STRATIFYING MODULAR REPRESENTATIONS OF FINITE GROUPS

DAVE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE

Abstract. We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.

Contents

1. Introduction 1 2. Strategy 3 3. Stratifications 6 4. Graded-injective dg modules 11 5. Graded polynomial algebras 18 6. Exterior algebras 19 7. A Koszul dg algebra 21 8. Elementary abelian groups 22 9. Finite groups 23 10. The main theorems 27 11. Applications 28 References 36

1. Introduction

Let G be a finite group and k a field of characteristic p, where p divides the order of G. Let Mod(kG) be the category of possibly infinite dimensional modules over the group algebra kG. In this article, a full subcategory C of Mod(kG) is said to be thick if it satisfies the following conditions.

• Any direct summand of a module in C is also in C.

1991 Mathematics Subject Classification. 20J06 (primary), 20C20, 13D45, 16E45, 18E30. Key words and phrases. local cohomology, local-global principle, modular representation the-

ory, stable module category, stratification, telescope conjecture.

• If 0 → M_{1} → M_{2} → M_{3} → 0 is an exact sequence of kG-modules, and two

of M_{1}, M_{2}, M_{3} are in C then so is the third.

A thick subcategory C is localising when, in addition, the following property holds:

• If {M_{α}} is a set of modules in C then ⨁_{α} M_{α} is in C. By a version of the Eilenberg swindle the first condition follows from the others.

There is a notion of support in this context, introduced by Benson, Carlson and Rickard [4]. It associates to each kG-module M a subset V_{G}(M) of the set Proj H^{∗}(G, k) of homogeneous prime ideals in the cohomology ring H^{∗}(G, k) other than the maximal ideal of positive degree elements.

Our main result is that Proj H^{∗}(G, k) stratifies Mod(kG), in the following sense.

Theorem 1.1. There is a natural one to one correspondence between non-zero localising subcategories of Mod(kG) that are closed under tensoring with simple kG-modules and subsets of Proj H^{∗}(G, k).

The localising subcategory corresponding to a subset V ⊆ Proj H^{∗}(G, k) is the full subcategory of modules M satisfying V_{G}(M) ⊆ V.

A more precise version of this result is given in Theorem 10.4. It is modelled on Neeman’s classification [25] of the localising subcategories of the unbounded derived category D(Mod R) of complexes of modules over a noetherian commutative ring R. Neeman’s work in turn was inspired by Hopkins’ classification [20] of the thick subcategories of the perfect complexes over R in terms of specialisation closed subsets of Spec R. The corresponding classification problem for the stable category stmod(kG) of finitely generated kG-modules was solved by Benson, Carlson and Rickard [5], at least in the case where k is algebraically closed. There is also a discussion in that paper as to why one demands that the subcategories are closed under tensor products with simple modules. For a p-group this condition is automatically satisfied, but for an arbitrary finite group there is a subvariety of Proj H^{∗}(G, k) called the nucleus, that encapsulates the obstruction to a classification of all localising subcategories, at least for the principal block.

Applications of Theorem 1.1 include a classification of smashing localisations of the stable category StMod(kG) of all kG-modules, a proof of the telescope conjecture in this setting, a classification of localising subcategories that are closed under products and duality, and a description of the left perpendicular category of a localising subcategory.

We also provide new proofs of the subgroup theorem, the tensor product theorem and the classification of thick subcategories of stmod(kG). The proofs of these results given in [4, 5] rely heavily on the use of cyclic shifted subgroups and an infinite dimensional version of Dade’s lemma, which play no role in our work.

As intermediate steps in the proof of the classification theorems for modules over kG, we establish analogous results on differential graded modules over polynomial rings, and over exterior algebras. These are of independent interest.

This paper is the third in a sequence devoted to supports and localising subcategories of triangulated categories. We have tried to make this paper as easy as possible to read without having to go through the first two papers [6, 7] in the series. In particular, some of the arguments in them have been repeated in this more restricted context for convenience.

Acknowledgments. The work of the second author was partly supported by NSF grant DMS 0602498. The first and second authors are grateful to the Humboldt Foundation for providing support which enabled them to make extended research visits to Paderborn to work with the third author. All three authors are grateful to the Mathematische Forschungsinstitut at Oberwolfach for support through a “Research in Pairs” visit. The authors also thank the referee for useful comments.

2. Strategy

The proof of Theorem 1.1 consists of a long chain of transitions from one category to another. In this section, we give an outline of the strategy.

The first step is to reduce to the stable module category StMod(kG). This category has the same objects as the module category Mod(kG), but the morphisms in StMod(kG) are given by quotienting out those morphisms in Mod(kG) that factor through a projective module. The category StMod(kG) is a triangulated category, in which the triangles come from the short exact sequences of kG-modules. See for example Theorem I.2.6 of Happel [19] or §5 of Carlson [11] for further details.

A thick subcategory of a triangulated category is a full triangulated subcategory that is closed under direct summands. A localising subcategory of a triangulated category is a full triangulated subcategory that is closed under direct sums. By an Eilenberg swindle, localising subcategories are also closed under direct summands, and hence thick.

Recall that, given kG-modules M and N, one considers M ⊗_{k} N as a kG-module with the diagonal G-action. The kG-modules M ⊗_{k} N and N ⊗_{k} M are isomorphic, and P ⊗_{k} N is projective for any projective kG-module P. Thus the tensor product on Mod(kG) passes down to a tensor product on StMod(kG). In either context, we say that a thick, or localising, subcategory C is tensor ideal if it satisfies the following condition.

• If M is in C and S is a simple kG-module then M ⊗_{k} S is in C. This condition is vacuous if G is a finite p-group since the only simple module is k with trivial G-action. Furthermore, every kG-module has a finite filtration whose subquotients are direct sums of simple modules (induced by the radical filtration of kG), so the condition above is equivalent to:

• If M is in C and N is any kG-module then M ⊗_{k} N is in C.

Proposition 2.1. Every non-zero tensor ideal localising subcategory of Mod(kG) contains the localising subcategory of projective modules.

The canonical functor from Mod(kG) to StMod(kG) induces a one to one correspondence between non-zero tensor ideal localising subcategories of Mod(kG) and tensor ideal localising subcategories of StMod(kG).

Proof. The tensor product of any module with kG is a direct sum of copies of kG. So if C is a non-zero tensor ideal localising subcategory of Mod(kG), it contains kG, and hence every projective kG-module. This proves the first statement of the proposition. The rest is now clear. □

There are some technical disadvantages to working in StMod(kG) that are solved by moving to a slightly larger triangulated category: K(Inj kG), the category whose objects are the complexes of injective kG-modules and whose morphisms are the homotopy classes of degree preserving maps of complexes. The tensor product of modules extends to complexes, and defines a tensor product on K(Inj kG). This category was investigated in detail by Benson and Krause [8]. Taking the Tate resolution tM = M ⊗_{k} tk of a kG-module M gives an equivalence of triangulated categories from the stable module category StMod(kG) to the full subcategory K_{ac}(Inj kG) of K(Inj kG) consisting of acyclic complexes. This equivalence preserves the tensor product. It suffices to check that tk ⊗_{k} tk ≅ tk, and an explicit isomorphism can be found in Section XII.4 of Cartan and Eilenberg [12].

The Verdier quotient of K(Inj kG) by K_{ac}(Inj kG) is the unbounded derived category D(Mod kG). There are left and right adjoints, forming a recollement

StMod(kG) ⟶^{∼} K_{ac}(Inj kG) \begin{matrix}Hom_{k}(tk,−) \\ ⟵ \\ ⟶ \\ ⟵ \\ −⊗_{k}tk \end{matrix} K(Inj kG) \begin{matrix}Hom_{k}(pk,−) \\ ⟵ \\ ⟶ \\ ⟵ \\ −⊗_{k}pk \end{matrix} D(Mod kG)

where pk denotes a projective resolution of k.

It is shown in [8] that the theory of supports for StMod(kG) developed in [4] extends in a natural way to K(Inj kG). Exactly one more prime ideal appears in the theory, namely the maximal ideal 𝔪 of positive degree elements in H^{∗}(G, k). We write Spec H^{∗}(G, k) for the set Proj H^{∗}(G, k) ∪ {𝔪} of all homogeneous prime ideals in H^{∗}(G, k). If X is an object in K(Inj kG) then there is associated to it a support V_{G}(X), which is a subset of Spec H^{∗}(G, k); see §3.

Proposition 2.2. For every tensor ideal localising subcategory C of StMod(kG) there are two tensor ideal localising subcategories of K(Inj kG). One is the image of C and the other is generated by this together with pk. This sets up a two to one correspondence between tensor ideal localising subcategories of K(Inj kG) and those of StMod(kG).

Proof. First we claim that for any non-zero object X in D(Mod kG), the tensor ideal localising subcategory generated by X is the whole of D(Mod kG).

Indeed, for any non-zero kG-module M the kG-module M ⊗_{k} kG is free and non-zero. Therefore, H^{∗}(X ⊗_{k} kG) is a non-zero direct sum of copies of kG, since it is isomorphic to H^{∗}(X) ⊗_{k} kG. This implies that in D(Mod kG) the complex X ⊗_{k} kG is isomorphic to a non-zero direct sum of copies of shifts of kG. Hence kG is in the tensor ideal localising subcategory generated by X. It remains to note that every object in D(Mod kG) is in the localising subcategory generated by kG.

Next observe that the full subcategory K_{ac}(Inj kG) of K(Inj kG) consisting of acyclic complexes is a tensor ideal localising subcategory. Thus the image in K(Inj kG) of every tensor ideal localising subcategory of StMod(kG) is a tensor ideal localising subcategory.

Now let D be a tensor ideal localising subcategory of K(Inj kG) that is not in the image of StMod(kG). Then D contains some object X which is not acyclic. The image of X in D(Mod kG) is non-zero, and hence generates D(Mod kG). Thus the tensor ideal containing X also contains pk. It follows that D is generated by pk and its intersection with the image of StMod(kG). □

A version of the main theorem for K(Inj kG) is as follows; see also Theorem 10.1.

Theorem 2.3. There is a natural one to one correspondence between tensor ideal localising subcategories C of K(Inj kG) and subsets of Spec H^{∗}(G, k).

The localising subcategory corresponding to a subset V of Spec H^{∗}(G, k) is the full subcategory of complexes X satisfying V_{G}(X) ⊆ V.

The proof of Theorems 1.1 and 2.3 occupies §§5–10. An outline of the proof is as follows. Using an appropriate version of the Quillen stratification theorem, it suffices to work with an elementary abelian p-group

E = ⟨g_{1}, …, g_{r}⟩.

Since k is of characteristic p the group algebra kE is isomorphic to the algebra

k[z_{1}, …, z_{r}]/(z_{1}^{p}, …, z_{r}^{p}),

where the (image of the element) z_{i} corresponds to g_{i} − 1.

