Fibrewise stratification of group representations

Given a finite cocommutative Hopf algebra $A$ over a commutative regular ring $R$, the lattice of localising tensor ideals of the stable category of Gorenstein projective $A$-modules is described in terms of the corresponding lattices for the fibres of $A$ over the spectrum of $R$. Under certain natural conditions on the cohomology of $A$ over $R$, this yields a stratification of the stable category. These results apply when $A$ is the group algebra over $R$ of a finite group, and also when $A$ is the exterior algebra on a finite free $R$-module.


Introduction
Following the seminal work of Hopkins [24] and Neeman [34] in stable homotopy theory and commutative algebra, much attention has been paid in the past few decades to the problem of classifying the thick subcategories of finite dimensional representations over various families of algebras, and also of the localising subcategories of all representations.In terms of the language and machinery developed in [11,13], the goal is to prove stratification theorems.For example, in the case of modular representations of a finite group, the thick tensor ideal subcategories of the small stable module category were classified in [8], while the tensor ideal localising subcategories of the large stable module category were classified in [12].These results were generalised to cover all finite group schemes over fields in [16].
In this paper we address the problem of change of coefficients, with a focus on representations of group algebras of finite group schemes, and in particular, of finite groups.Let A be the group algebra of a finite group scheme over a commutative noetherian ring R; in other words, A is a cocommutative Hopf algebra that is finitely generated and projective David John Benson et al. over R. For example, A could be the group algebra RG of a finite group G.The appropriate analogue of the stable category of finite dimensional representations over a finite group, or group scheme, is the singularity category of A, in the sense of Buchweitz [18] and Orlov [37].By a result of Buchweitz, this is equivalent to the stable category of Gorenstein projective A-modules: An A-module is said to be Gorenstein projective if it occurs as a syzygy in an acyclic complex of projective A-modules.The Gorenstein projective modules in mod A form a Frobenius category, with projective-injective objects the projective A-modules; Gproj A is the corresponding stable category.We also have to consider all Gorenstein projectives, not only the finitely generated ones, and the corresponding stable category GProj A. The category GProj A is triangulated and compactly generated; the subcategory of compact objects is equivalent to Gproj A.
From now on assume that R is regular, for example, R = Z.In this case, a finitely generated A-module is Gorenstein projective precisely when it is projective when viewed as an R-module.The same holds also for infinitely generated A-modules when in addition dim R is finite, that is to say, when R has finite global dimension; see Lemma 3.8.Since A is a Hopf algebra over R, the tensor product over R induces on Gproj A the structure of a tensor-triangulated category and also on GProj A. Moreover GProj A is rigidly compactly generated.Our goal is to classify the thick tensor ideals of Gproj A and the localising tensor ideals of GProj A.
One approach would be to extend methods developed in the case where R is a field to cover more general coefficient rings.In this work, we take a different tack, by viewing A as a family of Hopf algebras parameterised by Spec R, the Zariski spectrum of R. The fibre over each point p in Spec R is the finite dimensional Hopf algebra A k(p) := A ⊗ R k(p), where k(p) is the residue field at p.The results of [16] apply to yield a stratification of GProj A k(p) in terms of the projective spectrum of the cohomology ring of A k(p) .Then the task becomes one of "patching" these local stratifications to obtain a global stratification of GProj A in terms of the projective spectrum of the cohomology ring of A.
There are two aspects to this task: one representation theoretic and the other purely cohomological.The former is completely solved by the result below that can be viewed as a fibrewise criterion for detecting membership in localising tensor ideals.We deduce it from Theorem 4.5 that deals with the full homotopy category of projective A-modules.Theorem 1.1.Let R be a regular ring, A a finite cocommutative Hopf R-algebra, and M, N Gorenstein projective A-modules.The conditions below are equivalent.
(1) M ∈ Loc ⊗ (N ) in GProj A; (2) The cohomological aspect concerns the relationship between the fibres of the cohomology algebra S := Ext * A (R, R) of A and the cohomology algebra of the fibres of the R-algebra A. Namely, for each p in Spec R there is natural map of k(p)-algebras.The question is when this map induces a bijection on spectra.Its import is clear from the next result.
Theorem 1.2.Let R be a regular ring and A a finite cocommutative Hopf algebra over R. If the map κ p induces a bijection on spectra for each p in Spec R, then the tensortriangulated category GProj A is stratified by the action of S, and the support of GProj A is Proj S.
This result is proved at the end of Section 4, where we recall what it means for a tensor-triangulated category to be stratified by an action of a ring.For now it suffices to record that the result above yields the sought after classification of localising tensor ideals of GProj A and of the thick tensor ideals of Gproj A.
The hypothesis on κ p holds for the group algebra A = RG over a finite group. 1 This is a result of Benson and Habegger [10].The proof there is lacking detail, so we provide a full proof here in greater generality; see Theorem 5.4, and also the recent work of Lau [33, § 7].Putting this together with Theorem 1.2 yields the following stratification result.In fact we prove this stratification result for the slightly bigger compactly generated tensor-triangulated category K(Proj RG) consisting of complexes of projective RG-modules up to homotopy.The subcategory of compact objects identifies with the bounded derived category D b (mod RG).Another case where Theorem 1.2 applies is when A is an exterior algebra, over a finite free R-module, regarded as a Z/2-graded Hopf algebra; see Example 4.9.
So far we have focused on Hopf algebras, but in fact an appropriate version of Theorem 1.1 holds for any finite projective R-algebras A; see Theorem 2.1.In such contexts the natural cohomology ring to consider, vis-à-vis stratification, is the Hochschild cohomology of A over R. It is plausible that the analogue of the map κ p in that context is the key to patching fibrewise stratification, when available, to get a global stratification result for A.

Related works.
For G a finite group and R a regular ring, the classification of thick tensor ideals of D b (mod RG) (which follows from Theorem 1.3) has been obtained by Lau [33].Building on Lau's work in conjunction with developing novel homotopy theoretic methods Barthel [3,4] established a classification of localising tensor ideals of RG-modules that are projective as R-modules, which is closely related to Theorem 1.3 (see Remark 5.8 for more detail).
Our interest in the subject was propelled by asking a simple minded question: "Can one deduce the stratification of R-linear representations of G from that of representations over fields, where it is known?"-whichechoes the approach taken by Lau for the bounded derived category.As already indicated, in this work we argue that such a reduction is indeed possible and consists of two simple steps.The first is the fibrewise detection of Theorem 1.1 which works in a very general algebraic setting; and the second is the behaviour of the cohomological fibres which we handle in the case of finite groups as in Corollary 5.5 and in the case of exterior algebras.
Hence, for us the main aspects of this work are the general fibrewise criterion Theorem 1.1 and the general stratification Theorem 1.2 with the stratification for finite groups being an application of these general techniques.

