N↓_{G} for a finite dimensional G_{K}-module N. Clearly, N is endofinite in StMod G_{K} and (−)↓_{G} preserves products and coproducts, so it follows by Lemma 3.8 that M is endofinite in StMod G. By the same token, as the inclusion (StMod G)_{𝔭} → StMod G preserves products and coproducts, M is endofinite in (StMod G)_{𝔭} as well. Finally, the functor Γ_{V(𝔭)} is a right adjoint to the inclusion Γ_{𝔭}(StMod G) → (StMod G)_{𝔭}. It preserves products, being a right adjoint, and also coproducts. Thus M is endofinite in Γ_{𝔭}(StMod G), again by Lemma 3.8. □

4. G-modules and Tate duality

Now we turn to various dualities for modules over finite group schemes. We begin by recalling the construction of the transpose and the dual of a module over a finite group scheme, and certain functors associated with them. Our basic reference for this material is Skowroński and Yamagata [37, Chapter III].

Throughout G will be a finite group scheme over k. We write (−)^{∨} = Hom_{k}(−, k).

Transpose and dual. Let G^{op} be the opposite group scheme of G; it can be realised as the group scheme associated to the cocommutative Hopf algebra (kG)^{op}. Since kG is a G-bimodule, the assignment M ↦ Hom_{G}(M, kG) defines a functor

(−)^{t}: Mod G ⟶ Mod G^{op} .

Let now M be a finite dimensional G-module. Give a minimal projective presentation P_{1} ⟶^{f} P_{0} → M, the transpose of M is the G^{op}-module Tr M := Coker(f^{t}). By construction, there is an exact sequence of G^{op}-modules:

0 ⟶ M^{t} ⟶ P_{0}^{t} ⟶^{f^{t}} P_{1}^{t} ⟶ Tr M ⟶ 0 .

The P_{i}^{t} are projective G^{op}-modules, so this yields an isomorphism of G^{op}-modules

M^{t} ≅ Ω^{2} Tr M .

Given a G^{op}-module N, the k-vector space Hom_{k}(N, k) has a natural structure of a G-module, and the assignment N ↦ Hom_{k}(N, k) yields a functor

D := Hom_{k}(−, k): stmod G^{op} ⟶ stmod G .

The Auslander-Reiten translate. In what follows we write τ for the Auslander-Reiten translate of G:

τ := D ∘ Tr: stmod G → stmod G

Given an extension of fields K/k, for any finite dimensional G-module M there is a stable isomorphism of G_{K}-modules

(τM)_{K} ≅ τ(M_{K}) .

Nakayama functor. The Nakayama functor

ν: Mod G ⟶^{∼} Mod G

is given by the assignment

M ↦ D(kG) ⊗_{kG} M ≅ δ_{G} ⊗_{k} M .

where δ_{G} = ν(k) is the modular character of G; see [28, I.8.8]. Since the group of characters of G is finite, by [38, §2.1 & §2.2], there exists a positive integer d such that δ_{G}^{⊗d} ≅ k and hence as functors on Mod G there is an equality

(4.1)  ν^{d} = id .

When M is a finite dimensional G-module, there are natural stable isomorphisms

νM ≅ D(M^{t}) ≅ Ω^{−2}τM .

When in addition M is projective, one has

Hom_{G}(M, −)^{∨} ≅ (M^{t} ⊗_{kG} −)^{∨} ≅ Hom_{G}(−, νM).

Let K/k be an extension of fields. For any G-module M there is a natural isomorphism of G_{K}-modules

(4.2)  (νM)_{K} ≅ ν(M_{K}) .

This is clear for M = kG since

K ⊗_{k} Hom_{k}(kG, k) ≅ Hom_{k}(kG, K) ≅ Hom_{K}(K ⊗_{k} kG, K) ,

and the general case follows by taking a free presentation of M.

Remark 4.1. We have δ_{G} = k if and only if the algebra kG is symmetric. In particular, δ_{G} = k when G is a finite discrete group.

Tate duality. For finite groups, the duality theorem below is classical and due to Tate [19, Chapter XII, Theorem 6.4]. An argument for the extension to finite group schemes was sketched in [13, §2], and is reproduced here for readers convenience.

Theorem 4.2. Let G be a finite group scheme over a field k. For any G-modules M, N with M finite dimensional, there are natural isomorphisms

\underline{Hom}_{G}(M, N)^{∨} ≅ \underline{Hom}_{G}(N, Ω^{−1}τM) ≅ \underline{Hom}_{G}(N, ΩνM) .

Proof. A formula of Auslander and Reiten [1, Proposition I.3.4], see also [30, Corollary p. 269], yields the first isomorphism below

\underline{Hom}_{G}(M, N)^{∨} ≅ Ext_{G}^{1}(N, τM) ≅ \underline{Hom}_{G}(N, Ω^{−1}τM)

The second isomorphism is standard. It remains to recall that τM ≅ Ω^{2}νM. □

Restricted to finite dimensional G-modules, Tate duality is the statement that the k-linear category stmod G has Serre duality, with Serre functor Ων. A refinement of this Serre duality will be proved in Section 7.