We write A for the Koszul complex on kE with respect to z_{1}, …, z_{r}, and view it as a differential graded (dg) algebra. Thus, as a graded algebra A is the exterior algebra over kE on indeterminates y_{1}, …, y_{r}, with each y_{i} of degree −1, and the differential on A is determined by setting

d(z_{i}) = 0 and d(y_{i}) = z_{i}.

Observe that the elements z_{1}^{p−1}y_{1}, …, z_{r}^{p−1}y_{r} are cycles in A of degree −1. Let Λ be an exterior algebra over k on r generators ξ_{i} in degree −1, and view it as a dg algebra with zero differential. The map Λ → A given by

ξ_{i} ↦ z_{i}^{p−1}y_{i}

is a morphism of dg algebras and a quasi-isomorphism; see §7.

Let S be a graded polynomial ring k[x_{1}, …, x_{r}] where each variable x_{i} is of degree 2, and view it as a dg algebra with zero differential. One has an isomorphism of graded k-algebras S ≅ Ext_{Λ}^{∗}(k, k); see §6.

Let K(Inj A) and K(Inj Λ) denote the homotopy categories of graded-injective dg modules over A and Λ respectively; see §4. Our strategy is to establish first a classification of the localising subcategories of D(S), the derived category of dg Smodules, and then successively for K(Inj Λ), K(Inj A), K(Inj kE), K(Inj kG), and finally for StMod(kG) and Mod(kG). The following diagram provides an overview of the proof of Theorem 1.1.

D(S)^{§5} ⇝^{§6} K(Inj Λ) ⇝^{§7} K(Inj A) ⇝^{§8} K(Inj kE) ⇝^{§9} K(Inj kG) ⇝^{§10} Mod(kG).

Leitfaden

The passage from S to kE is modelled on the work of Avramov, Buchweitz, Iyengar, and Miller [1], where it is used to establish results on numerical invariants of complexes over commutative local rings by tracking them along a chain of categories as above. The focus here is on tracking structural information. A crucial idea in executing this passage from S to kG is that of a stratification of a tensor triangulated category, which allows one to focus on minimal localising subcategories. In §3 we describe the general theory of stratifications, and show that when a tensor triangulated category is stratified, one can classify its tensor ideal localising subcategories. This development is partly inspired by the work of Hovey, Palmieri, and Strickland [21].

In §11 we describe various applications, including a classification of smashing localisations of StMod(kG), and new proofs of the subgroup theorem, the tensor product theorem, and the classification of thick subcategories of stmod(kG).

3. Stratifications

In this section, we recall from [6, 7] those parts of the general theory of support varieties and stratifications that we wish to use in this paper.

Let T be a triangulated category admitting arbitrary coproducts. Usually, we denote Σ the shift on T. Given a subcategory C of T we write Loc(C) for the localising subcategory generated by C; this is the smallest localising subcategory of T containing C. The thick subcategory generated by C is denoted Thick(C). An object C of T is a generator if Loc(C) = T. The standing assumption in this article is that T is generated by a single compact object. Recall that an object C is compact if the functor Hom_{T}(C, −) commutes with coproducts.

For objects X, Y in T we write Hom_{T}^{∗}(X, Y) for the graded abelian group with degree n component Hom_{T}(X, Σ^{n}Y), and End_{T}^{∗}(X) for the graded ring Hom_{T}^{∗}(X, X).

Let R be a graded commutative noetherian ring; thus R is a ℤ-graded noetherian ring such that rs = (−1)^{|r||s|}sr for all homogeneous elements r, s in R. We say that the triangulated category T is R-linear, or that R acts on T, if we are given a homomorphism of graded rings ϕ: R → Z^{∗}T, to the graded centre of T. This means that for each object X there is a homomorphism of graded rings

ϕ_{X}: R → End_{T}^{∗}(X)

such that for each pair of objects X, Y the induced left and right actions of R on Hom_{T}^{∗}(X, Y) agree up to the usual sign; see [6, §4].

Let T be an R-linear triangulated category.

We write Spec R for the set of homogeneous prime ideals in R. For any ideal 𝔞 in R the subset {𝔭 ∈ Spec R | 𝔭 ⊇ 𝔞} is denoted V(𝔞). A subset V ⊆ Spec R is specialisation closed if 𝔭 ∈ V and 𝔮 ⊇ 𝔭 imply 𝔮 ∈ V. Given such a subset V, pick a compact generator C of T, and let T_{V} be the full subcategory

T_{V} = {X ∈ T | Hom_{T}^{∗}(C, X)_{𝔭} = 0 ∀ 𝔭 ∈ Spec R \ V}.

This is a localising subcategory of T. The following result is proved in [6, §4].

Proposition 3.1. The localising subcategory T_{V} depends only on V and not on the choice of compact generator C. Furthermore, there is a localisation functor L_{V}: T → T such that L_{V}X = 0 if and only if X ∈ T_{V}. □

This result and the theory of Bousfield localisation imply that there is an exact functor Γ_{V}: T → T and for each object X in T an exact localisation triangle

(3.2)  Γ_{V}X → X → L_{V}X → .

Proposition 3.1 has the following useful consequence.

Corollary 3.3. Let ϕ, ϕ′: R → Z^{∗}T be actions of R on T. If there exists a compact generator C for T for which the maps ϕ_{C}, ϕ′_{C}: R → End_{T}^{∗}(C) agree, then for any specialisation closed set V ⊆ Spec R the functors Γ_{V} and L_{V} defined through ϕ agree with those defined through ϕ′.

Proof. The definitions of Γ_{V} and L_{V} only depend on the action of R on Hom_{T}^{∗}(C, X), since C is a compact generator. The action factors through the map R → End_{T}^{∗}(C), which justifies the claim. □

In this work the principal source of an R-action on T is a tensor structure on it.

Tensor triangulated categories.

Let (T, ⊗, 𝟙) be a tensor triangulated category. By this we mean the following structure; see [6, §8] for details.

• T is a compactly generated triangulated category with coproducts.

• T is symmetric monoidal, with tensor product ⊗: T × T → T and unit 𝟙.

• The tensor product is exact in each variable and preserves coproducts.

• The unit 𝟙 is compact.

• Compact objects are strongly dualisable. In this article we make also the following assumption, for simplicity of exposition.

• T has a single compact generator. We write Loc^{⊗}(C) for the tensor ideal localising subcategory generated by a subcategory C of T. Observe that Loc(C) ⊆ Loc^{⊗}(C) and that the two categories coincide when the unit 𝟙 is a generator for T; this is the case in many of our contexts.

The stable category StMod(kG) of a finite group G is tensor triangulated, where the tensor product is M ⊗_{k} N with diagonal G-action and the unit is k. The homotopy category K(Inj kG) is also tensor triangulated with the same tensor product, but here the unit is ik, the injective resolution of k. In either case, the unit generates the category when G is a p-group; see [8] for details.

The ring End_{T}^{∗}(𝟙) is graded commutative and acts on T via homomorphisms

End_{T}^{∗}(𝟙) ⟶^{X⊗−} End_{T}^{∗}(X),

for each X in T. Thus any homomorphism of graded rings R → End_{T}^{∗}(𝟙) induces an R-action on T, and we call such an R-action canonical.

In the remainder of this section T denotes a tensor triangulated category and R a graded commutative noetherian ring that acts canonically on T.

For each specialisation closed subset V of Spec R there are natural isomorphisms

Γ_{V}X ≅ Γ_{V}𝟙 ⊗ X and L_{V}X ≅ L_{V}𝟙 ⊗ X.

Thus, the localisation triangle (3.2) is obtained by applying − ⊗ X to the triangle

Γ_{V}𝟙 → 𝟙 → L_{V}𝟙 → .

These are analogues of Rickard’s triangles [32] in StMod(kG); see [6, §§4,10].

Next we recall the construction of Koszul objects from [6, Definition 5.10]. Given a homogeneous element r of R and an object X in T, we write X//r for any object that appears in an exact triangle

X ⟶^{r} Σ^{|r|}X → X//r → .

Iterating this construction on a sequence of elements r = r_{1}, …, r_{n} one gets an object in T that is denoted X//r. Given an ideal 𝔞 in R, we write X//𝔞 for any Koszul object on a finite sequence of elements generating 𝔞.

While a Koszul object X//𝔞 depends on a choice of a generating sequence for 𝔞, the thick subcategory it generates does not. This follows from the second part of the following statement, where \sqrt{𝔞} denotes the radical of 𝔞 in R.

Lemma 3.4. Let 𝔞 be an ideal in R and X an object in T.

(1) X//𝔞 is in Thick(Γ_{V(𝔞)}X).

(2) X//𝔞 is in Thick(X//𝔟) for any ideal 𝔟 with \sqrt{𝔟} ⊆ \sqrt{𝔞}.

Proof. By construction it is clear that X//𝔞 is in Thick(X). Since the functor Γ_{V(𝔞)} is exact, this implies that Γ_{V(𝔞)}(X//𝔞) is in Thick(Γ_{V(𝔞)}X). It remains to note that Γ_{V(𝔞)}(X//𝔞) ≅ X//𝔞, for X//𝔞 is in T_{V(𝔞)} by [6, Corollary 5.11]. This proves (1).

The claim in (2) can be proved along the same lines as [21, Lemma 6.0.9]. □

The second part of the next result improves upon [6, Theorem 6.4].

Proposition 3.5. Let V be a specialisation closed subset of Spec R.

(1) If W ⊇ V is specialisation closed, then Γ_{V}X is in Loc^{⊗}(Γ_{W}X).

(2) Let C be a compact generator of T. For any decomposition V = ⋃_{i∈I} V(𝔞_{i}), where each 𝔞_{i} is an ideal in R, there are equalities

T_{V} = Loc({C//𝔞_{i} | i ∈ I}) = Loc({Γ_{V(𝔞_{i})}C | i ∈ I}).

Proof. (1) From [6, Proposition 6.1] one gets the first isomorphism below:

Γ_{V}X ≅ Γ_{V}(Γ_{W}X) ≅ Γ_{V}𝟙 ⊗ Γ_{W}X.

Thus Γ_{V}X is in Loc^{⊗}(Γ_{W}X).

(2) For each 𝔭 ∈ V there exists an i in I such that 𝔭 ∈ V(𝔞_{i}) holds, so that C//𝔭 is in Thick(C//𝔞_{i}), by Lemma 3.4(2). This fact and [6, Theorem 6.4] imply the last of the following inclusions:

Loc({C//𝔞_{i} | i ∈ I}) ⊆ Loc({Γ_{V(𝔞_{i})}C | i ∈ I}) ⊆ T_{V} ⊆ Loc({C//𝔞_{i} | i ∈ I}).

The first inclusion holds by Lemma 3.4(1). The second one holds as Γ_{V(𝔞_{i})}C is in T_{V} for each i, since T_{V(𝔞_{i})} ⊆ T_{V}. The inclusions above yield the desired result. □

Support.

Fix a point 𝔭 ∈ Spec R and let V and W be specialisation closed sets of primes such that V \ W = {𝔭}. It is shown in [6, Theorem 6.2] that the functor Γ_{𝔭} = Γ_{V}L_{W} is independent of choice of V and W with these properties, and coincides with L_{W}Γ_{V}. There are natural isomorphisms

Γ_{𝔭}X ≅ Γ_{𝔭}𝟙 ⊗ X.