Subsequent developments.
We are happy to report that the conclusion of Theorem 1.2 holds unconditionally.Building on recent work of van der Kallen [39], in joint work with Barthel [5] we prove that the map κ p is always a homeomorphism, and in particular bijective.One complication that arises is that the tensor-triangulated category in question is no longer the stable category of Gorenstein projective modules but a suitable category of lattices, and the arguments needed to deal with this, and other aspects, are rather more involved.

A fibrewise criterion for localising subcategories
Throughout R is a commutative noetherian ring and A a finite projective R-algebra; that is to say A is an R-algebra that is finitely generated and projective as an R-module.In particular, as a ring A is noetherian on the left and on the right.Given an A-complex X and a point p in Spec R, the Zariski spectrum of R, set viewed as a complex of A k(p) -modules.The assignment X → X k(p) is exact on the homotopy category of A-complexes, and hence also on any of its subcategories, in particular, on K(Proj A), the homotopy category of projective A-modules.The latter has a natural structure of a triangulated category, with arbitrary coproducts.Given an A-complex Y in K(Proj A) we write Loc(Y ) for the localising subcategory of the homotopy category generated by Y .The main result in this section is the following fibrewise criterion for detecting objects in Loc(Y ).
Theorem 2.1.Suppose that R is regular.Let A be a finite projective R-algebra, and let X, Y be objects in K(Proj A).The following conditions are equivalent. (1) The ring R is by definition regular if every finitely generated R-module has finite projective dimension.An equivalent condition is that the ring R p has finite global dimension for each p in Spec R. The global dimension of R is then equal to dim R, its Krull dimension.
The statement above is inspired by, and extends, an analogous statement for the derived category of A, established in [23].It yields also a statement about the stable category of Gorenstein projective A-modules; see Theorem 3.5.
The proof of the theorem above takes some preparation and is given towards the end of this section.We start by recalling some properties of the homotopy category of projective modules.
The homotopy category of projectives.For the moment, A can be any ring that is noetherian on both sides; that is to say, A is noetherian as a left and as a right A-module.For us, A-modules mean left A-modules, and A op -modules are identified with right Amodules.When A is an R-algebra for some commutative ring R, then it is convenient to write for any A-module the R-action on the opposite side.We denote by Mod A the (abelian) category of A-modules and by mod A its full subcategory consisting of finitely generated modules.The full subcategory of Mod A consisting of projective modules is denoted Proj A.
For any additive category A ⊆ Mod A, like the ones in the last paragraph, K(A) will denote the associated homotopy category, with its natural structure as a triangulated category.Morphisms in this category are denoted Hom K(A) (−, −).An object X in K(A) is acyclic if H * (X) = 0, and the full subcategory of acyclic objects in K(A) is denoted K ac (A).
Let D(Mod A) denote the derived category of A-modules and D b (mod A) the bounded derived category of mod A. Let q : K(Mod A) → D(Mod A) be the localisation functor; its kernel is K ac (Mod A).We write q also for its restriction to the homotopy category of projective modules.This functor has an adjoint: Our convention is to write the left adjoint above the corresponding right one.In what follows it is convenient to conflate p with p • q.The image of p, denoted K proj (A), consists precisely of the K-projective complexes, namely, those complexes P such that Hom K(A) (P, −) = 0 on acyclic complexes in K(Mod A).
Compact objects.The category K(Proj A) is triangulated, admits arbitrary direct sums, and is compactly generated.As in any triangulated category with arbitrary direct sums, an object X in K(Proj A) is compact if Hom K(A) (X, −) commutes with direct sums.The compact objects in K(Proj A) form a thick subcategory, denoted K c (Proj A).The assignment M → Hom A op (pM, A) induces an equivalence This result is due to Jørgensen [30,Theorem 3.2]; see also [27].
Regular rings.Recall that the ring R is regular if every finitely generated R-module has finite projective dimension, that is, every complex in D b (mod R) is perfect.We record a couple of basic facts for later use.
Proof.(1) The functor p in (2.1) restricted to compact objects embeds the perfect complexes over R into D b (mod R), via (2.2).This embedding is an equivalence if and only p is an equivalence.Clearly, p is an equivalence if and only if q is an equivalence.See also [26].
(2) The Koszul complex K on a minimal generating set for the maximal ideal of R provides a projective resolution K → k.Since Hom R (K, R) ∼ = Σ −d K, by [17, Proposition 1.6.10], the stated assertion follows.□ Fibres and the Koszul complex.We return to the context of finite projective algebras over a commutative ring R and wish to describe the functor − ⊗ R k(p) for each prime p in Spec R via the Koszul complex for p.We reduce to the local case.Let (R, m, k) be a local ring, with maximal ideal m and residue field k.In this paragraph, we write A k instead of A k(m) .Let π be the functor from K(Proj A) to K(Proj A k ) given by the assignment X → X k := X ⊗ R k.It is clear that π preserves coproducts and so has a right adjoint by Brown representability, say π r .Thus there is the adjoint pair  We need to understand the unit id → π r π and counit ππ r → id of the adjunction (2.3).To that end, consider the Koszul complex, K, on a minimal generating set for the ideal m.The map R → K of R-complexes induces a natural map On the other hand, the augmentation The result below is the key to the proof of Theorem 2.1 and also in other computations that follow.
Lemma 2.4.When R is regular, the following statements hold.
(1) Both π and π r preserve compact objects and arbitrary direct sums.