5. Local cohomology versus injective cohomology

Let k be a field and G a finite group scheme over k. In this section we establish the main result of this work; it identifies for a prime ideal 𝔭 in H^{∗}(G, k), up to some twist and some suspension, the local cohomology object Γ_{𝔭}(k) with the injective cohomology object T_{𝔭}(k).

Theorem 5.1. Fix a point 𝔭 in Proj H^{∗}(G, k) and let d be the Krull dimension of H^{∗}(G, k)/𝔭. There is a stable isomorphism of G-modules

Γ_{𝔭}(δ_{G}) ≅ Ω^{−d}T_{𝔭}(k) ;

equivalently, for any G-module M there is a natural isomorphism

\underline{Hom}_{G}(M, Ω^{d}Γ_{𝔭}(δ_{G})) ≅ Hom_{H^{∗}(G,k)}(H^{∗}(G, M), I(𝔭)) .

When G is the group scheme arising from a finite group the result above was proved by Benson and Greenlees [7, Theorem 2.4] using Gorenstein duality for cochains on BG, the classifying space of G. Benson [5, Theorem 2] gave a different proof by embedding G into a general linear group and exploiting the fact that its cohomology ring is a polynomial ring, as was proved by Quillen. There is an extension of these results to compact Lie groups; see [8, Theorem 6.10], and recent work of Barthel, Heard, and Valenzuela [2, Proposition 4.33].

Theorem 5.1 is established using (by necessity) completely different arguments, thereby giving yet another proof in the case of finite groups that is, in a sense, more elementary than the other ones for it is based on classical Tate duality.

A caveat: In [5, 7] it is asserted that Γ_{𝔭}(k) ≅ Ω^{d}T_{𝔭}(k). However, this is incorrect and the correct shift is the one in the preceding theorem. We illustrate this by computing these modules directly for the quaternions.

Example 5.2. Let G := Q_{8}, the quaternions, viewed as a group scheme over a field k of characteristic 2. In this case δ_{G} = k, the trivial character. The cohomology algebra of G is given by

H^{∗}(G, k) = k[z] ⊗_{k} B where B = k[x, y]/(x^{2} + xy + y^{2}, x^{2}y + xy^{2}) ,

with |x| = 1 = |y| and |z| = 4; see, for instance, [3, p. 186]. Thus Proj H^{∗}(G, k) consists of a single point, namely 𝔪 := (x, y). In particular, Γ_{𝔪}k = k, in StMod G.

Next we compute I(𝔪) as a module over H^{∗}(G, k)_{𝔪} ≅ k[z^{±1}] ⊗_{k} B, using Lemma A.3. The extension k[z^{±1}] ⊆ k[z^{±1}] ⊗_{k} B is evidently finite (and hence also residually finite). Since 𝔪 ∩ k[z^{±1}] = (0) and k[z^{±1}] is a graded field, from Lemma A.3 one gets an isomorphism of H^{∗}(G, k)_{𝔪}-modules.

I(𝔪) ≅ Hom_{k[z^{±1}]}(k[z^{±1}] ⊗_{k} B, k[z^{±1}])
≅ k[z^{±1}] ⊗_{k} Hom_{k}(B, k)
≅ k[z^{±1}] ⊗_{k} Σ^{3}B
≅ Σ^{3}H^{∗}(G, k)_{𝔪}

This yields the first isomorphism below of G-modules

T_{𝔪}(k) ≅ Ω^{−3}k ≅ Ω^{1}k ,

and the second one holds because Ω^{4}k ≅ k in StMod G.

Proof of Theorem 5.1. It follows from (2.7) that for any 𝔭-local H^{∗}(G, k)-module I, there is an isomorphism

Hom_{H^{∗}(G,k)}(H^{∗}(G, M), I) ≅ Hom_{H^{∗}(G,k)}(\underline{Hom}_{G}^{∗}(k, M), I).

Consequently, one can rephrase the defining isomorphism (2.1) for the object T_{𝔭}(k) as a natural isomorphism

\underline{Hom}_{G}(M, T_{𝔭}(k)) ≅ Hom_{H^{∗}(G,k)}(H^{∗}(G, M), I(𝔭)) .

Therefore, the main task is to prove that there is a stable isomorphism:

Γ_{𝔭}(νk) ≅ Ω^{−d}T_{𝔭}(k) .

Recall that νk = δ_{G}.

We first verify the isomorphism above for closed points of Proj H^{∗}(G, k) and then use a reduction to closed points. The proof uses the following simple observation: For any G-modules X and Y that are 𝔭-local and 𝔭-torsion, there is an isomorphism X ≅ Y in StMod G if and only if there is a natural isomorphism

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

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

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

Set A := H^{∗}(G, k) and R := A_{𝔪}. The injective hull, I(𝔪), of the A-module A/𝔪 is the same as that of the R-module k(𝔪), viewed as an A-module via restriction of scalars along the localisation map A → R. Thus I(𝔪) is the module I described in Lemma A.2. Let M be a G-module that is 𝔪-local and 𝔪-torsion. The claim is a consequence of the following computation:

\underline{Hom}_{G}(M, ΩΓ_{𝔪}(νk)) ≅ \underline{Hom}_{G}(M, Ωνk)
≅ \underline{Hom}_{G}(k, M)^{∨}
≅ Hom_{R}(\underline{Hom}_{G}^{∗}(k, M), I(𝔪))
≅ Hom_{A}(\underline{Hom}_{G}^{∗}(k, M), I(𝔪))
≅ \underline{Hom}_{G}(M, T_{𝔪}(k))

The first isomorphism holds because M is 𝔪-torsion; the second is Tate duality, Theorem 4.2, and the next one is by Lemma A.2, which applies because \underline{Hom}_{G}^{∗}(k, M) is 𝔪-local and 𝔪-torsion as an A-module.