Following [6, §5], we define the support of an object X in T to be

supp_{R} X = {𝔭 ∈ Spec R | Γ_{𝔭}X ≠ 0}.

If the R-module Hom_{T}^{∗}(C, X) is finitely generated for some compact generator C for T and 𝔞 is its annihilator ideal, then supp_{R} X equals the Zariski-closed subset V(𝔞); see [6, Theorem 5.5(1)].

The notion of support defined above is an analogue of the one for modular representations of a finite group G given in Benson, Carlson and Rickard [4]. Indeed, if T = StMod(kG) and 𝔭 is a non-maximal prime in H^{∗}(G, k), then Γ_{𝔭}𝟙 is the kappa module for 𝔭 defined in [4]; see [6, §10].

The following theorem is the “local-global principle”, studied in [7] for general triangulated categories.

Theorem 3.6. Let X be an object in T. Then

X ∈ Loc^{⊗}({Γ_{𝔭}X | 𝔭 ∈ supp_{R} X}).

Proof. It suffices to prove that 𝟙 is in the subcategory

C = Loc^{⊗}{Γ_{𝔭}𝟙 | 𝔭 ∈ Spec R}.

Indeed, one then gets the desired result by tensoring with X, keeping in mind that if 𝔭 ∉ supp_{R} X then Γ_{𝔭}𝟙 ⊗ X = Γ_{𝔭}X = 0.

Proposition 3.5(1) implies that the set

V = {𝔭 ∈ Spec R | Γ_{V(𝔭)}𝟙 ∈ C}.

is specialisation closed. We claim that V = Spec R. If not, then since the ring R is noetherian there exists a prime 𝔮 maximal in Spec R \ V. Setting Z = V(𝔮) \ {𝔮} one gets an exact triangle

Γ_{Z}𝟙 → Γ_{V(𝔮)}𝟙 → Γ_{𝔮}𝟙 → .

Since Z ⊆ V, it follows from Proposition 3.5(2) that

Γ_{Z}𝟙 ∈ T_{Z} ⊆ T_{V} = Loc{Γ_{V(𝔭)}C | 𝔭 ∈ V} ⊆ Loc^{⊗}{Γ_{V(𝔭)}𝟙 | 𝔭 ∈ V} ⊆ C,

where C denotes any compact generator of T. By definition, Γ_{𝔮}𝟙 is in C so it follows from the triangle above that Γ_{V(𝔮)}𝟙 is also in it. This contradicts the choice of 𝔮. Therefore V = Spec R holds. It remains to note that

𝟙 = Γ_{Spec R}𝟙 ∈ C

where the equality holds because Γ_{Spec R} is the identity on T, since T_{Spec R} = T. □

Stratification.

Let T be an R-linear tensor triangulated category. For each 𝔭 in Spec R, the full subcategory Γ_{𝔭}T is tensor ideal and localising. It consists of objects X in T with the property that the R-module Hom_{T}^{∗}(C, X) is 𝔭-local and 𝔭-torsion, where C is some compact generator for T; see [6, Corollary 4.10].

We say that T is stratified by R if Γ_{𝔭}T is either zero or minimal among tensor ideal localising subcategories for each 𝔭 in Spec R.

Given a localising subcategory C of T and a subset V of Spec R, we write

σ(C) = supp_{R} C = {𝔭 ∈ Spec R | Γ_{𝔭}C ≠ 0}
τ(V) = {X ∈ T | supp_{R} X ⊆ V}.

It follows from [6, §8] that the subcategory τ(V) is tensor ideal and localising, so these are maps

(3.7) \xymatrix{\{tensor ideal localising subcategories of T\} \ar[r]^{σ} & \{subsets of supp_{R} T\}. \ar[l]^{τ}}

The theorem below is the reason stratifications are relevant to this work.

Theorem 3.8. If T is stratified by R, then σ and τ are mutually inverse bijections between the tensor ideal localising subcategories of T and subsets of supp_{R} T.

Proof. It is clear that στ(V) = V for any subset V of supp_{R} T and that C ⊆ τσ(C) for any tensor ideal localising subcategory C. It remains to check that τσ(C) ⊆ C.

For each 𝔭 ∈ supp_{R} C, minimality of Γ_{𝔭}T implies the equality below

Γ_{𝔭}T = Γ_{𝔭}C ⊆ C

while the inclusion holds because Γ_{𝔭}X ≅ Γ_{𝔭}𝟙 ⊗ X. For any X in τσ(C) one has supp_{R} X ⊆ σ(C), so Theorem 3.6 and the inclusions above imply X ∈ C. □

The following characterisation of minimality is often useful.

Lemma 3.9. Let T be a tensor triangulated category and C a non-zero tensor ideal localising subcategory of T. The following statements are equivalent:

(a) The tensor ideal localising subcategory C is minimal.

(b) For all non-zero objects X and Y in C there exists an object Z in T such that Hom_{T}^{∗}(X ⊗ Z, Y) ≠ 0 holds.

(c) If C is a compact generator for T, then all non-zero objects X and Y in C satisfy Hom_{T}^{∗}(X ⊗ C, Y) ≠ 0.

Proof. (a) ⇒ (b): Let X be a non-zero object in C. Minimality of C implies that Loc^{⊗}(X) = C. Therefore, if there exists Y in C such that for all Z in T we have Hom_{T}^{∗}(X ⊗ Z, Y) = 0, then in particular Hom_{T}^{∗}(Y, Y) = 0 and hence Y = 0.

(b) ⇔ (c): This is clear, since Loc(C) = T for any generator C.

(c) ⇒ (a): Suppose that C is not minimal, so that it contains a non-zero proper tensor ideal localising subcategory, say C′. Let X be a non-zero object in C′. It follows from [27, Corollary 4.4.3] that there is a localisation functor with kernel Loc(X ⊗ C), so that for each object W in T there is a triangle W′ → W → W′′ → with W′ ∈ C′ and Hom_{T}^{∗}(X ⊗ C, W′′) = 0. Pick an object W in C but not in C′ and set Y = W′′. Since W′ and W are in C, so is Y and W is not in C′ one gets Y ≠ 0. Finally, we have Hom_{T}^{∗}(X ⊗ C, Y) = 0, a contradiction. □

We wish to transfer stratifications from one tensor triangulated category to another. In particular, we need to change the tensor product on a fixed triangulated category. It turns out to be inconvenient to have to keep track of the map from R into the graded centre of T. So we formulate the following principle.

Lemma 3.10. Let T be a triangulated category admitting two tensor triangulated structures with the same unit object, 𝟙, and let ϕ, ϕ′: R → Z^{∗}T be two actions. If 𝟙 generates T and the maps ϕ_{𝟙}, ϕ′_{𝟙} : R → End_{T}^{∗}(𝟙) agree, then T is stratified by R through ϕ if it is stratified by R through ϕ′.

Proof. It follows from Corollary 3.3 that Γ_{𝔭}X defined through ϕ and ϕ′ agree. This justifies the claim, since localising subcategories are tensor ideal. □

We require also a change of rings result for stratifications.

Lemma 3.11. Let T be an R-linear tensor triangulated category and ϕ: Q → R a homomorphism of rings. If T is stratified by the induced action of Q, then it is stratified by R.

Proof. Fix a prime 𝔭 in Spec R and set 𝔮 = ϕ^{−1}(𝔭). It is straightforward to verify that if an object X in T is 𝔭-torsion and 𝔭-local for the R-action, then it is 𝔮-torsion and 𝔮-local, for the Q-action. The claim is now obvious. □

4. Graded-injective dg modules

This section concerns a certain homotopy category of dg modules over a dg algebra. The development is based on the work of Avramov, Foxby, and Halperin [2], which is also our general reference for this material. The main results we prove here, Theorems 4.4 and 4.11, play a critical role in §§6–8 and are tailored for ready use in them; they are not the best one can do in that direction.

Hypothesis 4.1. Let A be a dg algebra over a field k with the following properties. (1) A^{i} = 0 for i > 0 and A is finite dimensional over k.

(2) A^{0} is a local ring with residue field k.

(3) A has a structure of a cocommutative dg Hopf k-algebra.

For the definition of a dg Hopf algebra see [16, §21]. The main consequence of the Hopf structure used in our work is that there is an isomorphism of dg A-modules

Hom_{k}(A, k) ≅ Σ^{d}A for some integer d.

This isomorphism can be verified as in [3, §3.1]. Note that A^{0} is also a cocommutative Hopf algebra and the inclusion A^{0} ⊆ A is compatible with the Hopf structure. In particular, A is free as a graded module over A^{0}; see [3, §3.3].

We write A^{♮} for the graded algebra underlying A. When M is a dg A-module, M^{♮} is the underlying graded A^{♮}-module. Following [2], we say that a dg A-module I is graded-injective if I^{♮} is an injective object in the category of graded A^{♮}-modules. Under Hypothesis 4.1 this condition is equivalent to saying that I^{♮} is a graded free A^{♮}-module. We write K(Inj A) for the homotopy category of graded-injective dg A-modules. The objects of this category are graded-injective dg A-modules and morphisms between such dg modules are identified if they are homotopic. The category K(Inj A) is given the usual structure of a triangulated category, where the exact triangles correspond to exact sequences of graded-injective dg modules. This is analogous to the case of the homotopy category of complexes over a ring, as described for instance in Verdier’s article in SGA4\frac{1}{2} [14]; see also [34].

K(Inj A) is compactly generated.

A dg A-module I is said to be semi-injective if it is graded-injective and the functor Hom_{K}(−, I), where K is the homotopy category of dg A-modules, takes quasi-isomorphisms to isomorphisms. Every dg module X admits a semi-injective resolution: a quasi-isomorphism X → I of dg A-modules with I semi-injective.

Lemma 4.2. Let A be a dg algebra satisfying Hypothesis 4.1. Each graded-injective dg A-module I has a family {I(n)}_{n∈ℤ} of dg submodules with the properties below.

(1) For each integer n, one has I(n − 1) ⊆ I(n).

(2) For each integer i, there exists an n_{i} with I(n_{i})^{⩾i} = I^{⩾i}.

(3) For each integer n, the dg module I(n) is semi-injective. When I^{i} = 0 for i ≪ 0 the dg A-module I is semi-injective.

Proof. Hypothesis 4.1 implies that the A^{♮}-module I^{♮} is free, so I^{♮} = ⨁_{j∈ℤ} A^{♮}U^{j}, where U^{j} is the set of basis elements in degree j. Set

I(n) = ⨁_{j≥−n} AU^{j}.

Since A^{i} = 0 for i > 0, by Hypothesis 4.1, it follows that d(U^{j}) ⊆ I(j − 1) holds for each j, and hence each I(n) is a dg A-submodule of I. Conditions (1) and (2) are immediate by construction, and it is evident that each I(n) is graded-injective with I(n)^{i} = 0 for i ≪ 0. It thus remains to verify the last claim of the lemma, for that would also imply that the I(n) are semi-injective.