(2) Restricting the adjunction (2.3) to compact objects gives the adjunction where ρ corresponds to π r and is induced by restriction along the map A → A k and d := dim R. The right adjoint k ⊗ L R − corresponds to π. (3) The maps (2.4) and (2.5) are the unit and the counit, respectively, of the adjunction (2.3).(4) For X ∈ K(Proj A) and Y ∈ K(Proj A k ), there are natural isomorphisms In particular, ππ r (Y ) is isomorphic to a finite direct sum of shifts of Y . Proof.
(1) and (2): We have already observed that π preserves arbitrary direct sums.We verify that π preserves compacts, equivalently that its right adjoint, π r , preserves arbitrary direct sums.Any compact object in This justifies the claim that π preserves compacts.Along the way we have established that In other words, restricted to compacts π is the functor 2).The functor Σ −d ρ is left adjoint to the functor above: The first isomorphism is standard adjunction, the second uses the fact that k is perfect as an R-complex, since R is regular, and the third one follows from the fact that RHom At this point we can apply Lemma 2.3 to deduce that π r also preserves compacts and is isomorphic to the functor Σ −d ρ when restricted to compact objects.This settles both (1) and ( 2). ( and this corresponds under the adjunction isomorphism (2.6) It suffices to prove that this is an isomorphism.We verify this when X is compact; the general case then follows as π and π r preserve arbitrary direct sums, by (1).
Since K is a finite free R-complex, the assignment ), and that the map (2.6) corresponds to the map and is hence an isomorphism.Here again we are using the hypothesis that R is regular.This completes the proof that (2.4) is the unit of the adjunction.The claim about (2.5) can be verified along the same lines.
David John Benson et al.
(4) The isomorphisms were verified along the way to verifying (3).The last assertion holds because V is a finite graded k-vector space with zero differential.□ Local cohomology and support.In the proof of Theorem 2.1, and later on in the sequel, we require the theory of local cohomology and support from [11], with respect to the action of the ring R on the homotopy category of projective modules.The analogue for the homotopy category of injective modules is described in [28, § 7].One could invoke that theory, for the two homotopy categories are equivalent, at least under certain minor additional constraints on A, but to keep this manuscript self-contained we develop the needed results for K(Proj A) directly.
For any pair of objects X, Y in K(Proj A) there is a natural R-module structure on Hom K(A) (X, Y ), so that K(Proj A) is an R-linear triangulated category, in the sense of [11, § 4].In particular, for each specialisation closed subset V of Spec R there is an exact triangle Here an object Y is by definition For any ideal I ⊂ R we consider the closed set Lemma 2.5.Let I ⊂ R be an ideal and K the Koszul complex on a finite generating set for the ideal I.For each X in K(Proj A) one has Proof.If K is the Koszul complex on a single element r in R, the complex X ⊗ R K is isomorphic to the mapping cone of the morphism X r − → X; in other words, X ⊗ R K is the complex denoted X/ /r in [13, § 2.5].The Koszul complex K on a sequence of elements r 1 , . . ., r n generating the ideal I can be constructed as an iterated mapping cone, so X⊗ R K represents X/ /I.Thus the stated result is a special case of [13,Proposition 2.11 (2)].□ Fix a point p in Spec R and consider the specialisation closed subset The localisation functor X → L Z(p) X models localisation at p in the sense that for each compact object C in K(Proj A) the map See [11,Theorem 4.7], and also [13,Proposition 2.3].
The localisation functor L Z(p) admits an alternative description which will be useful.For a complex X in K(Proj A) set viewed as a complex of A p -modules, where A p denotes the R p -algebra A ⊗ R R p .The assignment X → X p yields an adjoint pair of functors The right adjoint λ r exists as localisation preserves coproducts.
Lemma 2.6.The right adjoint λ r preserves coproducts.Moreover, for each X in K(Proj A), the unit X → λ r λ(X) of the adjunction (2.8) is naturally isomorphic to the localisation Proof.Observe that the functor λ also preserves compact objects, hence its right adjoint λ r preserves coproducts.As to the second claim, it suffices to verify that for any compact object C in K(Proj A) the map induced by the unit is localisation at p. Adjunction gives an isomorphism of R-modules, so the module on the left is p-local.Thus one gets an induced map and the desired result is that this is an isomorphism.Since C is compact, and localisation at p, and the functors λ r and λ preserve coproducts, it suffices to verify the map above is an isomorphism when X is also compact.Consider again the adjunction isomorphism Since C p and X p are compact in K(Proj A p ), by the description of compact objects in K(Proj A), the desired result is that for M, N in D b (mod A op ), the map The local cohomology functor at p is the functor Γ p on K(Proj A) given by where the isomorphism is the one from Lemma 2.6.The local cohomology functors reduce the description of localising subcategories to a local problem, because the local-to-global theorem says that (2.10) see [13, § 3] and also [38,Theorem 6.9].
Proof of Theorem 2.1.The implication (1)⇒( 2) is clear since for each p in Spec R the functor given by X → X k(p) is exact and preserves all coproducts.
(2)⇒(1) Let X, Y be objects in K(Proj A).By the local-to-global theorem (2.10) it suffices to verify for each p We denote by K the Koszul complex on a minimal generating set for the maximal ideal pR p of R p .We have by Lemma 2.4.This means by Lemma 2.5.It remains to apply the functor λ r .Thus ( The preceding result applied with N = A implies that M is perfect if and only if it is fibrewise perfect.Here is a more precise result, for later use.Lemma 2.9.Let R be a commutative noetherian ring, A a finite projective R-algebra, and M a finitely generated A-module.When M is projective as an R-module, there is an equality Moreover, it suffices to take the supremum over the maximal ideals in R. Proof where L 1 , . . ., L r are the simple A-modules.The L j are modules over A k := A/mA and M is projective over R, so adjunction yields isomorphisms Since the L j are the simple modules over A k the desired result follows.□