Let 𝔭 be a point in Proj H^{∗}(G, k) that is not closed, and let K, b, and 𝔪 be as in Construction 3.3. Recall that 𝔪 is a closed point in H^{∗}(G_{K}, K) lying over 𝔭.

Claim. There is a stable isomorphism of G-modules

(5.1)  (T_{𝔪}(K)//b)↓_{G} ≅ Ω^{−d+1}T_{𝔭}(k)

where d is the Krull dimension of H^{∗}(G, k)/𝔭.

Let M be a G-module that is 𝔭-local and 𝔭-torsion. Then Theorem 3.4 applies and yields isomorphisms of G-modules.

(M_{K}//b)↓_{G} ≅ (M_{K} ⊗_{K} K//b)↓_{G} ≅ M .

This gives the sixth isomorphism below.

\underline{Hom}_{G}(M, Ω^{d−1}(T_{𝔪}(K)//b)↓_{G}) ≅ \underline{Hom}_{G_{K}} (M_{K}, Ω^{d−1}(T_{𝔪}(K)//b))
≅ \underline{Hom}_{G_{K}} (M_{K}//b, T_{𝔪}(K))
≅ Hom_{H^{∗}(G_{K},K)}(\underline{Hom}_{G_{K}}^{∗} (K, M_{K}//b), I(𝔪))
≅ Hom_{H^{∗}(G,k)}(\underline{Hom}_{G_{K}}^{∗} (K, M_{K}//b), I(𝔭))
≅ Hom_{H^{∗}(G,k)}(\underline{Hom}_{G}^{∗}(k, (M_{K}//b)↓_{G}), I(𝔭))
≅ Hom_{H^{∗}(G,k)}(\underline{Hom}_{G}^{∗}(k, M), I(𝔭))
≅ \underline{Hom}_{G}(M, T_{𝔭}(k))

The first and the fifth isomorphisms are by adjunction. The second isomorphism is a direct computation using (2.3) and (3.1). The next one is by definition and the fourth isomorphism is by Lemma A.3 applied to the canonical homomorphism H^{∗}(G, k) → H^{∗}(G_{K}, K); note that the H^{∗}(G_{K}, K)-module \underline{Hom}_{G_{K}}^{∗} (K, M_{K}//b) is 𝔪-torsion. This justifies the claim.

We are now ready to wrap up the proof of the theorem. Since 𝔪 is a closed point in Proj H^{∗}(G_{K}, K), the first claim yields that the G_{K}-modules Γ_{𝔪}(νK) and Ω^{−1}T_{𝔪}(K) are isomorphic. This then gives an isomorphism of G_{K}-modules.

νK ⊗_{K} Γ_{𝔪}(K//b) ≅ Γ_{𝔪}(νK)//b ≅ Ω^{−1}T_{𝔪}(K)//b

Restricting to G and applying (5.1) gives the last of the following isomorphisms of G-modules.

Γ_{𝔭}(νk) ≅ ((νk)_{K} ⊗_{K} Γ_{𝔪}(K//b)↓_{G}
≅ (νK ⊗_{K} Γ_{𝔪}(K//b))↓_{G}
≅ Ω^{−d}T_{𝔭}(k)

The first one is by Theorem 3.4 and the second by (4.2). □

The following consequences of Theorem 5.1 were anticipated in [4, pp. 203]. For a graded module M = ⨁_{n∈ℤ} M^{n} and i ∈ ℤ the twist M(i) is the graded module with M(i)^{n} = M^{n+i}.

Corollary 5.3. Fix 𝔭 in Proj H^{∗}(G, k). With d the Krull dimension of H^{∗}(G, k)/𝔭 there are isomorphisms of H^{∗}(G, k)-modules

H^{∗}(G, Γ_{𝔭}(k)) ≅ Hom_{H^{∗}(G,k)}^{∗}(H^{∗}(G, δ_{G}), I(𝔭))(d)

and

H^{∗}(G, End_{k}(Γ_{𝔭}(k))) ≅ (H^{∗}(G, k)_{𝔭})^{∧}

where (−)^{∧} denotes completion with respect to the 𝔭-adic topology.

Proof. Remark 2.3 will be used repeatedly and without comment. Set R = H^{∗}(G, k). Since Γ_{𝔭}(k) is 𝔭-local, the R-modules H^{∗}(G, Γ_{𝔭}(k)) and \underline{Hom}_{G}^{∗}(k, Γ_{𝔭}(k)) are isomorphic. The first of the stated isomorphisms is a composition of the following isomorphisms of R-modules.