Assume I^{i} = 0 for i ≪ 0 and let I(n) be as above. There are canonical surjections I → I/I(n) for each n and I ≅ lim_{n} I/I(n) as dg modules over A. It thus suffices to prove that each I/I(n) is semi-injective. So we may assume that I^{♮} = ⨁_{j∈ℤ} A^{♮}U^{j} with U^{j} = ∅ for |j| ≫ 0. By induction on the number of non-empty U^{j}, it suffices to consider the case when I is a free dg A-module. Thus I has the form A ⊗_{k} V with V a graded k-vector space with zero differential. The self-duality of A then implies that I is isomorphic to a shift of Hom_{k}(A, V) and hence semi-injective. □

The statement below is a special case of a result from [2]. We thank the authors for allowing us to reproduce the proof here.

Lemma 4.3. Let A be a dg algebra satisfying Hypothesis 4.1, and let 𝔪 be the dg ideal Ker(A → k). Each graded-injective dg A-module I is isomorphic in K(Inj A) to a graded-injective dg A-module J whose differential satisfies d(J) ⊆ 𝔪J.

Proof. We repeatedly use the fact that graded-injective dg A-modules are gradedfree. For any dg module F we write cone(F) for the mapping cone of the identity map F ⟶^{=} F. When F is graded-free, cone(F) has the following lifting property: If α: M → N is a surjective morphism of dg A-modules, then for any morphism β: cone(F) → N there is a morphism γ: cone(F) → M such that αγ = β. This is readily verified by a diagram chase. This observation is used below.

Since A/𝔪A is the field k, the complex of k-vector spaces I/𝔪I is isomorphic to H^{∗}(I/𝔪I) ⊕ cone(V), where V is a graded k-vector space with zero differential. Pick a surjective morphism F → V of dg A-modules with F a free dg A-module and F/𝔪F ≅ V; here V is viewed as a dg A-module via the morphism A → k. One thus gets a surjective morphism cone(F) → cone(V) of dg A-modules. The composed morphism cone(F) → cone(V) → I/𝔪I then lifts to a morphism γ: cone(F) → I of dg A-modules. It follows by construction that the map γ ⊗_{A} k is injective, which implies that γ itself is split-injective. Hence its cokernel, say, J, is graded-free. It is also not hard to verify that d(J) ⊆ 𝔪J holds. Finally, cone(F) ≅ 0 in K(Inj A) and hence I is homotopy equivalent to J. □

The result below is an analogue of [24, Proposition 2.3] for dg algebras. In it K^{c}(Inj A) denotes the full subcategory of compact objects in K(Inj A). As usual, D(A) stands for the derived category of dg modules over A. We write D^{f}(A) for the full subcategory whose objects are dg modules M such that the H^{∗}(A)-module H^{∗}(M) is finitely generated; equivalently, H^{∗}(M) is finite dimensional over k.

Theorem 4.4. Let A be a dg algebra satisfying Hypothesis 4.1 and ik a semiinjective resolution of the dg A-module k. The triangulated category K(Inj A) is compactly generated by ik and the canonical functor K(Inj A) → D(A) restricts to an equivalence

K^{c}(Inj A) ⟶^{∼} D^{f}(A).

In particular, Thick(ik) = K^{c}(Inj A) and Loc(ik) = K(Inj A).

Proof. The dg A-module k has a semi-injective resolution I with I^{j} = 0 for j < 0. One way to construct it is to start with a semi-free resolution F → k with F^{i} = 0 for i > 0 and apply Hom_{k}(−, k); note that Hom_{k}(F, k) is semi-injective, by adjunction. Semi-injective resolutions of ik are isomorphic in K(Inj A) so we may assume that ik is concentrated in non-negative degrees.

We have to prove that ik is compact in K(Inj A) and that it generates K(Inj A). Let K denote the homotopy category of dg A-modules with the usual triangulated structure. Identifying K(Inj A) with a full subcategory of K one gets an identification of Hom_{K(Inj A)}(−, −) with Hom_{K}(−, −).

We claim that for any graded-injective module I, the natural map k → ik induces an isomorphism Hom_{K}(ik, I) ≅ Hom_{K}(k, I). Indeed, the mapping cone of the canonical inclusion k → ik gives rise to an exact triangle k → ik → C → in K with C^{j} = 0 for j < 0. The desired result is that

Hom_{K}(C, I) = 0 = Hom_{K}(Σ^{−1}C, I).

By Lemma 4.2, there exists a semi-injective dg module J ⊆ I with J^{⩾−2} = I^{⩾−2}. One then has isomorphisms

0 ≅ Hom_{K}(Σ^{n}C, J) ≅ Hom_{K}(Σ^{n}C, I) for each n ≤ 0,

where the second one holds for degree reasons, and the first one because H^{∗}(C) = 0 and J is semi-injective. This proves the claim.

If {I_{α}} is a set of graded-injective dg modules over A then the claim yields the first and third isomorphisms below:

Hom_{K} (ik, ⨁_{α} I_{α}) ≅ Hom_{K} (k, ⨁_{α} I_{α})

≅ ⨁_{α} Hom_{K}(k, I_{α})

≅ ⨁_{α} Hom_{K}(ik, I_{α}).

The second isomorphism holds because the A^{♮}-module k is finitely generated. Therefore the dg module ik is compact in K(Inj A).

Suppose I is a graded-injective dg module with Hom_{K}^{∗}(ik, I) = 0. We wish to verify that I is homotopy equivalent to 0. We may assume that d(I) ⊆ 𝔪I, by Lemma 4.3, and hence that the differential on the dg module Hom_{A}(k, I) is zero. This explains the first isomorphism below:

Hom_{A}(k, I) ≅ H^{∗}(Hom_{A}(k, I)) ≅ Hom_{K}^{∗}(k, I) ≅ Hom_{K}^{∗}(ik, I) = 0.

The second isomorphism is standard and the third one holds by the claim established above. The equality is by hypothesis, and it follows that I = 0. □

The following test for equivalence of triangulated categories is implicit in [22, §4.2]. The proof uses a standard dévissage argument. Recall that T^{c} denotes the subcategory of compact objects in T.

Lemma 4.5. Let F: S → T be an exact functor between compactly generated triangulated categories with coproducts. If F preserves coproducts and restricts to an equivalence S^{c} ⟶^{∼} T^{c}, then F is an equivalence of categories.

In particular, if there exists a compact generator C of S such that F (C) is a compact generator of T and the induced map End_{S}^{∗}(C) → End_{T}^{∗}(F C) is an isomorphism, then F is an equivalence.

Proof. Fix a compact object D of S and let S_{D} be the full subcategory with objects X in S for which the induced map F_{D,X}: Hom_{S}(D, X) → Hom_{T}(F D, F X) is a bijection. This is a localising subcategory and contains S^{c} by the assumption on F. Therefore S_{D} = S. Given this, a similar argument shows that for any object Y in S the subcategory {X ∈ S | F_{X,Y} is bijective} equals S. Thus F is fully faithful. The essential image of F is a localising subcategory of T and contains a set of compact generators. We conclude that F is an equivalence. □

This is used in the proof of the following result.

Proposition 4.6. Let φ: A → B be a morphism of dg k-algebras where A and B satisfy Hypothesis 4.1. One has an exact functor Hom_{A}(B, −): K(Inj A) → K(Inj B) of triangulated categories. If φ is a quasi-isomorphism, then this functor is an equivalence and sends a semi-injective resolution of k over A to a semi-injective resolution of k over B.

Proof. When I is a graded-injective dg module over A, the adjunction isomorphism

Hom_{B}(−, Hom_{A}(B, I)) ≅ Hom_{A}(−, I)

implies that Hom_{A}(B, I) is a graded-injective over B. Since Hom_{A}(B, −) is additive it defines an exact functor at the level of homotopy categories. The isomorphism above also implies that when I preserves quasi-isomorphisms so does Hom_{A}(−, I). Hence, when I is semi-injective so is Hom_{A}(B, I).

Suppose φ is a quasi-isomorphism and let ik be a semi-injective resolution of k over A. The dg B-module Hom_{A}(B, ik) is then semi-injective and has cohomology k. It is hence a semi-injective resolution of k over B. In view of Theorem 4.4 one thus gets the following commutative diagram.

\xymatrix{K^{c}(Inj A) \ar[r]^{Hom_{A}(B,−)} \ar[d]^{∼} & K^{c}(Inj B) \ar[d]^{∼} \\ D^{f}(A) \ar[r]^{RHom_{A}(B,−)} & D^{f}(B)}

The functor RHom_{A}(B, −) is an equivalence, because φ is a quasi-isomorphism. Finally, since B^{♮} is finite dimensional over k, it is finite when viewed as a module over A^{♮} via φ, so the functor Hom_{A}(B, −) preserves coproducts. It remains to apply Lemma 4.5 to deduce that Hom_{A}(B, −) is an equivalence. □

In the remainder of this section we discuss stratification for homotopy categories of graded-injective dg modules.

K(Inj A) is tensor triangulated.

Given a dg Hopf algebra A and dg A-modules M and N, there is a dg A-module structure on M ⊗_{k} N, obtained by restricting the natural action of A ⊗_{k} A along the comultiplication A → A ⊗_{k} A. This is the diagonal action of A on M ⊗_{k} N.

Proposition 4.7. Let A be a dg k-algebra satisfying Hypothesis 4.1 and ik a semiinjective resolution of k over A. The tensor product ⊗_{k} with diagonal A-action endows K(Inj A) with a structure of a tensor triangulated category with unit ik.

Proof. Standard arguments show that for any dg A-module M the graded A^{♮}- modules underlying M ⊗_{k} A and A ⊗_{k} M are free; see [3, §3.1]. Graded-injective dg A-modules are graded-free as A^{♮}-modules and direct sums of graded-injectives are graded-injectives, since A^{♮} is noetherian. Therefore if I and J are gradedinjective dg A-modules, then so is the dg A-module I ⊗_{k} J. Hence one does get a tensor product on K(Inj A).

We claim that the morphism k → ik induces an isomorphism

I ≅ k ⊗_{k} I ⟶^{∼} ik ⊗_{k} I

in K(Inj A), so that ik is the unit of the tensor product on K(Inj A). Indeed, it is an isomorphism when I = ik because the morphism ik → ik ⊗_{k} ik is a quasiisomorphism and both ik and ik ⊗_{k} ik are semi-injective dg modules, the first by construction and the second by Lemma 4.2. Therefore the map above is an isomorphism for any I in Loc(ik), which is all of K(Inj A), by Theorem 4.4. For an alternative argument, see [8, Proposition 5.3].

The other requirements of a tensor triangulated structure are readily verified. □

Remark 4.8. Let A be a dg k-algebra satisfying Hypothesis 4.1.