Gorenstein algebras
Let R be a commutative noetherian ring.Following [23,28], we say A is a Gorenstein R-algebra when it is a finite projective R-algebra such that for each p in supp R A the ring A p has finite injective dimension on the left and on the right; that is to say, it is Iwanaga-Gorenstein.When this holds R p is Gorenstein for each p in supp R A. Here is a characterisation of the Gorenstein property that is in the spirit of this work; see also [ is perfect on both sides, for each p in Spec R; this follows from Corollary 2.8 applied with M := Hom R (A, R) and N := A. It remains to observe that this latter condition is equivalent to A k(p) being Iwanaga-Gorenstein.□ Remark 3.2.One consequence of the Gorenstein condition is that acyclic complexes in K(Proj A) are totally acyclic, namely each complex X ∈ K ac (Proj A) satisfies Hom K(A) (X, P ) = 0 for any projective A-module P .
See [28,Theorem 5.6] for a proof.The functors induce a recollement of triangulated categories The functor t, left adjoint to the inclusion of the acyclic complexes of projectives, associates to each complex its complete resolution.From [28, Theorem 4.6] one gets an equivalence: Composing this with the equivalence (2.2) yields a canonical equivalence Gorenstein projective modules.An A-module M is Gorenstein projective if it occurs as a syzygy module in an acyclic complex of projective A-modules.Thus, there is some ).We write GProj A for the full subcategory of Mod A consisting of the Gorenstein projective modules, and Gproj A for its subcategory of finitely generated modules.Both these are Frobenius categories, with projective and injective objects the projective modules in the corresponding categories; see for example [32,Proposition 7.2].The corresponding stable categories are denoted GProj A and Gproj A, respectively.The first part of the result below was proved by Buchweitz [18,Theorem 4.4.1]when A is Iwanaga-Gorenstein, but the same argument carries over to this context.
Moreover, these categories are compactly generated, and Gproj A identifies with the full subcategory of compact objects of GProj A.
Proof.In the dual setting of Gorenstein injectives the first assertion is [32,Proposition 7.2].In fact, we have an equivalence K(Inj A) ∼ − → K(Proj A) by [28,Theorem 5.6] and then the second assertion follows from [28,Theorem 6.5]. □ The Gorenstein projectivity of a module is inherited by its fibres.Without further restrictions, the converse need not hold.

Lemma 3.4. Let R be a regular ring and
Proof.Let X be an acyclic complex of projective A-modules in which M is a syzygy.Since R is regular and X is in K ac (Proj R), it is null-homotopic as an R-complex by Lemma 2.2.Thus X k(p) = X ⊗ R k(p) is also null-homotopic, and in particular acyclic.It consists of projective A k(p) -modules and M k(p) is a syzygy module in it, so the latter is Gorenstein projective. □ In view of Theorem 3.3 and the preceding result, one gets an analogue of Theorem 2.1 for Gorenstein projective modules.

Theorem 3.5. Let R be a regular ring and A a Gorenstein R-algebra. For Gorenstein projective R-modules X, Y we have in GProj
In the remainder of this section, we focus on a class of Gorenstein algebras for which it is easy to describe the Gorenstein projective modules.

Fibrewise self-injective algebras. The dualising bimodule of a finite projective Ralgebra
As noted in the proof of Proposition 3.1, when A is a Gorenstein R-algebra, ω A/R is perfect on either side, though not necessarily as a bimodule.Moreover, this property characterises the Gorenstein property of A when R is Gorenstein; see [28,Theorem 4.6].In the sequel, Gorenstein algebras for which the dualising bimodule is projective on either side play a prominent role.The result below characterises these algebras in terms of their fibres.An R-algebra A is fibrewise self-injective if it is a finite projective R-algebra such that the finite dimensional algebra A k(p) is self-injective for each p in supp R A. Our primary example is the group algebra of a finite group scheme over R.
For a module M we denote by add(M ) the full subcategory of finite direct sums of copies of M plus their direct summands.Lemma 3.6.Let R be a Gorenstein ring and A a finite projective R-algebra.The conditions below are equivalent.
(1) The R-algebra A is fibrewise self-injective; (2) add(ω A/R ) = add(A) and add(ω A op /R ) = add(A op ); (3) The dualising bimodule ω A/R is projective on the left and on the right.

If they hold the R-algebra A is Gorenstein, and one has an equivalences of categories
with inverse Hom A (ω A/R , −), and similarly for Proj A op .
Proof.In what follows we use the observation that for each p in Spec R one has an isomorphism of A k(p) -bimodules: Hence the A-bimodule ω A/R is projective on either side if and only if the A k(p) -bimodule ω A k(p) /k(p) is projective on either side for each p in Spec R; see Lemma 2.9.
(1)⇒( 2) It suffices to prove that A is in add(ω A/R ), equivalently that the map given by evaluation, is surjective.Since an R-module M is zero if and only if M k(p) is zero for each p in Spec R, it suffices to check the surjectivity of the map on the fibres, that is to say, the map is surjective.This holds as the k(p)-algebra A k(p) is self-injective.
(3)⇒(1) Given the isomorphism above, condition (3) yields that the dualising bimodule of It remains to verify the last part of the statement.The Gorenstein property follows from Proposition 3.1.The equivalence follows from the fact that ω A/R is projective on both sides and the isomorphism See also [28,Theorem 4.5].

□
In what follows dim R A denotes the Krull dimension of A viewed as an R-module.
Proposition 3.7.Let R be a Gorenstein ring and A a fibrewise self-injective R-algebra.
Proof.It suffices to verify the equality for A; the one for A op holds, by symmetry.Since the injective dimension of A is detected by the vanishing of Ext i A (−, A) on finitely generated A-modules, it suffices to verify that inj.dimAp A p = dim Rp A p for each p in supp R A.
We can replace R and A by their localisations at p so that R is local, and hence A is semi-local, and dim R A = dim R. For any simple A-module L one has This justifies the stated equalities.□ In the result below, the converse statement need not hold for general Gorenstein algebras, as can be seen by contemplating the case when R is a field.