\underline{Hom}_{G}^{∗}(k, Γ_{𝔭}(k)) ≅ \underline{Hom}_{G}^{∗}(δ_{G}, Γ_{𝔭}(δ_{G}))
≅ \underline{Hom}_{G}^{∗}(δ_{G}, Ω^{−d}T_{𝔭}(k))
≅ \underline{Hom}_{G}^{∗}(δ_{G}, T_{𝔭}(k))(d)
≅ \underline{Hom}_{R}^{∗}(H^{∗}(G, δ_{G}), I(𝔭))(d)

The second isomorphism is by Theorem 5.1, the one after is standard, while the last one is by the definition of T_{𝔭}(k).

In the same vein, one has the following chain of isomorphisms.

\underline{Hom}_{G}^{∗}(k, End_{k}(Γ_{𝔭}(k))) ≅ \underline{Hom}_{G}^{∗}(Γ_{𝔭}(k), Γ_{𝔭}(k))
≅ \underline{Hom}_{G}^{∗}(T_{𝔭}(k), T_{𝔭}(k))
≅ Hom_{R}^{∗}(\underline{Hom}_{G}^{∗}(k, T_{𝔭}(k)), I(𝔭))
≅ Hom_{R}^{∗}(I(𝔭), I(𝔭))
≅ (R_{𝔭})^{∧}

The second isomorphism is by Theorem 5.1 and the rest are standard. □

Remark 5.4. Another consequence of Theorem 5.1 is that Γ_{𝔭}(δ_{G}), and hence also Γ_{𝔭}k, is an indecomposable pure injective object in StMod G; see [15, Theorem 5.1].

6. The Gorenstein property

In this section we introduce a notion of a Gorenstein triangulated category and reinterpret Theorem 5.1 to mean that StMod G has this property. The rationale for doing so will become clear in the next section.

The Gorenstein property. Let R be a graded commutative noetherian ring and T a compactly generated R-linear triangulated category. The support of T is by definition the set

supp_{R}(T) = {𝔭 ∈ Spec R | Γ_{𝔭} ≠ 0} .

We say that T is Gorenstein if there is an R-linear triangle equivalence

F: T^{c} ⟶^{∼} T^{c}

and for every 𝔭 in supp_{R}(T) there is an integer d(𝔭) and a natural isomorphism

Γ_{𝔭} ∘ F ≅ Σ^{d(𝔭)} ∘ T_{𝔭}

of functors T^{c} → T. In this context we call F a global Serre functor, because in Proposition 7.3 we show that localising at 𝔭 induces a Serre functor Σ^{−d(𝔭)}F_{𝔭} in the sense of Bondal and Kapranov [16].

Let now T = (T, ⊗, 𝟙) be a tensor triangulated category such that R acts on T via a homomorphism of graded rings R → End_{T}^{∗}(𝟙). In that case the Gorenstein property is captured by the existence of a distinguished object.

Lemma 6.1. The R-linear triangulated category T is Gorenstein if and only if there exists a compact object W with the following properties:

(1) There is a compact object W^{−1} such that W ⊗ W^{−1} ≅ 𝟙;

(2) For each 𝔭 in supp_{R}(T) there exists an integer d(𝔭) and an isomorphism

Γ_{𝔭}W ≅ Σ^{d(𝔭)}T_{𝔭}(𝟙) .

One may think of W as a dualising object in T.

Proof. Since Γ_{𝔭} and T_{𝔭} are tensor functors, by (2.2), the functor F: T^{c} → T^{c} and the object W determine each other: W := F (𝟙) and F := W ⊗ −. □

The example below justifies the language of Gorenstein triangulated categories.

Example 6.2. Let A be a commutative noetherian ring and D the derived category of A-modules. This is an A-linear compactly generated tensor triangulated category, with compact objects the perfect complexes of A-modules, that is to say, complexes quasi-isomorphic to bounded complexes of finitely generated projective A-modules.

Recall that the ring A is Gorenstein if for each 𝔭 ∈ Spec A the injective dimension of A_{𝔭}, as a module over itself, is finite; see [17, 3.1]. By Grothendieck’s local duality theorem [17, §3.5], this is equivalent to an isomorphism of A_{𝔭}-modules

Γ_{𝔭}A ≅ Σ^{− dim A_{𝔭}}I(𝔭) .

Thus D is Gorenstein with dualising object A and d(𝔭) = − dim A_{𝔭}; see Lemma 6.1. Conversely, it is not difficult to check that D is Gorenstein only if A is Gorenstein.

There are other examples (notably, differential graded algebras) that fit into this context but we defer discussing these to another occasion. For a finite group scheme G over a field k, the Gorenstein property for StMod G is basically a reformulation of Theorem 5.1.

Corollary 6.3. As an H^{∗}(G, k)-linear triangulated category, StMod G is Gorenstein, with F = δ_{G} ⊗_{k} − the Nakayama functor and d(𝔭) = dim H^{∗}(G, k)/𝔭 for each 𝔭 in Proj H^{∗}(G, k). □

Next we discuss the Gorenstein property for K(Inj G) for a finite group scheme G. To this end observe that the assignment X ↦ νX induces triangle equivalences

K(Inj G) ⟶^{∼} K(Inj G)  and  D^{b}(mod G) ⟶^{∼} D^{b}(mod G).