The k-algebra Ext_{A}^{∗}(k, k) is graded commutative, as A is a Hopf k-algebra. Identifying Ext_{A}^{∗}(k, k) with End_{K}^{∗}(ik) there is a canonical action on K(Inj A) given by

Ext_{A}^{∗}(k, k) ⟶^{X⊗_{k}−} Hom_{K(Inj A)}^{∗}(X, X)

for X in K(Inj A). If the k-algebra Ext_{A}^{∗}(k, k) is finitely generated, and hence noetherian, the theory of localisation and support described in §3 applies to K(Inj A). Finite generation holds, for instance, when the differential on A is zero, by a result of Friedlander and Suslin [17].

Given a dg k-algebra A satisfying Hypothesis 4.1, the structure of K(Inj A) which is most relevant for us does not depend on the choice of a comultiplication on A. This is made precise in the following proposition which is an immediate consequence of Corollary 3.3 and Lemma 3.10.

Proposition 4.9. Let A be a dg k-algebra satisfying Hypothesis 4.1. The following structures of K(Inj A) do not depend on the choice of a comultiplication on A:

(1) the functors Γ_{V}, L_{V}, and Γ_{𝔭};

(2) the maps σ and τ defined in (3.7);

(3) stratification of K(Inj A) via the canonical action of Ext_{A}^{∗}(k, k). □

Transfer of stratification.

The next results deal with transfer of stratification between homotopy categories of graded injective dg modules of dg algebras.

Proposition 4.10. Let φ: A → B be a quasi-isomorphism of dg k-algebras where A, B satisfy Hypothesis 4.1. Then K(Inj A) is stratified by the canonical action of Ext_{A}^{∗}(k, k) if and only if K(Inj B) is stratified by the canonical action of Ext_{B}^{∗}(k, k).

Observe: φ is not required to commute with the comultiplications on A and B.

Proof. Proposition 4.6 yields that Hom_{A}(B, −): K(Inj A) → K(Inj B) is an equivalence sending a semi-injective resolution of k over A to a semi-injective resolution of k over B. Thus Hom_{A}(B, −) induces an isomorphism μ: Ext_{A}^{∗}(k, k) ⟶^{∼} Ext_{B}^{∗}(k, k). Observe that K(Inj B) admits two actions of Ext_{A}^{∗}(k, k). The first is the canonical action of Ext_{B}^{∗}(k, k) composed with μ and the other is the canonical action on K(Inj A) composed with the equivalence Hom_{A}(B, −).

Suppose K(Inj A) is stratified by the canonical action of Ext_{A}^{∗}(k, k). Then K(Inj B) is stratified by Ext_{A}^{∗}(k, k) via the second action because Hom_{A}(B, −) is an equivalence. It follows from Lemma 3.10 that K(Inj B) is stratified via the first action, and hence the canonical action of Ext_{B}^{∗}(k, k) stratifies K(Inj B) since μ is an isomorphism. This argument can be reversed by using a quasi-inverse of Hom_{A}(B, −). □

Theorem 4.11. Let A be a dg k-algebra satisfying Hypothesis 4.1. If K(Inj A) is stratified by the canonical action of Ext_{A}^{∗}(k, k), then K(Inj A^{0}) is stratified by the canonical action of Ext_{A^{0}}^{∗}(k, k).

Proof. We write K(A^{0}) and K(A) for K(Inj A^{0}) and K(Inj A), respectively. Hypothesis 4.1 implies that A^{♮}, the graded module underlying A, is free of finite rank over A^{0}. Therefore when I is a graded-injective (respectively, semi-injective) dg Amodule the adjunction isomorphism Hom_{A^{0}}(−, I) ≅ Hom_{A}(A ⊗_{A^{0}} −, I) yields that I is also graded-injective (respectively, semi-injective) as a dg A^{0}-module. In particular, the inclusion A^{0} → A gives rise to a restriction functor

(−)↓: K(A) → K(A^{0}).

Let ik be a semi-injective resolution of k over A. The dg A^{0}-module ik↓ is then a semi-injective resolution of k over A^{0}, so restriction induces a homomorphism of graded k-algebras Ext_{A}^{∗}(k, k) → Ext_{A^{0}}^{∗}(k, k). In view of Lemma 3.11 it thus suffices to prove that K(A^{0}) is stratified by the action of Ext_{A}^{∗}(k, k).

The restriction functor has a right adjoint

Hom_{A^{0}}(A, −): K(A^{0}) → K(A).

Fix a prime 𝔭 in Spec Ext_{A}^{∗}(k, k), and let X and Y be objects in Γ_{𝔭}K(A^{0}) with Hom_{K(A^{0})}^{∗}(X, Y) = 0. It suffices to prove that X or Y is zero; see Lemma 3.9.

As an A^{0}-module A^{♮} is free of finite rank therefore the dg A^{0}-module A↓ is in Thick(A^{0}), the thick subcategory of K(A^{0}) generated by A^{0}. Thus the dg A^{0}-module Hom_{A^{0}}(A, X)↓ is in Thick(X). This justifies the second isomorphism below:

Hom_{K(A)}^{∗}(Hom_{A^{0}}(A, X), Hom_{A^{0}}(A, Y)) ≅ Hom_{K(A^{0})}^{∗}(Hom_{A^{0}}(A, X)↓, Y) = 0.

The first one is adjunction. Observe that the adjunction isomorphism

Hom_{K(A^{0})}^{∗}(ik↓, X) ≅ Hom_{K(A)}^{∗}(ik, Hom_{A^{0}}(A, X))

is compatible with the action of Ext_{A}^{∗}(k, k). It thus follows that both Hom_{A^{0}}(A, X) and Hom_{A^{0}}(A, Y) are in Γ_{𝔭}K(A), and hence that one of them is zero, since K(A) is stratified by Ext_{A}^{∗}(k, k). Assume, without loss of generality, that Hom_{A^{0}}(A, X) = 0. The isomorphism above then yields Hom_{K(A^{0})}^{∗}(ik↓, X) = 0. By Theorem 4.4 the dg A^{0}-module ik↓ generates K(A^{0}), hence one gets that X = 0. □

5. Graded polynomial algebras

Let k be a field and S a graded polynomial k-algebra on finitely many indeterminates. We assume that each variable is of even degree if the characteristic of k is not 2, so that S is strictly commutative. The algebra S is viewed as a dg algebra with zero differential and we write D(S) for the derived category of dg S-modules. The objects of this category are dg S-modules and the morphisms are obtained by inverting the quasi-isomorphisms; see for example [22].

The main result of this section is a classification of the localising subcategories of D(S). This is a graded analogue of the theorem of Neeman [25]. In [7], we establish this result for general graded commutative noetherian rings. The argument presented here for S is simpler because we make use of the fact that the Koszul complex of a regular local ring is quasi-isomorphic to its residue field.

The category D(S) is tensor triangulated where the tensor product is ⊗_{S}^{L}, the derived tensor product, and the unit is S. Observe that S is compact and that it generates D(S). In particular, localising subcategories of D(S) are also tensor ideal. There is a canonical action of the ring S on D(S), where the homomorphism S → End_{D(S)}^{∗}(X) is given by multiplication. The theory of localisation and support described in §3 thus applies.

In this context, one has also the following useful identification.

Lemma 5.1. Let 𝔭 be a point in Spec S and set Z = {𝔮 ∈ Spec S | 𝔮 ⊈ 𝔭}. For each M in D(S) there is a natural isomorphism L_{Z}M ≅ M_{𝔭}.

Proof. The functor on D(S) defined by M ↦ M_{𝔭} is a localisation functor and has the same acyclic objects as L_{Z}, since H^{∗}(L_{Z}M) ≅ H^{∗}(M)_{𝔭}, by [6, Theorem 4.7]. This implies the desired result. □

Theorem 5.2. The category D(S) is stratified by the canonical S-action. In particular, the maps

\xymatrix{\{localising subcategories of D(S)\} \ar[r]^{σ} & \{subsets of Spec S\} \ar[l]^{τ}}

described in (3.7) are mutually inverse bijections.

Proof. By Theorem 3.8, the second part of the statement follows from the first since localising subcategories of D(S) are tensor ideal.

Fix a point 𝔭 ∈ Spec S. Since Hom_{D(S)}^{∗}(S, X) equals H^{∗}(X), the subcategory Γ_{𝔭}D(S) consists of dg modules whose cohomology is 𝔭-local and 𝔭-torsion. Let k(𝔭) be the homogeneous localisation of S/𝔭 at 𝔭; it is a graded field. Evidently k(𝔭) is in Γ_{𝔭}D(S), so for the desired result it suffices to prove that there is an equality

Loc(M) = Loc(k(𝔭)).

for any non-zero dg module M in Γ_{𝔭}D(S). We verify this first for M = Γ_{𝔭}S.

Let s = s_{1}, …, s_{d} be a sequence of elements in S whose images in S_{𝔭} are a minimal set of generators for the ideal 𝔭S_{𝔭} in the ring S_{𝔭}. Let V denote the set of primes in Spec S containing s and let Z = {𝔮 ∈ Spec S | 𝔮 ⊈ 𝔭}; these are specialisation closed subsets. Observe that V \ Z = {𝔭}, so one gets the first equality below:

Loc(Γ_{𝔭}S) = Loc(L_{Z}Γ_{V}S) = Loc(L_{Z}(S//s)) = Loc((S//s)_{𝔭}) = Loc(k(𝔭)).

The second equality is a consequence of Proposition 3.5(2), since the functor L_{Z} preserves arbitrary coproducts, while the third one follows from Lemma 5.1. The last equality holds because the dg module (S//s)_{𝔭} is quasi-isomorphic to k(𝔭) by the choice of s, since it is a regular sequence in S_{𝔭}; see [10, Corollary 1.6.14].

We now know that Loc(Γ_{𝔭}S) = Loc(k(𝔭)) holds. Applying the functor − ⊗_{S}^{L} M to it yields the second equality below:

Loc(M) = Loc(Γ_{𝔭}M) = Loc(k(𝔭) ⊗_{S}^{L} M).

The first one holds because M is in Γ_{𝔭}D(S). The action of S on k(𝔭) ⊗_{S}^{L} M factors through the graded field k(𝔭), as S is commutative. The equality above implies that H^{∗}(k(𝔭)⊗_{S}^{L}M) is non-zero, so one deduces that k(𝔭)⊗_{S}^{L}M is quasi-isomorphic to a direct sum of shifts of k(𝔭). Hence Loc(k(𝔭) ⊗_{S}^{L} M) = Loc(k(𝔭)). Combining this equality with the one above yields the desired result. □

6. Exterior algebras

Let k be a field and let Λ be the graded exterior algebra over k on indeterminates ξ_{1}, …, ξ_{c} of negative odd degree. We view Λ as a dg algebra with zero differential. The main result of this section is a classification of the localising subcategories of the homotopy category of graded-injective dg Λ-modules. It will be deduced from Theorem 5.2, via a dg Bernstein-Gelfand-Gelfand correspondence from [1, §7].