Lemma 3.8. Let A be a Gorenstein R-algebra and M an A-module. When M is Gorenstein projective, it is Gorenstein projective also as an R-module. The converse holds when
A is fibrewise self-injective and either M is finitely generated or dim R A is finite.
Proof.Since A is finite projective as an R-module, any projective A-module is also projective as an R-module, and hence any acyclic complex of projective A-modules is an acyclic complex of projective R-modules.It follows that any Gorenstein projective A-module is Gorenstein projective also as an R-module.
Suppose that A is fibrewise self-injective and that M is Gorenstein projective as an Rmodule.We verify that Ext i A (M, −) = 0 for i ≥ 1 and on Proj A. Given this, if M is finitely generated one can apply [28,Lemma 6.3] to conclude that it is Gorenstein projective also as an A-module.We can draw the same conclusion from [19,Corollary 11.5.3] for a general M when we also know dim R A is finite, for then A is Iwanaga-Gorenstein, by Proposition 3.7.
As to the vanishing of Ext, any projective A-module is a direct summand of a free Amodule, and any free A-module is of the form A⊗ R F for some free R-module F .Therefore it suffices to verify that Thus it suffices to verify that Ext i A (M, Hom R (A, F )) = 0 for i ≥ 1.This follows from the adjunction isomorphism ) and the hypothesis that M is Gorenstein projective as an R-module.□ Remark 3.9.Let R be a regular ring of finite Krull dimension, G a finite group, and RG the group algebra.As noted earlier the R-algebra RG is fibrewise self-injective.Let M be an RG-module that is projective as an R-module.It follows from Lemma 3.8 and Theorem 3.5 that M is projective as an RG-module if and only if for each p in Spec R the k(p)G-module k(p) ⊗ R M is projective.This result appears to be in conflict with the example constructed by Benson and Goodearl in [9, Section 8].However there is no conflict because the claim in [9] that the module k ⊗ R M is kG-projective is not correct: the image of multiplication by 1 − g is strictly contained in the kernel of the multiplication by 1 − g.Indeed, in the example in question one has R := k[|t|], the ring of power series in the variable t, over a field k of characteristic two, and G = Z/2.Thus RG ∼ = R[x]/(x 2 ), with 1 + x representing the generator of G.The module M in question can be realised as the free R-module R[u] ⊕ R ⊕ R[s] with the action of x given by x • (f (u), r, g(u)) = (0, 0, f (u) + r) for f (u), g(u) in R[u], and r ∈ R. Evidently the element (1, 1, 0) is in the kernel of multiplication by x but not in its image.
The fibrewise test for projectivity stated above holds over any noetherian commutative ring R; see [5,Proposition 3.5].

Cocommutative Hopf algebras
Throughout this section R will be a regular commutative noetherian ring and A a finite cocommutative Hopf algebra over R; this includes the condition that A is projective as an R-module.Then K(Proj A) has a natural structure of a tensor-triangulated category, and one has an analogue of the fibrewise criterion from Section 2 that takes into account this additional structure.With some further assumption on the cohomology of A we are then able to stratify the tensor-triangulated category K(Proj A) via the action of the cohomology ring of A.
Tensor structure.Given A-modules X and Y , there is a natural diagonal A-module structure on X ⊗ R Y , obtained by restricting its A ⊗ R A-module structure along the coalgebra map ∆ : Lemma 4.1.Let P, Q be A-modules.If P is projective over A and Q is projective over R, then the A-module P ⊗ R Q, with the usual diagonal action, is projective.
Proof.This follows from the standard adjunction isomorphism The preceding result implies that − ⊗ R − induces a tensor product on K(Proj A).A) is tensor-triangulated, with product − ⊗ R − and unit the A-complex

Lemma 4.2. The triangulated category K(Proj
Proof.We have already seen that − ⊗ R − provides a tensor product on K(Proj A), and it remains to verify the assertion about the unit.Since A op is noetherian, one can assume that for any M in D b (mod A op ), its projective resolution p A op M is finitely generated in each degree and that (p M op ) i = 0 for i ≪ 0. This fact will be used multiple times in what follows.
The augmentation ε : For each X in K(Proj A) this induces the map and the desired result is that this map is an isomorphism.
Since K(Proj A) is compactly generated, and the functors involved preserves coproducts, it suffices to verify the claim when X is compact, that is to say, of the form Hom A (p A op M, A) for some M in mod A op ; see (2.

2). Consider the diagram Hom
where the vertical map is the natural one; it is an isomorphism because p A op A and p A op M are degreewise finitely generated.The map α is the obvious composition.It suffices to check that α is an isomorphism.One can verify that the map α is obtained from the map by applying Hom A (−, A).Since ε is a quasi-isomorphism so is the map above.Since the source and target consist of projective A op -modules, and are bounded to the right, they are semi-projective complexes over A op .Thus the map above is a homotopy equivalence in K(Proj A op ).Therefore applying Hom A (−, A) induces an isomorphism in K(Proj A).This is the desired result.□ Rigidity.The tensor-triangulated category K(Proj A), which is always compactly generated, is also rigid when R is regular.We recall briefly the notion of rigidity in tensortriangulated categories and refer to [25, A.2] for details.
Let T be a compactly generated tensor-triangulated category, with product ⊗ and unit 1.We assume that 1 is compact and that ⊗ preserves coproducts.Being a compactly generated tensor-triangulated category, T has an internal function object, Hom(−, −), defined by the property that for X, Y and Z in T.There is a natural map and X is rigid if this map is an isomorphism for all Y .Since 1 is compact, every rigid object is compact, and one says that T is rigidly-compactly generated when the converse holds: compact objects and rigid objects coincide.It is straightforward to verify that this property holds if and only if the conditions below hold: (1) the subcategory of compact objects is closed under ⊗; (2) (4.1) is an isomorphism when X, Y are compact.Back to the tensor-triangulated category K(Proj A) with product ⊗ R , where A is a cocommutative Hopf algebra over R. In this case, the internal function object can be described quite concretely as where j is the right adjoint to the inclusion of K(Proj A) into the homotopy category of flat A-modules; see [27,Proposition 2.4].Here is the pertinent result; one can also prove that K(Proj A) is not rigid when R is not regular.Proof.As noted above, it suffices to verify that for any X, Y in K c (Proj A), the complex X ⊗ R Y is also in K c (Proj A) and that (4.1) is an isomorphism.Given the identification (3.2) of compact objects in K(Proj A), this is tantamount to verifying that for M, N 3) is a tensor functor which fits -together with its adjoints -into the framework discussed in [2].