We are ready to establish the Gorenstein property for K(Inj G).

Theorem 6.4. Let G be a finite group scheme over a field k. Then K(Inj G) is Gorenstein with F induced by the Nakayama functor and d(𝔭) = dim H^{∗}(G, k)/𝔭 for each 𝔭 ∈ Spec H^{∗}(G, k).

Proof. Set 𝔪 := H^{⩾1}(G, k). By Lemma 2.6, one has Γ_{𝔪} = pk⊗_{k}−, where pk denotes a projective resolution of the trivial representation. The isomorphism Γ_{𝔪} ∘ F ≅ T_{𝔪} follows from [32, Theorem 3.4].

For a prime ideal 𝔭 ≠ 𝔪, the assertion follows from Theorem 5.1, for localisation at 𝔭 yields a triangle equivalence Γ_{𝔭}(K(Inj G)) ⟶^{∼} Γ_{𝔭}(StMod G) by Lemma 2.6. □

7. Local Serre duality

In this section we introduce a notion of local Serre duality for an essentially small R-linear triangulated category and link it to the Gorenstein property from Section 6. We use the concept of a Serre functor for a triangulated category which is due to Bondal and Kapranov [16]; this provides a conceptual way to formulate classical Serre duality and Grothendieck’s local duality in a triangulated setting.

In the second part of this section we discuss the existence of Auslander-Reiten triangles. These were introduced by Happel for derived categories of finite dimensional algebras [24], and in [25] he established the connection with the Gorenstein property, while Reiten and Van den Bergh [36] discovered the connection between Auslander-Reiten triangles and the existence of a Serre functor.

Small triangulated categories with central action. Let C be an essentially small R-linear triangulated category. Fix 𝔭 ∈ Spec R and let C_{𝔭} denote the triangulated category that is obtained from C by keeping the objects of C and setting

Hom_{C_{𝔭}}^{∗}(X, Y) := Hom_{C}^{∗}(X, Y)_{𝔭} .

Then C_{𝔭} is an R_{𝔭}-linear triangulated category and localising the morphisms induces an exact functor C → C_{𝔭}.

Let γ_{𝔭}C be the full subcategory of 𝔭-torsion objects in C_{𝔭}, namely

γ_{𝔭}C := {X ∈ C_{𝔭} | End_{C_{𝔭}}^{∗}(X) is 𝔭-torsion}.

This is a thick subcategory of C_{𝔭}. In [12] this category is denoted Γ_{𝔭}C. The notation has been changed to avoid confusion.

Remark 7.1. Let F: C → C be an R-linear equivalence. It is straightforward to check that this induces triangle equivalences F_{𝔭}: C_{𝔭} ⟶^{∼} C_{𝔭} and γ_{𝔭}C ⟶^{∼} γ_{𝔭}C making the following diagram commutative.

\begin{matrix}C & C_{𝔭} & γ_{𝔭}C \\ C & C_{𝔭} & γ_{𝔭}C \end{matrix} F F_{𝔭} F_{𝔭}

Remark 7.2. Let T be a compactly generated R-linear triangulated category. Set C := T^{c} and fix 𝔭 ∈ Spec R. The triangulated categories T_{𝔭} and Γ_{𝔭}T are compactly generated. The left adjoint of the inclusion T_{𝔭} → T induces (up to direct summands) a triangle equivalence C_{𝔭} ⟶^{∼} (T_{𝔭})^{c} and restricts to a triangle equivalence

γ_{𝔭}C ⟶^{∼} (Γ_{𝔭}T)^{c} .

This follows from the fact that the localisation functor T → T_{𝔭} taking X to X_{𝔭} preserves compactness and that for compact objects X, Y in T

Hom_{T}^{∗}(X, Y)_{𝔭} ⟶^{∼} Hom_{T_{𝔭}}^{∗}(X_{𝔭}, Y_{𝔭}) .

For details we refer to [12].

Local Serre duality. Let R be a graded commutative ring that is local; thus there is a unique homogeneous maximal ideal, say 𝔪. Extrapolating from Bondal and Kapranov [16, §3], we call an R-linear triangle equivalence F: C ⟶^{∼} C a Serre functor if for all objects X, Y in C there is a natural isomorphism

(7.1)  Hom_{R}(Hom_{C}^{∗}(X, Y), I(𝔪)) ⟶^{∼} Hom_{C}(Y, F X) .

The situation when R is a field was the one considered in [16]. For a general ring R, the appearance of Hom_{R}^{∗}(−, I(𝔪)), which is the Matlis duality functor, is natural for it is an extension of vector-space duality; see also Lemma A.2.