Definition 6.1. Let S be a graded polynomial algebra over k on indeterminates x_{1}, …, x_{c} with |x_{i}| = −|ξ_{i}| + 1 for each i. The k-algebra Λ ⊗_{k} S is graded commutative; view it as a dg algebra with zero differential. In it consider the element

δ = ∑_{i=1}^{c} ξ_{i} ⊗_{k} x_{i}

of degree 1. It is easy to verify that δ^{2} = 0 holds. In what follows J denotes the dg module over Λ ⊗_{k} S with underlying graded module and differential given by

J^{♮} = Hom_{k}(Λ, k) ⊗_{k} S and d(e) = δe.

Observe that since J is a dg module over Λ ⊗_{k} S, for each dg module M over Λ there is an induced structure of a dg S-module on Hom_{Λ}(J, M).

The result below builds on [1, Theorem 7.4]; see Remark 6.3 below. Recall that K(Inj Λ) is the homotopy category of graded-injective dg modules over Λ and D(S) is the derived category of dg modules over S.

Theorem 6.2. The dg (Λ ⊗_{k} S)-module J in Definition 6.1 has these properties: (1) There is a quasi-isomorphism k → J of dg Λ-modules and J is semi-injective.

(2) The map S → Hom_{Λ}(J, J) induced by right multiplication is a morphism of

dg k-algebras and a quasi-isomorphism. For any dg (Λ ⊗_{k} S)-module J satisfying these conditions the functor

Hom_{Λ}(J, −): K(Inj Λ) → D(S)

is an equivalence of triangulated categories.

Proof. The surjection Λ → k is a morphism of dg Λ-modules and hence so is its dual k → Hom_{k}(Λ, k). Combined with the map of k-vector spaces k → S one gets a morphism k → Hom_{k}(Λ, k)⊗_{k} S = J of dg Λ-modules. The module J is precisely the dg module X from [1, §7.3]. It thus follows from [1, §§7.6.2,7.6.5] that k → J is a quasi-isomorphism and that J is semi-injective as a dg Λ-module. Moreover, the map S → Hom_{Λ}(J, J) is a quasi-isomorphism by [1, Theorem 7.4]. The module J thus has the stated properties.

Let J be any dg (Λ ⊗_{k} S)-module satisfying conditions (1) and (2) in the statement of the theorem. It is easy to verify that the functor Hom_{Λ}(J, −) from K(Inj Λ) to D(S) is exact. We claim that J is compact and generates the triangulated category K(Inj Λ). One way to prove this is to note that Λ satisfies Hypothesis 4.1, with comultiplication defined by ξ_{i} ↦ ξ_{i} ⊗ 1 + 1 ⊗ ξ_{i}, so that Theorem 4.4 applies. Compactness yields that the functor Hom_{Λ}(J, −) preserves coproducts, since a quasi-isomorphism between dg S-modules is an isomorphism in D(S). Furthermore, as S is a compact generator for D(S), condition (2) provides the hypotheses required to apply Lemma 4.5, which yields that Hom_{Λ}(J, −) is an equivalence. □

Remark 6.3. Let F: D(S) → K(S) be a left adjoint to the canonical localisation functor K(S) → D(S), and − ⊗_{S}^{L} J the composite functor

D(S) ⟶^{F} K(S) ⟶^{−⊗_{S}J} K(Inj Λ).

The proof of Theorem 6.2 shows that − ⊗_{S}^{L} J is left adjoint to Hom_{Λ}(J, −), so restricting the equivalence in Theorem 6.2 to compact objects yields the equivalence D^{f}(Λ) ⟶^{∼} D^{f}(S) contained in [1, Theorem 7.4].

As in the proof of Theorem 6.2 consider Λ as dg Hopf k-algebra with

Δ(ξ_{i}) = ξ_{i} ⊗ 1 + 1 ⊗ ξ_{i}.

Observe that Λ satisfies Hypothesis 4.1, so the triangulated category K(Inj Λ) has a canonical tensor triangulated structure, by Proposition 4.7, and hence a canonical action of Ext_{Λ}^{∗}(k, k); see Remark 4.8. Moreover, the k-algebra Ext_{Λ}^{∗}(k, k) is noetherian, by Theorem 6.2(2).

Theorem 6.4. The tensor triangulated category K(Inj Λ) is stratified by the canonical action of Ext_{Λ}^{∗}(k, k). Therefore the maps

{ localising subcategories of K(Inj Λ)} [figure: σ; τ]{subsets of Spec Ext_{Λ}^{∗}(k, k)}

described in (3.7) are mutually inverse bijections.

Proof. Let S and J be as in Definition 6.1. In view of Theorem 6.2(1) we identify Ext_{Λ}^{∗}(k, k) and Hom_{K(Inj Λ)}(J, J). Theorem 5.2 applies to S and yields that D(S) is stratified by the canonical S-action on it, and hence also by an action of Ext_{Λ}^{∗}(k, k) obtained from the isomorphism S ≅ Ext_{Λ}^{∗}(k, k) in Theorem 6.2(2). Theorem 6.2 provides an equivalence of triangulated categories K(Inj Λ) → D(S) that sends J, the compact generator of K(Inj Λ), to S, the compact generator of D(S). Hence K(Inj Λ) is stratified by the canonical action of Ext_{Λ}^{∗}(k, k), by Lemma 3.10.

By Theorem 3.8 the stated bijection is a consequence of the stratification. □

7. A Koszul dg algebra

In this section we classify the localising subcategories of the homotopy category of graded-injective dg modules over the Koszul dg algebra of the group algebra of an elementary abelian group. To this end we establish an equivalence with the corresponding homotopy category over an exterior algebra, covered by Theorem 6.4.

Let E be an elementary abelian p-group of rank r and let k be a field of characteristic p. The group algebra kE is thus of the form

kE = k[z_{1}, …, z_{r}]/(z_{1}^{p}, …, z_{r}^{p}).

Let A be the dg algebra with A^{♮} an exterior algebra over kE on generators y_{1}, …, y_{r} each of degree −1, and differential defined by d(y_{i}) = z_{i}, so that d(z_{i}) = 0. This is the Koszul complex of A, viewed as a dg algebra; see [10, §1.6].

The group algebra kE is an example of a complete intersection, and the Koszul dg algebra of such a ring is formal, with cohomology an exterior algebra; see [10, §2.3]. This computation is straightforward for the case of kE and is given below.

Lemma 7.1. Let Λ be an exterior algebra over k on indeterminates ξ_{1}, …, ξ_{r} of degree −1, viewed as a dg algebra with zero differential. The morphism φ: Λ → A of dg k-algebras defined by φ(ξ_{i}) = z_{i}^{p−1}y_{i} is a quasi-isomorphism. In particular, the k-algebra Ext_{A}^{∗}(k, k) is a polynomial ring in r indeterminates of degree 2.

Proof. A routine calculation shows that φ is a morphism of dg k-algebras. What needs to be verified is that it is a quasi-isomorphism, and this is determined only by the structure of Λ and A as complexes of k-vector spaces.

Let Λ(i) be the exterior algebra on the variable ξ_{i} and A(i) the Koszul dg algebra over k[z_{i}]/(z_{i}^{p}), with exterior generator y_{i}. Observe that φ = φ(1) ⊗_{k} ⋯ ⊗_{k} φ(r) where φ(i): Λ(i) → A(i) is the morphism of complexes mapping ξ_{i} to z_{i}^{p−1}y_{i}. Each φ(i) is a quasi-isomorphism, by inspection, and hence so is φ.

Since φ is a quasi-isomorphism the k-algebras Ext_{Λ}^{∗}(k, k) and Ext_{A}^{∗}(k, k) are isomorphic. Theorem 6.2(2) implies Ext_{A}^{∗}(k, k) has the stated structure. □

We endow the dg algebra A with a comultiplication

Δ(z_{i}) = z_{i} ⊗ 1 + 1 ⊗ z_{i} and Δ(y_{i}) = y_{i} ⊗ 1 + 1 ⊗ y_{i}.

With this structure A satisfies Hypothesis 4.1. The homotopy category of gradedinjective dg A-modules K(Inj A) has thus a tensor triangulated structure and a canonical action of Ext_{A}^{∗}(k, k); see Proposition 4.7 and Remark 4.8.

Theorem 7.2. The tensor triangulated category K(Inj A) is stratified by the canonical action of Ext_{A}^{∗}(k, k). Therefore the maps

{ localising subcategories of K(Inj A)} [figure: σ; τ]{subsets of Spec Ext_{A}^{∗}(k, k)}

described in (3.7) are mutually inverse bijections.

Proof. Let Λ be the exterior algebra from Lemma 7.1. With comultiplication defined by ξ_{i} ↦ ξ_{i} ⊗ 1 + 1 ⊗ ξ_{i} this dg algebra satisfies Hypothesis 4.1. The desired result thus follows from Lemma 7.1, Theorem 6.4 and Proposition 4.10. □

8. Elementary abelian groups

We classify the localising subcategories of the homotopy category of injective modules over the group algebra of an elementary abelian group, by deducing it from the corresponding statement for its Koszul dg algebra in Theorem 7.2.

Let E be an elementary abelian p-group of rank r and k a field of characteristic p. In this section we view its group algebra kE, which is isomorphic to

k[z_{1}, …, z_{r}]/(z_{1}^{p}, …, z_{r}^{p})

as a dg algebra over k with zero differential. The diagonal map of E endows kE with a structure of a dg Hopf k-algebra with comultiplication

Δ(z_{i}) = z_{i} ⊗ 1 + z_{i} ⊗ z_{i} + 1 ⊗ z_{i}.

We call this the group Hopf structure on kE; it is evidently cocommutative. This Hopf structure is needed in §9 for the passage from E to a general finite group.

We also have to consider a different cocommutative dg Hopf algebra structure on kE, where the comultiplication is defined by

Δ(z_{i}) = z_{i} ⊗ 1 + 1 ⊗ z_{i}.

We call this the Lie Hopf structure on kE. It comes from regarding kE as the restricted universal enveloping algebra of an abelian p-restricted Lie algebra with zero p-power map. The Lie Hopf structure has the advantage that the inclusion of kE into its Koszul dg algebra is a map of dg Hopf algebras; this is exploited in the proof of Theorem 8.1 below.

The group Hopf structure and the Lie Hopf structure on kE both satisfy Hypothesis 4.1, and thus give rise to two tensor triangulated structures on K(Inj kE). Thus there are two actions of Ext_{kE}^{∗}(k, k) on K(Inj kE), which we call the group action and the Lie action, respectively; see Proposition 4.7 and Remark 4.8. The k-algebra Ext_{kE}^{∗}(k, k) is finitely generated and hence noetherian. Thus the theory of localisation and support described in §3 applies.

Theorem 8.1. The triangulated category K(Inj kE) is stratified by both the group action and the Lie action of Ext_{kE}^{∗}(k, k). The maps σ and τ described in (3.7) do not depend on the action used, and give mutually inverse bijections:

\xymatrix{\{localising subcategories of K(Inj kE)\} \ar[r]^{σ} & \{subsets of Spec Ext_{kE}^{∗}(k, k)\}. \ar[l]^{τ}}

Proof. Proposition 4.9 justifies the statement about independence of actions. Hence it suffices to consider the Lie Hopf structure on kE.