Fibrewise criterion.
Here is the analogue of the fibrewise criterion for detecting membership in localising subcategories, Theorem 2.1, in the presence of the tensor product.In the statement Loc ⊗ (Y ) denotes the tensor ideal localising subcategory generated by Y .
Theorem 4.5.Let R be a regular ring and A a finite cocommutative Hopf R-algebra.Let X, Y be objects in K(Proj A).The following conditions are equivalent. (1) Proof.The argument follows the same lines as that for Theorem 2.1, using in addition that K(Proj A) is a rigidly generated tensor-triangulated category by Lemma 4.3.Again it is straightforward to verify (1)⇒( 2), once one observes that the functor π from (2.3) respects the tensor products: For X, Y in K(Proj A), there is a natural isomorphism For the implication (2)⇒(1) we use the version of the local-to-global theorem (2.10) for tensor-triangulated categories in [13,Theorem 7.2].Then the task reduces to proving for X, Y in K(Proj A) that when (R, m, k) is a local ring and X k is in Loc ⊗ (Y k ), the complex π r π(X) is in Loc ⊗ (π r π(Y )).When reducing to the local case, one uses that for each p in Spec R the category K(Proj A p ) identifies with a localising tensor ideal of K(Proj A) via (2.8).
Whilst the functor π r need not respect tensor products, the following projection formula holds.For U in K(Proj A) and V in K(Proj A k ), there is a natural isomorphism One can verify this directly, but this is a general fact about tensor functors and their right adjoints between rigidly-compactly generated tensor-triangulated categories; see [2,Theorem 1.3].This formula and the fact that, up to direct summands, π is surjective on objects -see Lemma 2.4 -yield the desired result.□ Finite generation.Let R be a commutative noetherian ring and A a finite cocommutative Hopf algebra over R. Set S := Ext * A (R, R) .This is a graded-commutative R-algebra.Van der Kallen [39] has proved that S is finitely generated as an R-algebra; equivalently, that it is noetherian.This generalises earlier work of Friedlander and Suslin [22] that dealt with the case where R is a field.The ring S can be realised as the graded-ring of morphisms Since 1 is the unit of the tensor product on K(Proj A), it has a natural S-linear action on it.For each p in Spec R one has that A k(p) is a finite dimensional cocommutative Hopf algebra over k(p).Set S(p) Cohomological support.Let R be a commutative noetherian ring, A a finite cocommutative Hopf algebra over R, and S the cohomology ring introduced above.We write Spec S for the homogenous prime ideals in S. Following [11, § 5], the action of S on K(Proj A) gives rise to a notion of support for objects in K(Proj A).Namely for each q in Spec S there is a local cohomology functor functor Γ q , defined akin to (2.9).The support of an object X ∈ K(Proj A) is by definition the set and for any class of objects X we set Support and fibres.Theorem 4.5 yields a stratification -see the discussion further below -of K(Proj A) in terms of subsets of the space where to each X in K(Proj A) we associate the subset The task is to relate this to supp S X, viewed as a subset of Spec S.
To that end consider the structure map η : R → S, which induces a map η a : Spec S −→ Spec R .
The fibre of this map over p is which we identify with a subset of Spec S in the usual way.The functor

This induces the map of graded k(p)-algebras
Even in the best of cases, one does not expect this to be an isomorphism.Consider the induced map on spectra: The result below tracks the behavior of supports as we pass to the fibres.
Lemma 4.6.Let R be a regular ring and A a finite cocommutative Hopf R-algebra.Fix p in Spec R and let κ a p be the map in (4.3).For each X in K(Proj A) there is an equality Proof.We can reduce to the case where (R, m, k) is a local ring and p = m, the maximal ideal of R. Set S k := S ⊗ R k = S/mS.Via the map κ m : S k → S(m) the S(m)-action on K(Proj A k ) induces an S k -action.Applying [14,Corollary 7.8 (1)] to the identity functor on K(Proj A k ) yields an equality It thus suffices to work with the subset on the right.
Consider the adjunction (2.3).For any compact object C in K(Proj A) one has isomorphisms of graded S k -modules where the last isomorphism is by Lemma 2.4.Any compact object in K(Proj A k ) is a direct summand of πC for some compact object C in K(Proj A) by Lemma 2.4.By [11, Theorem 5.2], one can compute supp S k (X k ) from the support of the S k -modules Hom * K(A k ) (πC, X k ).This gives the first equality below The second one holds because X ⊗ R K represents X/ /mS; see [13,Lemma 2.6].□

Stratification.
Let S be a graded commutative noetherian ring and T a rigidly-compactly generated tensor-triangulated category.We denote by T c the full subcategory of compact objects.We say that T is S-linear to mean that S acts on T via a map of graded rings S −→ End * T (1) ; see [13, § 7].In the context above, one says that the tensor-triangulated category T is stratified by S if for each q in Spec S the category Γ q T, consisting of the q-local and qtorsion objects in T, is either zero or minimal, in that, it has no proper localising tensor ideals; see [13, § 7.2].When this holds one has a bijection When in addition the graded S-module Ext * T (X, X) is finitely generated for each X ∈ T c , the subset supp S X of Spec S is closed and, by [ Here is a slightly different perspective on the stratification property.
Lemma 4.7.Let T be an S-linear tensor-triangulated category as above.Then the tensortriangulated category T is stratified by S if and only if for any X, Y in T there are equivalences David John Benson et al.
Proof.We use the fact that for each q in Spec S we have by [11,Corollary 5.9].Evidently when the stated property holds the localising tensor ideal Γ q T of T is minimal for each q in Spec S, so T is stratified by S as a tensor-triangulated category.The converse is equally clear.□ We focus on the case for a cocommutative Hopf algebra A over R, and S its cohomology algebra.Here is one of the main results of our work.When it applies2 , one gets a classification of the localising tensor ideals of K(Proj A) and also the thick tensor ideals of its subcategory of compact objects, which identifies with D b (mod A).Proof.The main task is to verify that when X, Y are objects in K(Proj A) with supp S X ⊆ supp S Y , the complex X is in Loc ⊗ (Y ); see Lemma 4.7.Since κ a p is a homeomorphism for each p in Spec R, Lemma 4.6 yields an inclusion supp Since A k(p) is a finite dimensional cocommutative Hopf algebra over k(p), the triangulated category K(Proj A k(p) ) is stratified by the action of it cohomology algebra, S(p); this is the main result of [16].Thus the inclusion above implies This holds for each p in Spec R, so we can apply Theorem 4.5 to deduce that X is in Loc ⊗ (Y ) as desired.
It remains to observe that supp S 1 = Spec S, as follows, from example, from Lemma 4.6, for 1 k(p) is the unit of K(Proj A k(p) ) and its support is Spec S(p).□ In Section 5 we prove that group algebras of finite groups satisfy the hypotheses of the preceding result.Here is one more family of examples to which it applies.Example 4.9.Let R be a regular ring.Set A := ∧ R F , the exterior algebra on a finite free R-module F .We view it as a Z/2-graded Hopf algebra, with coalgebra structure defined by ∆(x) = x ⊗ 1 + 1 ⊗ x for x ∈ F .In this case Ext * A (R, R) is the symmetric algebra on Hom R (F, R).Given this it is clear that the hypotheses of Theorem 4.8 are satisfied in this case.
Here is another family of examples: Suppose k is field, R a k-algebra, and that the Hopf algebra A is of the form R ⊗ k A ′ where A ′ is a finite dimensional cocommutative Hopf algebra over k.
With this, it is easy to verify that A falls under the purview of Theorem 4.8.
Next we prepare to prove Theorem 1.2 stated in the introduction.
David John Benson et al. localising tensor ideal of K(Proj A) by Lemma 4.11.In particular, the support of GProj A is precisely Proj S. □