For an arbitrary graded commutative ring R, we say that an R-linear triangulated category C satisfies local Serre duality if there exists an R-linear triangle equivalence F: C ⟶^{∼} C such that for every 𝔭 ∈ Spec R and some integer d(𝔭) the induced functor Σ^{−d(𝔭)}F_{𝔭}: γ_{𝔭}C ⟶^{∼} γ_{𝔭}C is a Serre functor for the R_{𝔭}-linear category γ_{𝔭}C. Thus for all objects X, Y in γ_{𝔭}C there is a natural isomorphism

Hom_{R}(Hom_{C_{𝔭}}^{∗}(X, Y), I(𝔭)) ⟶^{∼} Hom_{C_{𝔭}}(Y, Σ^{−d(𝔭)}F_{𝔭}X) .

For a compactly generated triangulated category, the Gorenstein condition from Section 6 is linked to local Serre duality for the subcategory of compact objects.

Proposition 7.3. Let R be a graded commutative noetherian ring and T a compactly generated R-linear triangulated category. Suppose that T is Gorenstein, with global Serre functor F and shifts {d(𝔭)}. Then for each 𝔭 ∈ supp_{R}(T), object X ∈ (Γ_{𝔭}T)^{c} and Y ∈ T_{𝔭} there is a natural isomorphism

Hom_{R}(Hom_{T}^{∗}(X, Y), I(𝔭)) ≅ Hom_{T}(Y, Σ^{−d(𝔭)}F_{𝔭}(X)) .

Proof. Given Remark 7.2 we can assume X = C_{𝔭} for a 𝔭-torsion compact object C in T. The desired isomorphism is a concatenation of the following natural ones:

Hom_{R}(Hom_{T}^{∗}(C_{𝔭}, Y), I(𝔭)) ≅ Hom_{R}(Hom_{T}^{∗}(C, Y), I(𝔭))
≅ Hom_{T}(Y, T_{𝔭}(C))
≅ Hom_{T}(Y, Σ^{−d(𝔭)}Γ_{𝔭}F (C))
≅ Hom_{T}(Y, Σ^{−d(𝔭)}Γ_{V(𝔭)}F (C)_{𝔭})
≅ Hom_{T}(Y, Σ^{−d(𝔭)}Γ_{V(𝔭)}F_{𝔭}(C_{𝔭}))
≅ Hom_{T}(Y, Σ^{−d(𝔭)}F_{𝔭}(C_{𝔭}))

In this chain, the first map is induced by the localisation C ↦ C_{𝔭} and is an isomorphism because Y is 𝔭-local. The second one is by the definition of T_{𝔭}(C); the third is by the Gorenstein property of T; the fourth is by the definition of Γ_{𝔭}; the last two are explained by Remark 7.2, where for the last one uses also the fact that C_{𝔭}, and hence also F_{𝔭}(C_{𝔭}), is 𝔭-torsion. □

Corollary 7.4. Let R be a graded commutative noetherian ring and T a compactly generated R-linear triangulated category. If T is Gorenstein, then T^{c} satisfies local Serre duality.

Proof. Combine Proposition 7.3 with Remark 7.2. □

Example 7.5. In the notation of Example 6.2, when A is a (commutative noetherian) Gorenstein ring, local Serre duality reads: For each 𝔭 ∈ Spec A and integer n there are natural isomorphisms

Hom_{A_{𝔭}}(Ext_{A_{𝔭}}^{n}(X, Y), I(𝔭)) ≅ Ext_{A_{𝔭}}^{n+dim A_{𝔭}}(Y, X)

where X is a perfect complexes of A_{𝔭}-modules with finite length cohomology, and Y is a complex of A_{𝔭}-modules.

Auslander-Reiten triangles. Let C be an essentially small triangulated category. Following Happel [24], an exact triangle X ⟶^{α} Y ⟶^{β} Z ⟶^{γ} ΣX in C is an Auslander-Reiten triangle if

(1) any morphism X → X′ that is not a split monomorphism factors through α;

(2) any morphism Z′ → Z that is not a split epimorphism factors through β;

(3) γ ≠ 0. In this case, the endomorphism rings of X and Z are local, and in particular the objects are indecomposable. Moreover, each of X and Z determines the AR-triangle up to isomorphism. Assuming conditions (2) and (3), the condition (1) is equivalent to the following: (1′) The endomorphism ring of X is local. See [29, §2] for details.

Let C be a Krull-Schmidt category, that is, each object decomposes into a finite direct sum of objects with local endomorphism rings. We say that C has ARtriangles if for every indecomposable object X there are AR-triangles

V → W → X → ΣV and X → Y → Z → ΣX .

The next proposition establishes the existence of AR-triangles; it is the analogue of a result of Reiten and Van den Bergh [36, I.2] for triangulated categories that are Hom-finite over a field.

Proposition 7.6. Let R be a graded commutative ring that is local, and let C be an essentially small R-linear triangulated category that is Krull-Schmidt. If C has a Serre functor, then it has AR-triangles.

Proof. Let F be a Serre functor for C and X an indecomposable object in C. The ring End_{C}(X) is thus local; let J be its maximal ideal and I the right ideal of End_{C}^{∗}(X) that it generates. Choose a nonzero morphism Hom_{C}^{∗}(X, X)/I → I(𝔪) and let γ: X → F X be the corresponding morphism in C provided by Serre duality (7.1). We claim that the induced exact triangle

Σ^{−1}F X → W → X ⟶^{γ} F X

is an AR-triangle. Indeed, by construction, if X′ → X is not a split epimorphism, the composition X′ → X ⟶^{γ} F X is zero. Moreover X is indecomposable so is F X.

