

Local duality for representations of
finite group schemes

Dave Benson, Srikanth B. Iyengar, Henning Krause
and Julia Pevtsova
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
Henning Krause
Fakultät für Mathematik
Universität Bielefeld
33501 Bielefeld
Julia Pevtsova
Department of Mathematics
University of Washington
Seattle, WA 98195
12 July 2017

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 -local and -torsion subcategories of the stable category, for each homogeneous prime ideal in the cohomology ring of the group scheme.

Key words and phrases:
Serre duality, local duality, finite group scheme, stable module category, Auslander-Reiten triangle
2010 Mathematics Subject Classification:
16G10 (primary); 20C20, 20G10 20J06, 18E30
SBI was partly supported by NSF grant DMS-1503044 and JP was partly supported by NSF grants DMS-0953011 and DMS-1501146.

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 over a field is the category obtained from the (abelian) category of finite dimensional -modules by annihilating morphisms that factor through a projective module; we denote it , and write for the morphisms in this category. The category is triangulated with suspension , and Tate duality translates to the statement that for all finite dimensional -modules and there are natural isomorphisms

Here is the modular character of , described in Jantzen [28, §I.8.8]; it is isomorphic to the trivial representation when 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 has Serre duality with Serre functor . 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 of on the graded abelian group

The ring is graded commutative, and also finitely generated as a -algebra, by a result of Friedlander and Suslin [23]. Fix a homogeneous prime ideal not containing and consider the triangulated category that is obtained from by localising the graded morphisms at and then taking the full subcategory of objects such that the graded endomorphisms are -torsion; see Section 7 for details. Our interest in the subcategories stems from the fact that they are the building blocks of 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 -local spectra in stable homotopy theory that give the chromatic filtration of a spectrum; see [35].

Our local Serre duality statement reads:

Theorem 1.1.

Let and the Krull dimension of . The assignment induces a Serre functor for . Thus for all in there are natural isomorphisms

where is the injective hull of the graded -module .

One of the corollaries is that has Auslander-Reiten triangles. One can thus bring to bear the techniques of AR theory to the study of -modules. These results are contained in Theorem 7.8.

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

Theorem 1.2.

For any -module and there is a natural isomorphism

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

see, for example, Hartshorne [27].

When is a finite group is the Rickard idempotent module , introduced by Benson, Carlson, and Rickard [6], that is associated to the irreducible subvariety of defined by . 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 , the following consequences of Theorem 1.2 have been anticipated in [4] when is a finite group.

Theorem 1.3.

Suppose that . Then the -module is injective and up to a twist isomorphic to . Also, there is an isomorphism of -algebras

where denotes completion with respect to the -adic topology, and the -module is pure injective.

Theorem 1.2 can be interpreted to mean that the category is Gorenstein, for it is analogous to Grothendieck’s result that a commutative noetherian ring is Gorenstein if, and only if, is the injective hull of , up to suspension, for each in . 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 , there is a purely transcendental extension of and a closed point of lying above the point in . Here, denotes the group scheme that is obtained from by extending the field to . The crux is that one can choose such that the following statement holds.

Theorem 1.4.

Restriction of scalars induces a dense exact functor

This result is proved in Section 3, building on our work in [14]. It gives a remarkable description of the compact objects in : they are obtained from the finite dimensional objects in 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 are endofinite -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 be a triangulated category with suspension . Given objects and in , consider the graded abelian groups

Composition makes a graded ring and a left- right- module.

Let be a graded commutative noetherian ring. In what follows we will only be concerned with homogeneous elements and ideals in . In this spirit, ‘localisation’ will mean homogeneous localisation, and will denote the set of homogeneous prime ideals in .

We say that a triangulated category is -linear if for each in there is a homomorphism of graded rings such that the induced left and right actions of on are compatible in the following sense: For any and , one has

An exact functor between -linear triangulated categories is -linear if the induced map

of graded abelian groups is -linear for all objects in .

In what follows, we fix a compactly generated -linear triangulated category and write for its full subcategory of compact objects.


Fix an ideal in . An -module is -torsion if for all in with . Analogously, an object in is -torsion if the -module is -torsion for all . The full subcategory of -torsion objects

is localising and the inclusion admits a right adjoint, denoted .

Fix a in . An -module is -local if the localisation map is invertible, and an object in is -local if the -module is -local for all . Consider the full subcategory of of -local objects

and the full subcategory of -local and -torsion objects

Note that are localising subcategories. The inclusion admits a left adjoint while the inclusion admits a right adjoint. We denote by the composition of those adjoints; it is the local cohomology functor with respect to ; see [9, 10] for the construction of this functor.

The following observation is clear.

Lemma 2.1.