Finite groups
Let G be a finite group.The main result of this section is that for A = RG, the group algebra of G over any commutative noetherian ring R, the map (4.3) is a homeomorphism for all primes in the spectrum of R. As a consequence we get a stratification theorem when R is regular; see Theorem 5.6.In this case the cohomology algebra Ext * RG (R, R) is the group cohomology algebra.This is usually denoted H * (G, R), and we follow suit.

Cup products.
Let R be a commutative ring, not necessarily noetherian, and M an Rmodule, both viewed as G-modules with trivial action.The cup product makes H * (G, R) an R-algebra and H * (G, M ) a module over H * (G, R).These are defined as follows: Let P be a projective resolution of the trivial ZG-module Z and ∆ : P → P ⊗ Z P a diagonal approximation.Given classes x ∈ H * (G, R) and y ∈ H * (G, M ), represented by cocycles x ∈ Hom ZG (P, R) and y ∈ Hom ZG (P, M ) the cup product x ∪ y is represented by the class of the composition of maps Infinitesimal deformations of coefficients.Let R → R ′ be a surjective map of commutative rings whose kernel, say I, satisfies I 2 = 0; thus I is an R ′ -module.One thinks of R as an infinitesimal deformation of R ′ .The exact sequence where µ is the multiplication map.To compute the effect of the connecting homomorphism, we first lift x and y to cochains x and y in Hom ZG (P, R).Since d x = 0 = d y, the elements In what follows, we say that an abelian group M is p-local, for a prime number p, if the natural map M → M (p) is an isomorphism.Lemma 5.2.Let p be a prime dividing |G| and π : R → R ′ a map of p-local rings, with Ker(π) nilpotent.Then the map H * (G, π) has nilpotent kernel, and there exists an integer n such that for any element x ∈ H ⩾1 (G, R ′ ), the element x p n is in the image of the map H * (G, π).
Proof.Set I := Ker(π).Since this ideal is nilpotent, so is H * (G, I), under cup products.The claim about nilpotence is clear because, by the exact sequence in cohomology arising from (5.1), the kernel of H * (G, π) is the image of the map which respects cup products.
As to the second part of the statement, it suffices to consider the case where I 2 = 0. Let n be the largest integer such that p n divides |G|.Since |G| annihilates H ⩾1 (G, R ′ ), and the ring R ′ and hence also H * (G, R ′ ) is p-local, one gets that Thus if also p is odd, then x 2 = 0, since we are in the p-local situation.Thus we can suppose either |x| is even or p = 2.In either case, a repeated application of Lemma 5.1 yields δ(x i ) = ix i−1 δ(x) for each i ≥ 1.In particular δ(x p n ) = 0.It then follows from the exact sequence in group cohomology arising from (5.1) that x p n is in the image of the map H * (G, R) → H * (G, R ′ ).□ Modules with bounded torsion.Let M is an abelian group such that its torsionsubgroup, denoted tors(M ) is bounded; that is to say, there exists an integer n such that n • tors(M ) = 0. Fomin [21] proved that inclusion tors(M ) ⊆ M splits; see also [31,Corollary pp. 134].This result will be used below.
Lemma 5.3.Let p be a prime dividing |G| and M a p-local abelian group such that tors(M ) is bounded.For all integers s ≫ 0 the map Proof.Since M is p-local, the only torsion is p-torsion.Choose s ≫ 0 such that The equality on the left means that the sequence below, where the map M → p s M is given by m → p s m, is exact: This is split-exact, by Fomin's result recalled above, so the induced map where the one on the right is inclusion.By the choice of p s , the composition of the induced maps is zero in degrees ≥ 1.Since the map on the left is surjective, it follows that the one on the right is zero in degrees ≥ 1.Then the cohomology exact sequence arising from the exact sequence 0 −→ p s M −→ M −→ M/p s M −→ 0 yields the desired statement.□ Noetherian ring of coefficients.The result below was proved by Benson and Habegger [10] when R = Z; the argument given here is modeled on their proof.A general result, allowing non-trivial G-action on R, was proved by Lau [33, § 7].
Recall that a map of rings f : S → T containing a field of positive characteristic p is an F-isomorphism if ker(f ) is nilpotent, and for each t ∈ T there exists an n such that t p n is in the image of f .Theorem 5.4.Let G be a finite group and R a commutative noetherian ring.For each prime number p, the map Thus replacing R by R (p) we can assume R is p-local.
For any finitely generated R-module M , the (additive) torsion submodule tors(M ) is an R-submodule of M , and hence finitely generated as an R-module, as R is noetherian.It follows that tors(M ) is bounded, so Lemma 5.3  where δ 1 and δ 2 are the connecting maps.The choice of s ensures that π * and θ * are injective in positive degrees; see Lemma 5.3.Since π ⩾ 1 is injective and the kernel of ι 2 is nilpotent, by Lemma 5.2, so is the kernel of ι 1 in positive degrees.This map factors through the map H * (G, R) ⊗ R R/pR −→ H * (G, R/pR) so the latter is one-to-one up to nilpotence.
Fix x ∈ H ⩾1 (G, R/pR).Applying Lemma 5.2 to the map R/p s R → R/pR yields that for some n ≥ 1 the element x p n is in the image of ι 2 .So in the diagram above, we have δ 2 (x p n ) = 0. Since θ ⩾ 1 is injective we have δ 1 (x p n ) = 0.It follows from the exactness of the top row of the diagram that x p n is in the image of ι 1 .This implies that the map in the statement of the theorem is F-onto.□ Here is a consequence of the preceding theorem.Proof.The R-algebra H * (G, R) is finitely generated by a result of Evens [20] and Venkov [40].Thus the result follows from Theorem 4.8 and Corollary 5.5.□ An immediate, and standard, consequence of the stratification is a classification of thick tensor ideals in the bounded derived category (and also the small stable module category) of RG-modules.An analogous classification with Proj holds for Gproj.Corollary 5.7.Let G be a finite group and R be a regular local ring.There is a one-to-one correspondence between thick tensor ideal subcategories in D b (mod RG) and specialisation closed subsets in Spec(H * (G, R)), the spectrum of homogeneous prime ideals in H * (G, R).Remark 5.8.Theorem 5.6 yields classifications of thick and localising tensor ideals which have recent predecessors.Corollary 5.7 is established as a main theorem in Lau's work [33] where algebraic techniques similar to ours are used.Lau only works with finitely generated modules computing the Balmer spectrum of the category of perfect complexes on the Deligne Mumford stack [(Spec R)/G], that is the derived category of bounded complexes of G-equivariant finitely generated projective R-modules.In the case of a regular ring R this is precisely the derived category D b (mod RG).Though Lau stays in the realms of small categories, he allows for non-trivial action of G on the ring R so his result is more general in that setting.
In the series of papers [3,4], Barthel takes a very different approach to the stratification of representations of a finite group G over a ring R, by developing powerful homotopy theory machinery, with an eye towards applications to many other topological situations.