Let A be the Koszul dg algebra of kE with structure of dg Hopf k-algebra introduced in §7. Observe that kE = A^{0} and that the inclusion kE → A is compatible with the Lie Hopf structure; this is the reason for working with this Hopf structure on kE. It now remains to apply Theorems 7.2 and 4.11. □

9. Finite groups

In this section we prove that the tensor ideal localising subcategories of the homotopy category of complexes of injective modules over the group algebra of a finite group are stratified by the cohomology of the group. This is achieved by descending to the case of an elementary abelian group, covered by Theorem 8.1. The crucial new input required here is Quillen’s stratification theorem [28, 29].

Let G be a finite group and k a field of characteristic dividing the order of G. The group algebra kG is a Hopf algebra where the comultiplication is defined by Δ(g) = g ⊗ g for each g ∈ G. We consider the homotopy category of complexes of injective kG-modules K(Inj kG). The diagonal action of G induces a tensor triangulated structure with unit the injective resolution ik of k; see [8, §5] and also Proposition 4.7. As is customary, H^{∗}(G, k) denotes the cohomology of G, which is the k-algebra Ext_{kG}^{∗}(k, k). This algebra is finitely generated, and hence noetherian, by a result of Evens and Venkov [15, 33]; see also Golod [18]. It acts on K(Inj kG) via the canonical action, and the theory described in §3 applies. We write V_{G} for Spec H^{∗}(G, k) and V_{G}(X) for the support of any complex X in K(Inj kG).

For each subgroup H of G restriction yields a homomorphism of graded rings res_{G,H}: H^{∗}(G, k) → H^{∗}(H, k), and hence a map on Spec:

res_{G,H}^{∗}: V_{H} → V_{G}.

Part of Quillen’s theorem^{1} is that for each 𝔭 in V_{G} there exists an elementary abelian subgroup E of G such that 𝔭 is in the image of res_{G,E}^{∗}. We say that 𝔭 originates in such an E if there does not exist a proper subgroup E′ of E such that 𝔭 is in the image of res_{G,E′}^{∗}. In this language, [29, Theorem 10.2] reads:

Theorem 9.1. For each 𝔭 ∈ V_{G}, the pairs (E, 𝔮) where 𝔭 = res_{G,E}^{∗}(𝔮) and such that 𝔭 originates in E are all G-conjugate. This sets up a one to one correspondence between primes 𝔭 in V_{G} and G-conjugacy classes of such pairs (E, 𝔮). □

To make use of this we need a subgroup theorem for elementary abelian groups. As usual given a subgroup H ≤ G there are exact functors

(−)↓_{H}: K(Inj kG) → K(Inj kH) and (−)↑^{G}: K(Inj kH) → K(Inj kG)

defined by restriction and induction, − ⊗_{kH} kG, respectively. The functor (−)↑^{G} is faithful and left adjoint to (−)↓_{H}. For each object X in K(Inj kG) and object Y in K(Inj kH) there is a natural isomorphism

(9.2)  (X↓_{H} ⊗_{k} Y)↑^{G} ≅ X ⊗_{k} Y ↑^{G}

in K(Inj kG), where an element (x ⊗ y) ⊗ g is mapped to xg ⊗ (y ⊗ g).

Lemma 9.3. Let H be a subgroup of G. Fix 𝔭 ∈ V_{G} and set U = (res_{G,H}^{∗})^{−1}{𝔭}. (1) For any X ∈ K(Inj kG) there is an isomorphism (Γ_{𝔭}X)↓_{H} ≅ ⨁_{𝔮∈U} Γ_{𝔮}(X↓_{H}).

(2) For any Y ∈ K(Inj kH) there is an isomorphism Γ_{𝔭}(Y ↑^{G}) ≅ ⨁_{𝔮∈U}(Γ_{𝔮}Y)↑^{G}.

Proof. (1) For X = ik, this follows as in [4, Lemma 8.2 and Proposition 8.4 (iv)]. The general case is deduced as follows:

(Γ_{𝔭}X)↓_{H} ≅ (Γ_{𝔭}ik ⊗_{k} X)↓_{H} ≅ (Γ_{𝔭}ik)↓_{H} ⊗_{k} X↓_{H}

≅ ⨁_{𝔮∈U}Γ_{𝔮}ik ⊗_{k} X↓_{H} ≅ ⨁_{𝔮∈U}Γ_{𝔮}(X↓_{H}).

(2) Using part (1) and (9.2), one gets isomorphisms

Γ_{𝔭}(Y ↑^{G}) ≅ Γ_{𝔭}ik ⊗_{k} Y ↑^{G} ≅ ((Γ_{𝔭}ik)↓_{H} ⊗_{k} Y)↑^{G}
≅ ⨁_{𝔮∈U}(Γ_{𝔮}ik ⊗_{k} Y)↑^{G} ≅ ⨁_{𝔮∈U}(Γ_{𝔮}Y)↑^{G}.  □

Proposition 9.4. Let H be a subgroup of G. For any object X in K(Inj kG) and object Y in K(Inj kH) one has

V_{G}(X↓_{H}↑^{G}) ⊆ V_{G}(X) and V_{G}(Y ↑^{G}) = res_{G,H}^{∗} V_{H}(Y).

Proof. Let W be an injective resolution of the permutation module, k(G/H), on the cosets of H in G. We claim that the natural map k(G/H) → W of complexes of kG-modules induces an isomorphism X↓_{H}↑^{G} ≅ X ⊗_{k} W in K(Inj kG).

Indeed, this is easy to check when X = iM, the injective resolution of a finite dimensional kG-module M, for then both iM↓_{H}↑^{G} and iM ⊗_{k} W are injective resolutions of M ⊗_{k} k(G/H). It then remains to note that complexes of the form iM are a set of compactly generators for K(Inj kG); see [24, Lemmas 2.1 and 2.2].

1See the discussion following Proposition 11.2 of [29].

When 𝔭 ∈ V_{G}(X ⊗_{k} W) holds, since Γ_{𝔭}(X ⊗_{k} W) ≅ Γ_{𝔭}X ⊗_{k} W one gets Γ_{𝔭}X ≠ 0, that is to say, 𝔭 ∈ V_{G}(X), as desired.

By Lemma 9.3(2) the condition 𝔭 ∈ V_{G}(Y ↑^{G}) is equivalent to: there exists 𝔮 ∈ V_{H} such that res_{G,H}^{∗}(𝔮) = 𝔭 and Γ_{𝔮}Y ≠ 0. Hence V_{G}(Y ↑^{G}) = res_{G,H}^{∗} V_{H}(Y). □

The result below is an analogue of the subgroup theorem for an elementary abelian group E. Its proof is based on the classification of localising subcategories of K(Inj kE). The full version of the subgroup theorem, Theorem 11.2, will be a consequence of the classification theorem for K(Inj kG).

Theorem 9.5. Let E′ ≤ E be elementary abelian p-groups. For any object X in K(Inj kE) there is an equality

V_{E′}(X↓_{E′}) = (res_{E,E′}^{∗})^{−1}V_{E}(X).

Proof. Fix a prime 𝔮 in V_{E′} and set 𝔭 = res_{E,E′}^{∗}(𝔮). Proposition 9.4 yields an equality V_{E}(Γ_{𝔮}ik↑^{E}) = {𝔭} = V_{E}(Γ_{𝔭}ik). It thus follows from the classification of localising subcategories for kE in Theorem 8.1 that there is an equality

Loc(Γ_{𝔮}ik↑^{E}) = Loc(Γ_{𝔭}ik).

This implies that Γ_{𝔮}ik↑^{E} ⊗_{k} X ≠ 0 holds if and only if Γ_{𝔭}ik ⊗_{k} X ≠ 0. The desired result follows from the chain of implications:

Γ_{𝔮}(X↓_{E′}) ≠ 0 ⟺ Γ_{𝔮}ik ⊗_{k} X↓_{E′} ≠ 0 ⟺
Γ_{𝔮}ik↑^{E} ⊗_{k} X ≠ 0 ⟺ Γ_{𝔭}ik ⊗_{k} X ≠ 0 ⟺ Γ_{𝔭}X ≠ 0.

The second implication follows from (9.2) and the fact that (−)↑^{E} is faithful. □

Next we formulate a version of Chouinard’s theorem for K(Inj kG). Recall that for any ring A, a complex P of projective A-modules is semi-projective if the functor Hom_{K}(P, −), where K is the homotopy category of complexes of A-modules, takes quasi-isomorphisms to isomorphisms.

Proposition 9.6. Let G be a finite group and 𝔪 the ideal of positive degree elements of H^{∗}(G, k). The following statements hold for each X in K(Inj kG).

(1) V_{G}(X) ⊆ Spec H^{∗}(G, k) \ {𝔪} if and only if X is acyclic.

(2) V_{G}(X) ⊆ {𝔪} if and only if X is semi-projective.

(3) X = 0 if and only if X↓_{E} = 0 for every elementary abelian subgroup E ≤ G.

Proof. We write pk for a projective resolution and tk for a Tate resolution of the kG-module k. These fit into an exact triangle pk → ik → tk → which is the localisation triangle (3.2) in K(Inj kG) for ik with respect to the closed subset {𝔪}. In particular Γ_{𝔪}ik = pk, so that Γ_{𝔪}X ≅ pk ⊗_{k} X.

Claim. The induced map pk ⊗_{k} X → X is a semi-projective resolution of X.

Indeed, since pk is a complex of projectives, so is pk ⊗_{k} X; semi-projectivity follows from the isomorphism Hom_{K}(pk ⊗_{k} X, −) ≅ Hom_{K}(pk, Hom_{k}(X, −)).

Statements (1) and (2) are immediate from the preceding claim.

(3) Suppose X is non-zero in K(Inj kG). In view of the localisation triangle pk ⊗_{k} X → X → tk ⊗_{k} X → we may assume that pk ⊗_{k} X ≠ 0 or tk ⊗_{k} X ≠ 0.

Assume that pk ⊗_{k} X is non-zero; equivalently, that it is not acyclic, by the preceding claim. Observe that for each subgroup H ≤ G the restriction functor (−)↓_{H} sends semi-projectives to semi-projectives. Therefore (pk ⊗_{k} X)↓_{H}, and hence X↓_{H}, is non-zero.

Now assume that tk⊗_{k} X is non-zero. Chouinard’s theorem [13] applies, because one can identify the subcategory of acyclic complexes in K(Inj kG) with StMod(kG), and this identification is compatible with restrictions. This yields an elementary abelian subgroup E ≤ G with (tk ⊗_{k} X)↓_{E}, and hence also X↓_{E}, non-zero. □

The following result is a culmination of the development in §§4–9. Its applications are deferred to ensuing sections.

Theorem 9.7. Let G be a finite group. The triangulated category K(Inj kG) is stratified by the canonical action of the cohomology algebra H^{∗}(G, k).