Applying this construction to F^{−1}ΣX yields an AR-triangle starting at X. □

Let R be a graded commutative ring. An R-linear triangulated category C is noetherian if the R-module Hom_{C}^{∗}(X, Y) is noetherian for all X, Y in C. This property implies that for an object X in γ_{𝔭}C the R_{𝔭}-module E^{∗} := End_{γ_{𝔭}C}^{∗}(X) is of finite length. Therefore the graded ring E^{∗} is artinian, and so E^{0} is an Artin algebra over R_{𝔭}^{0}; see [17, Theorem 1.5.5]. In particular, the idempotent completion of γ_{𝔭}C is a Krull-Schmidt category.

Corollary 7.7. Let T be a compactly generated R-linear triangulated category with T^{c} noetherian. If T is Gorenstein, then (Γ_{𝔭}T)^{c} has AR-triangles for 𝔭 in Spec R.

Proof. As explained above, the assumption that T^{c} is noetherian implies that (Γ_{𝔭}T)^{c} is a Krull-Schmidt category. Thus the assertion follows by combining Propositions 7.3 and 7.6. □

Next we consider local Serre duality for D^{b}(mod G) and stmod G. Recall from Lemma 2.6 that localisation at 𝔭 ∈ Proj H^{∗}(G, k) induces a triangle equivalence Γ_{𝔭}(K(Inj G)) ⟶^{∼} Γ_{𝔭}(StMod G). Using Remark 7.2, this yields (up to direct summands) triangle equivalences

γ_{𝔭}(D^{b}(mod G)) ⟶^{∼} Γ_{𝔭}(K(Inj G))^{c} ⟶^{∼} Γ_{𝔭}(StMod G)^{c} ⟶^{∼} γ_{𝔭}(stmod G).

The result below contains Theorem 1.1 in the introduction.

Theorem 7.8. Let G be a finite group scheme over a field k. Then the H^{∗}(G, k)- linear triangulated category D := D^{b}(mod G) satisfies local Serre duality. Said otherwise, given 𝔭 in Spec R and with d the Krull dimension of H^{∗}(G, k)/𝔭, for each M in γ_{𝔭}D and N in D_{𝔭}, there are natural isomorphisms

Hom_{H^{∗}(G,k)}(Hom_{D_{𝔭}}^{∗}(M, N), I(𝔭)) ≅ Hom_{D_{𝔭}}(N, Ω^{d}δ_{G} ⊗_{k} M) .

In particular, γ_{𝔭}D has AR-triangles.

Proof. The first assertion follows from Theorem 6.4 and Proposition 7.3. The existence of AR-triangles then follows from Corollary 7.7, as D is noetherian. □

AR-components and periodicity. The existence of AR-triangles for a triangulated category C gives rise to an AR-quiver; see for example [26, 34]. The vertices are given by the isomorphism classes of indecomposable objects in C and an arrow [X] → [Y] exists if there is an irreducible morphism X → Y.

In the context of stmod G, one can describe part of the structure of the AR-quiver of the 𝔭-local 𝔭-torsion objects as the Serre functor is periodic.

Proposition 7.9. Let G be a finite group scheme over a field k. Fix a point 𝔭 in Proj H^{∗}(G, k) and set d = dim H^{∗}(G, k)/𝔭. Then the Serre functor

Ω^{d}ν: γ_{𝔭}(stmod G) ⟶^{∼} γ_{𝔭}(stmod G)

is periodic, that is, (Ω^{d}ν)^{r} = id for some positive integer r.

Proof. Lemma 2.1 and (4.1) provide an integer r ≥ 0 such that ν^{r}M ≅ M and Ω^{r}M ≅ M for M in γ_{𝔭}(stmod G). Thus (Ω^{d}ν)^{r} = id, since ν and Ω commute. □

This has the following consequence.

Corollary 7.10. Every connected component of the AR-quiver of γ_{𝔭}(stmod G) is a stable tube in case it is infinite; and otherwise, it is of the form ℤΔ/U, where Δ is a quiver of Dynkin type and U is a group of automorphisms of ℤΔ.

Proof. Since the Serre functor on γ_{𝔭}(stmod G) is periodic, the desired result follows from [34, Theorem 5.5]; see also [26]. □

The preceding result may be seen as a first step in the direction of extending the results of Farnsteiner’s [22, §3] concerning stmod G to γ_{𝔭}(stmod G) for a general (meaning, not necessarily closed) point 𝔭 of Proj H^{∗}(G, k).

Appendix A. Injective modules at closed points

In this section we collect some remarks concerning injective hulls over graded rings, for use in Section 5. Throughout k will be a field and A := ⨁_{i⩾0} A^{i} will be a finitely generated graded commutative k-algebra with A^{0} = k; we have in mind H^{∗}(G, k), for a finite group scheme G over k.