David John Benson et al.
In particular, in contrast to our fibrewise approach, Quillen's philosophy of reducing the question to elementary abelian groups enters into Barthel's homotopy theoretic methods.To make a direct comparison of our specific applications, Barthel's [4, Theorem C], which classifies localising tensor ideals in the (exact) category of R-linear representations of a finite group G whose underlying R-module is projective, can be seen as a direct consequence of Theorem 5.6, under the assumption that R is regular.We arrive to this application though through entirely different routes.Remark 5.9.For a group algebra RG there are other possible versions of a stable module category.For instance, one can endow the category of RG-modules with the exact structure given by those short exact sequences which are split exact when restricted to the trivial subgroup.Then the category mod RG of finitely presented RG-modules is a Frobenius category, and the corresponding stable category stmod RG is in fact a tensor triangulated category [15].For general R this category is different from Gproj RG, as can be seen from the discussion in [15, § 7] for R = Z. π

2 . 7 .Corollary 2 . 8 .
The derived category D(Mod A) identifies with a localising subcategory of K(Proj A) via the left adjoint of the canonical functor K(Proj A) → D(Mod A).Thus Theorem 2.1 implies the analogous description of localising subcategories of D(Mod A) from[23].Here is another noteworthy consequence.Let R be a regular ring, A a finite projective R-algebra, and M, N in D b (mod A).The following conditions are equivalent.

Lemma 4 . 3 .
If R is regular, the tensor-triangulated category K(Proj A) is rigid.

Theorem 4 . 8 .
Let R be a regular ring and A a finite cocommutative Hopf algebra over R. Let S denote the finitely generated R-algebra Ext * A (R, R).If the map κ a p in (4.3) is bijective for each p in Spec R, then the tensor-triangulated category K(Proj A) is stratified by the action of S, and supp S K(Proj A) = Spec S.

Lemma 5 . 1 .
via the cup product.The statement of the result below, and its proof, are a variation on[7, Lemma 4.3.3].In the context above, for x, y ∈ H * (G, R ′ ) one hasδ(x ∪ y) = δ(x) ∪ y + (−1) |x| x ∪ δ(y) .Proof.As in the proof of [7, Lemma 4.3.3],let P be the projective resolution of the trivial ZG-module Z, and ∆ : P → P ⊗ Z P a diagonal approximation.The exact sequence (5.1) induces the exact sequence of complexes 0 −→ Hom ZG (P, I) −→ Hom ZG (P, R) −→ Hom ZG (P, R ′ ) −→ 0 .Represent x and y by cocycles x and y on the right hand side of this sequence.Then x ∪ y is represented by the composite x ∪ y : P

d x and d
y lie in I.The element x ∪ y lifts to x ∪ y, andd( x ∪ y) = d x ∪ y + (−1) |x| x ∪ d y = d x ∪ y + (−1) |x| x ∪ d yThe second equality holds as d x and d y lie in I.This gives the stated equality.□ Proof.Set R (p) := Z (p) ⊗ Z R. As R/pR is p-local, the map R → R/pR factors through R (p) .As localisation is an exact functor, there are natural isomorphismsH * (G, R) ⊗ R R/pR ∼ = H * G, R (p) ⊗ R (p) R/pR H * (G, R/pR) ∼ = H * G, R (p) /pR (p) .

Corollary 5 . 5 .Theorem 5 . 6 .
Let G be a finite group and R a commutative noetherian ring.For any map of rings R → k with k a field of positive characteristic, the natural mapH * (G, R) ⊗ R k −→ H * (G, k) is an F -isomorphism,and hence the induced map on spectraSpec H * (G, k) −→ Spec (H * (G, R) ⊗ R k) is a homeomorphism.Proof.Let p be the characteristic of k.The map R → k factors through R/pR.Applying − ⊗ R k to the F-isomorphism in Theorem 5.4 yields the F-isomorphism H * (G, R) ⊗ R k −→ H * (G, R/pR) ⊗ R/pR k .It remains to observe that the right hand side is isomorphic to H * (G, k).□With G a finite group and R a regular ring, the tensor-triangulated category K(Proj RG) is stratified by the action of H * (G, R).

Suppose that R is regular and local of Krull dimension d with residue field k. Then
Let R be a commutative Gorenstein ring and A a finite projective Ralgebra.Then A is Gorenstein if and only the finite dimensional algebra A k(p) is Iwanaga-Gorenstein for each p in Spec R.Proof.By [28, Theorem 4.6], the R-algebra A is Gorenstein if and only if the A-bimodule Hom R (A, R) is perfect on both sides.Since the A-module Hom R (A, R) is finitely generated on both sides, it is perfect if and only if the David JohnBenson etal.Hence inj.dimA A ≤ dim R by [18, Lemma B.3.1].For the converse equality, with k the residue field of R, one has Ext d R (k, R) ∼ = k; see Lemma 2.2.Since A is a non-zero finite free R-module, one gets Ext applies.Choose an integer s large enough that the conclusion of op.cit.applies to the R-modules R and pR.Consider the commutative diagram of coefficients * (G, pR) H * G, R/p s+1 R H * (G, R/pR) H * G, pR/p s+1 R