For any element in , say of degree , and -local object , the natural map is an isomorphism. ∎

The functor commutes with exact functors preserving coproducts.

Lemma 2.2.

Let be an exact functor between -linear compactly generated triangulated categories such that is -linear and preserves coproducts. Suppose that the action of on factors through a homomorphism of graded commutative rings. For any ideal of there is a natural isomorphism

of functors , where denotes the ideal of that is generated by .


The statement follows from an explicit description of in terms of homotopy colimits; see [10, Proposition 2.9]. ∎

Injective cohomology objects

Given an object in and an injective -module , Brown representability yields an object in such that


This yields a functor

For each in , we write for the injective hull of and set

viewed as a functor .

Tensor triangulated categories

Let now be a tensor triangulated category such that acts on via a homomorphism of graded rings . It is easy to verify that in this case the functors and are tensor functors:


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

The stable module category

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

We write for the cohomology algebra, , of . This is a graded commutative -algebra, because is a Hopf algebra, and acts on , for any -modules . Moreover, the -algebra is finitely generated, and, when are finite dimensional, is finitely generated over it; this is by a theorem of Friedlander and Suslin [23].

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

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

We use the notation for the -sets in . The cohomology algebra acts on via a homomorphism of -algebras

Thus, the preceding discussion on localisation functors on triangulated categories applies to the -linear category .

Koszul objects

An element in corresponds to a morphism in ; let denote its mapping cone. This gives a morphism . For a sequence of elements in and a -module , we set

It is easy to check that for a -module and , there is an isomorphism


Let denote the ideal of generated by . By abuse of notation we set . If is a finite set of elements in such that , then, by [12, Proposition 3.10], for any in there is an equality


Fix in . We will repeatedly use the fact that is generated as a triangulated category by the family of objects with in ; see [10, Proposition 3.9]. In fact, if denotes the direct sum of a representative set of simple -modules, then there is an equality


It turns out that one has where denotes , the ideal of elements of positive degree; see Lemma 2.5 below. For this reason, it is customary to focus on , the set of homogeneous prime ideals not containing , when dealing with .

Tate cohomology

By construction, the action of on factors through , the graded ring of endomorphisms of the identity. The latter ring is not noetherian in general, which is one reason to work with . In any case, there is little difference, vis a vis their action on , as the next remarks should make clear.

Remark 2.3.

Let and be -modules. The map induces a map of -modules. This map is surjective in degree zero, with kernel , the maps from to that factor through a projective -module. It is bijective in positive degrees and hence one gets an exact sequence of graded -modules


with for . For degree reasons, the -modules and are -torsion. Consequently, for in the induced localised map is an isomorphism:


More generally, for each in localisation induces an isomorphism

of -modules. This means that 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 are finite dimensional -modules, is a finitely generated -module for any . Moreover the -module

is finitely generated for each in .

Lemma 2.5.

One has , where .


Given (2.5) it suffices to check that in , where is the direct sum of representative set of simple -modules. For any -module , the -module is -torsion; see [9, Lemma 5.11(1)]. Thus, when is finite dimensional, the -module is -torsion and finitely generated, by Remark 2.4, so it follows that for . This implies that is projective, since is self-injective. ∎

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

The homotopy category of injectives

Let and denote the homotopy category of and the derived category of , respectively. These are also -linear compactly generated tensor triangulated category, with the tensor product over . The unit of the tensor product on is an injective resolution of the trivial -module , while that of is . The canonical quotient functor induces an equivalence of triangulated category , where the target is the bounded derived category of ; see [31, Proposition 2.3].

Taking Tate resolutions identifies with , the full subcategory of acyclic complexes in . In detail, let and be a projective and an injective resolution of the trivial -module , respectively, and let be the mapping cone of the composed morphism ; this is a Tate resolution of . Since projective and injective -modules coincide, one gets the exact triangle


in . For a -module , the complex is a Tate resolution of and the assignment induces an equivalence of categories

with quasi-inverse , the submodule of cycles in degree . Assigning in to is a left adjoint of the inclusion . These results are contained in [31, Theorem 8.2]. Consider the composed functor

A straightforward verification yields that these functors are -linear. The result below is the categorical underpinning of Remark 2.3 and Lemma 2.5.

Lemma 2.6.

There is a natural isomorphism for . For each in , the functor induces triangle equivalences


We identify with . This entails , by Lemma 2.5. It is easy to check that is -torsion, and hence so is , for it is in the localising subcategory generated by , and the class of -torsion objects in is a tensor ideal localising subcategory; see, for instance, [9, Section 8]. Thus, applying to the exact triangle (2.8) yields . It then follows from (2.2) that for any in .