As usual Proj A denotes the homogeneous prime ideals in A that do not contain the ideal A^{⩾1}. Given a point 𝔭 in Proj A, we write k(𝔭) for the graded residue field at 𝔭; this is the homogeneous field of fractions of the graded domain A/𝔭. Observe that k(𝔭)^{0} is a field extension of k and k(𝔭) is of the form k(𝔭)^{0}[t^{±1}] for some indeterminate t over k(𝔭)^{0}; see, for example, [17, Lemma 1.5.7].

Lemma A.1. The degree of k(𝔪)^{0}/k is finite for any closed point 𝔪 in Proj A.

Proof. One way to verify this is as follows: The Krull dimension of A/𝔪 is one so, by Noetherian normalisation, there exists a subalgebra k[t] of A/𝔪 where t is an indeterminate over k and the A/𝔪 is finitely generated k[t]-module. Thus, inverting t, one gets that (A/𝔪)_{t} is a finitely generated module over the graded field k[t^{±1}], and hence isomorphic to k(𝔪). The finiteness of the extension k(𝔪)/k[t^{±1}] implies that the extension k(𝔪)^{0}/k of fields is finite. □

The result below is familiar; confer [17, Proposition 3.6.16].

Lemma A.2. Let A be as above, let 𝔪 be a closed point in Proj A and set R := A_{𝔪}. The R-submodule I := ⋃_{i⩾0} Hom_{k}^{∗}(R/𝔪^{i}, k) of Hom_{k}^{∗}(R, k) is the injective hull of k(𝔪), and for any 𝔪-torsion R-module N, there is a natural isomorphism

Hom_{R}(N, I) ≅ Hom_{k}(N, k) .

Proof. Set K = k(𝔪)^{0} and recall that k(𝔪) = K[t^{±1}], for some indeterminate t over K. Thus, one has isomorphisms of graded k(𝔪)-modules

(A.1)  Hom_{k}^{∗}(k(𝔪), k) ≅ Hom_{K}^{∗}(k(𝔪), Hom_{k}(K, k))
≅ Hom_{K}^{∗}(k(𝔪), K)
≅ k(𝔪) .

The first isomorphism is adjunction, the second holds because rank_{k} K is finite, by Lemma A.1, and the last one is a direct verification.

The R-module Hom_{k}^{∗}(R, k) is injective and hence so is its 𝔪-torsion submodule

⋃_{i⩾0} Hom_{R}^{∗}(R/𝔪^{i}, Hom_{k}^{∗}(R, k)) .

This is precisely the R-module I, by standard adjunction. Thus I must be a direct sum of shifts of injective hulls of k(𝔪). It remains to verify that I is in fact just the injective hull of k(𝔪). To this end, note that for any 𝔪-torsion R-module N, one has isomorphisms of graded k(𝔪)-modules

Hom_{R}^{∗}(N, I) ≅ Hom_{R}^{∗}(N, Hom_{k}^{∗}(R, k))
≅ Hom_{k}^{∗}(N, k) .

This settles the last assertion in the desired result and also yields the first isomorphism below of graded k(𝔪)-modules.

Hom_{R}^{∗}(k(𝔪), I) ≅ Hom_{k}^{∗}(k(𝔪), k) ≅ k(𝔪)

The second one is by (A.1). It follows that I is the injective hull of k(𝔪). □

The next result, whose proof is rather similar to the one above, gives yet another way to get to the injective hull at a closed point of Proj.

Recall that I(𝔭) denotes the injective hull of A/𝔭 for any 𝔭 in Spec A.

Lemma A.3. Let A → B be a homomorphism of graded commutative algebras, let 𝔪 be a closed point in Proj B, and set 𝔭 := 𝔪 ∩ A. If the extension of residue fields k(𝔭) ⊆ k(𝔪) is finite, then the B-module Γ_{V(𝔪)} Hom_{A}^{∗}(B, I(𝔭)) is the injective hull of B/𝔪, and for any 𝔪-torsion B-module N, adjunction induces an isomorphism

Hom_{B}(N, I(𝔪)) ≅ Hom_{A}(N, I(𝔭)) .

Proof. The B-module I := Γ_{V(𝔪)} Hom_{A}^{∗}(B, I(𝔭)) is injective, for it is the 𝔪-torsion submodule of the injective B-module Hom_{A}^{∗}(B, I(𝔭)). As 𝔪 is a closed point, I is a direct sum of shifts of copies of I(𝔪). It remains to make the computation below:

Hom_{B}^{∗}(k(𝔪), I) ≅ Hom_{B}^{∗}(k(𝔪), Hom_{A}^{∗}(B, I(𝔭)))
≅ Hom_{A}^{∗}(k(𝔪), I(𝔭))
≅ Hom_{k(𝔭)}^{∗}(k(𝔪), Hom_{A}^{∗}(k(𝔭), I(𝔭))
≅ Hom_{k(𝔭)}^{∗}(k(𝔪), k(𝔭))
≅ k(𝔪)

These are all isomorphisms of k(𝔪)-modules. The last one is where the hypothesis that k(𝔪)/k(𝔭) is finite is required. This implies that I ≅ I(𝔪). Given this, the last isomorphism follows by standard adjunction. □

Acknowledgements. It is a pleasure to thank Tobias Barthel for detailed comments on an earlier version of this manuscript.

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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.