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

These are precisely the -torsion objects in , by the already established part of the result. Said otherwise, is acyclic if and only if . It follows that contains the subcategory of -local objects, for each in . On the other hand, the inclusion preserves coproducts, so its left adjoint preserves compactness of objects and all compacts of are in the image of . Given this a simple calculation shows that contains . Thus . ∎

3. Passage to closed points

Let be a finite group scheme over a field of positive characteristic. In this section we describe a technique that relates the -local -torsion objects in , for a point in , to the corresponding modules at a closed point defined over a field extension of . Recall that a point is closed when it is maximal with respect to inclusion: implies for all in . In what follows, denotes the graded residue field of at .

For a field extension extension of scalars and restriction give exact functors

Moreover, since as -algebras, yields a homomorphism of rings. It induces a map

with mapping to . We say that lies over to indicate this. The main objective of this section is the proof of the following result.

Theorem 3.1.

Fix in and a purely transcendental extension of degree . There exists a closed point in lying over with such that the functor restricts to functors

that are dense.

The proof of the theorem yields more: There is a subcategory of on which is full and dense; ditto for the category of compact objects. However, the functor need not be full on all of ; see Example 3.7.

Here is one consequence of Theorem 3.1.

Corollary 3.2.

The compact objects in are precisely the restrictions of finite dimensional -modules in .


By [9, Theorem 6.4], for any ideal in , we have

Applying this observation to the ideal of and noting that , since 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 as is explained in the construction below, from [14, §7].

Construction 3.3.

Fix in ; the following construction is relevant only when is not a closed point. Choose elements in of the same degree such that their image in is algebraically independent over and is finitely generated as a module over the subalgebra . Thus the Krull dimension of is . Set , the field of rational functions in indeterminates and

viewed as elements in . Let denote the extension of to , and set

It is proved as part of [14, Theorem 7.7] that the ideal is a closed point in with the property that . 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

The sequence of elements in yields a morphism , where , and composing its restriction to with the canonical morphism gives in a morphism

Since the are not in , Lemma 2.1 yields a natural stable isomorphism


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

By [13, Lemma 2.1], for any -module there is a natural isomorphism


The result below extends [14, Theorem 8.8]; the latter is the case .

Theorem 3.4.

For any -module , the morphism induces a natural stable isomorphism of -modules

When is -torsion, these induce natural stable isomorphisms


We begin by verifying the second set of isomorphisms. As is -torsion so is and then it is clear that the natural map is an isomorphism. The third of the desired isomorphisms follows from (3.2). It thus remains to check that induces an isomorphism

It is easy to verify that the modules having this property form a tensor ideal localising subcategory of . Keeping in mind  (3.1), from [14, Theorem 8.8] one obtains that this subcategory contains . The desired assertion follows since the -torsion modules form a tensor ideal localising subcategory of that is generated by ; 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 be an arbitrary -module, and let be as in Construction 3.3. Since is -torsion, the already established isomorphism yields the second one below.

The third one is by Lemma 2.2, applied to the functor from to . The next one is standard while the penultimate one holds because is -torsion. This completes the proof. ∎

In the next remark we recast part of Theorem 3.4.

Remark 3.5.

Fix a point in , and let , and be as in Construction 3.3. Consider the following adjoint pair of functors.

It is easy to check that this induces an adjoint pair

Theorem 3.4 implies that for any in .

Proof of Theorem 3.1.

Let , , and be as in Construction 3.3. As noted there, is a closed point in lying over and . The modules in are precisely those with support contained in . It then follows from [14, Proposition 6.2] that restricts to a functor

This functor is dense because for any -module that is -local and -torsion one has where is the functor from Remark 3.5.

Consider the restriction of to compact objects in . First we verify that its image is contained in the compact objects of . To this end, it suffices to check that there exists a generator of , as a thick subcategory, whose restriction is in .

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

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

It remains to note that is in , again by (2.5).

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

Theorem 3.4 yields that for any morphism in ; in particular, the restriction functor is full and dense on the subcategory of consisting of objects of the form , where is a -local -torsion -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 be a field and the polynomial ring in an indeterminate . Let denote its derived category; it is -linear in an obvious way. For the prime of the -local -torsion subcategory is naturally identified with the derived category of , the field of rational functions in .

With denoting the field of rational functions in an indeterminate , the maximal ideal of lies over the prime ideal of . The inclusion induces an isomorphism . The analogue of Theorem 3.4 is that restriction of scalars along the inclusion induces a dense functor

This property can be checked directly: The -torsion module restricts to , and each object in is a direct sum of shifts of . This functor is however not full: For , the -module is -torsion, and satisfies

where stands for the appropriate derived category. In particular, if , the canonical map