Proof. For any subgroup H ≤ G we abbreviate K(Inj kH) to K(kH). We have to prove that for each 𝔭 ∈ V_{G}, the subcategory Γ_{𝔭}K(kG) is minimal among tensor ideal localising subcategories of K(kG).

Let X be a non-zero object in Γ_{𝔭}K(kG). Proposition 9.6 provides an elementary abelian subgroup E_{0} of G such that X↓_{E_{0}} is non-zero. Choose a prime 𝔮_{0} in V_{E_{0}}(X↓_{E_{0}}). Using Proposition 9.4 one thus obtains

res_{G,E_{0}}^{∗}(𝔮_{0}) ∈ V_{G}(X↓_{E_{0}}↑^{G}) ⊆ V_{G}(X) = {𝔭}.

Hence res_{G,E_{0}}^{∗}(𝔮_{0}) = 𝔭, so that E_{0} ≥ E and 𝔮_{0} = res_{E_{0},E}^{∗}(𝔮) for some pair (E, 𝔮) corresponding to 𝔭 as in Theorem 9.1. Thus 𝔮 ∈ V_{E}(X↓_{E}), by Theorem 9.5.

By Theorem 9.1 all pairs (E, 𝔮) where 𝔭 originates in E are conjugate, hence if we choose one, then each non-zero X ∈ Γ_{𝔭}K(kG) has Γ_{𝔮}X↓_{E} ≠ 0.

Now let Y be another non-zero object in Γ_{𝔭}K(kG) and set Z to be the injective resolution of k(G/E), the permutation module. As noted in the proof of Proposition 9.4, the complexes X↓_{E}↑^{G} and X ⊗_{k} Z are isomorphic in K(Inj kG). This gives the first isomorphism below:

Hom_{K(kG)}^{∗}(X ⊗_{k} Z, Y) ≅ Hom_{K(kG)}^{∗}(X↓_{E}↑^{G}, Y) ≅ Hom_{K(kE)}^{∗}(X↓_{E}, Y ↓_{E}).

The second isomorphism is by Frobenius reciprocity. Lemma 9.3 implies that Γ_{𝔮}X↓_{E} and Γ_{𝔮}Y ↓_{E} are non-zero direct summands of X↓_{E} and Y ↓_{E}, respectively. It follows from Theorem 8.1 and the isomorphism above that Hom_{K(kG)}^{∗}(X ⊗_{k} Z, Y) is non-zero, so Γ_{𝔭}K(kG) is minimal, by Lemma 3.9. □

10. The main theorems

Let k be a field of characteristic p and G be a finite group, where p divides the order of G. The following result implies Theorem 2.3.

Theorem 10.1. The tensor triangulated category K(Inj kG) is stratified by the canonical action of H^{∗}(G, k). The maps

\{ \begin{matrix}tensor ideal localising \\ subcategories of K(Inj kG) \end{matrix}\} [figure: σ; τ]\{subsets of Spec H^{∗}(G, k)\}

described in (3.7) are mutually inverse bijections.

Proof. This follows from Theorems 3.8 and 9.7. Note that the support of K(Inj kG) equals Spec H^{∗}(G, k) since V_{G}(k) = Spec H^{∗}(G, k). □

Next we prove our main result about the stable module category StMod(kG). We identify it with K_{ac}(Inj kG), the full subcategory of K(Inj kG) consisting of acyclic complexes. Observe that K_{ac}(Inj kG) is the tensor ideal localising subcategory of K(Inj kG) corresponding to the subset Proj H^{∗}(G, k) of non-maximal primes; see Proposition 9.6. This provides one way of classifying all tensor ideal localising subcategories of StMod(kG) in terms of subsets of Proj H^{∗}(G, k); see Proposition 2.2.

Now we define an action of the cohomology algebra H^{∗}(G, k) on StMod(kG). The equivalence K_{ac}(Inj kG) ⟶^{∼} StMod(kG) sends a complex X to its cycles Z^{0}X in degree zero. Restriction to the category of acyclic complexes induces therefore the following H^{∗}(G, k)-action

H^{∗}(G, k) → Z^{∗}K(Inj kG) ⟶^{res} Z^{∗}K_{ac}(Inj kG) ⟶^{∼} Z^{∗} StMod(kG),

where the first map is given by the canonical action of H^{∗}(G, k) on K(Inj kG). For each kG-module M, this action induces the map

(10.2) H^{∗}(G, k) ↪ Êxt_{kG}^{∗}(k, k) = End_{StMod(kG)}^{∗}(k) ⟶^{M⊗_{k}−} End_{StMod(kG)}^{∗}(M).

Thus we are in the setting of §3. The support of any module M in StMod(kG) is denoted V_{G}(M); it is a subset of Spec H^{∗}(G, k). By construction, this coincides with the support of a Tate resolution of M in K(Inj kG). Moreover, V_{G}(M) is the support defined in [4]. Indeed if 𝔭 is a non-maximal prime 𝔭 in H^{∗}(G, k), then Γ_{𝔭}k equals the kappa module for 𝔭 defined in [4]; see [6, §10].

Theorem 10.3. The tensor triangulated category StMod(kG) is stratified by the action of H^{∗}(G, k) given by (10.2). The maps

\{ \begin{matrix}tensor ideal localising \\ subcategories of StMod(kG) \end{matrix}\} [figure: σ; τ]\{subsets of Proj H^{∗}(G, k)\}

described in (3.7) are mutually inverse bijections.

Proof. Using the identification of StMod(kG) with K_{ac}(Inj kG), the assertion follows from Theorem 10.1 and Proposition 2.2. □

The result below implies Theorem 1.1 from the introduction.

Theorem 10.4. The maps

\{ \begin{matrix}non-zero tensor ideal localising \\ subcategories of Mod kG \end{matrix}\} [figure: σ; τ]\{subsets of Proj H^{∗}(G, k)\}

given by the obvious analogue of (3.7) are mutually inverse bijections.

Proof. This follows from Theorem 10.3 and Proposition 2.1. □

In [8], it was proved that if G is a finite p-group then there is an equivalence of triangulated categories

K(Inj kG) ⟶^{∼} D(C^{∗}(BG; k))

where C^{∗}(BG; k) denotes the dg algebra of cochains on the classifying space BG of G. This gives D(C^{∗}(BG; k)) the structure of a tensor triangulated category. Composing with the canonical map one gets an action

(10.5)  H^{∗}(G, k) → Z^{∗}K(Inj kG) ⟶^{∼} Z^{∗}D(C^{∗}(BG; k)).

Theorem 10.6. If G is a finite p-group then the tensor triangulated category D(C^{∗}(BG; k)) is stratified by the action of H^{∗}(G, k) given by (10.5). The maps

{localising subcategories of D(C^{∗}(BG; k))} [figure: σ; τ]{subsets of Spec H^{∗}(G, k)}

described in (3.7) are mutually inverse bijections.

Proof. This follows from Theorem 10.1 and the observation that every localising subcategory is tensor ideal. □

11. Applications

In this section we deduce the principal theorems of Benson, Carlson, and Rickard [4, 5] from Theorem 10.1 without using shifted subgroups, any form of Dade’s lemma, or algebraic closure of the field. Then we make various other deductions from the main theorem. We classify localising subcategories closed under products, and show that these are the same as those closed under Brown-Comenetz duality. We classify the smashing subcategories, and show that the telescope conjecture holds for StMod(kG) and K(Inj kG). Finally, we find the left perpendicular categories to localising subcategories. Note that similar applications can be formulated for dg modules over graded polynomial algebras and graded exterior algebras; this is left to the interested reader; see [7].

Throughout this section, we abbreviate Hom_{K(Inj kG)}(−, −) to Hom_{K(kG)}(−, −).

The tensor product theorem.

Part (2) of the result below was proved by Benson, Carlson, and Rickard [4] under the additional hypothesis that k is algebraically closed.

Theorem 11.1. Let G be a finite group and k be a field of characteristic p. (1) If X, Y are objects in K(Inj kG), then V_{G}(X ⊗_{k} Y) = V_{G}(X) ∩ V_{G}(Y).

(2) If M, N are objects in StMod(kG), then V_{G}(M ⊗_{k} N) = V_{G}(M) ∩ V_{G}(N).

Proof. Part (2) follows from (1) via the identification of StMod kG with K_{ac}(Inj kG).

(1) One has an isomorphism Γ_{𝔭}(X⊗_{k}Y) ≅ Γ_{𝔭}X⊗_{k}Γ_{𝔭}Y, which yields an inclusion V_{G}(X ⊗_{k} Y) ⊆ V_{G}(X) ∩ V_{G}(Y). Conversely, if 𝔭 ∈ V_{G}(X) ∩ V_{G}(Y) then Γ_{𝔭}X ≠ 0 and Γ_{𝔭}Y ≠ 0. Theorem 10.1 implies that Γ_{𝔭}ik is in Loc^{⊗}(Γ_{𝔭}X), hence that Γ_{𝔭}Y is in Loc^{⊗}(Γ_{𝔭}(X ⊗_{k} Y)). Since Γ_{𝔭}Y ≠ 0 it follows that Γ_{𝔭}(X ⊗_{k} Y) ≠ 0. □

The subgroup theorem.

The theorem below strengthens Theorem 9.5 from elementary abelian p-groups to all finite groups. For StMod(kG) and k algebraically closed it was proved in [4].

Theorem 11.2. Let H ≤ G be a subgroup. For any complex X in K(Inj kG) there is an equality

V_{H}(X↓_{H}) = (res_{G,H}^{∗})^{−1}V_{G}(X).

The analogous statement where K(Inj kG) is replaced by StMod(kG) also holds.

Proof. The argument for K(Inj kG) is the same as the proof of Theorem 9.5 except that one uses Theorem 10.1 instead of Theorem 8.1. The statement about StMod(kG) is deduced by identifying it with the acyclic complexes in K(Inj kG). □

The thick subcategory theorem.

Recall from [24, Proposition 2.3] that the compact objects in K(Inj kG) are the semi-injective resolutions of bounded complexes of finitely generated kG-modules, and that the canonical functor K(Inj kG) → D(Mod kG) induces an equivalence of triangulated categories K^{c}(Inj kG) ⟶^{∼} D^{b}(mod kG); we view this as an identification.

Lemma 11.3. Let C be a direct sum of the simple kG-modules. For any complex X in D^{b}(mod kG) one has V_{G}(X) = V(𝔞) where 𝔞 is the annihilator of the H^{∗}(G, k)- module H^{∗}(G, C ⊗_{k} X).

Proof. This is a restatement of [6, Theorem 5.5 (1)] in this context. □

In view of the equivalence of tensor triangulated categories

D^{b}(mod kG)/D^{b}(proj kG) ⟶^{∼} stmod(kG)

given by [31, Theorem 2.1], the following theorem generalises, with a new proof, a classification of the tensor ideal thick subcategories of stmod(kG) from [5].

Theorem 11.4. There is a one to one correspondence between tensor ideal thick subcategories of D^{b}(mod kG) and specialisation closed subsets V of V_{G}.

The thick subcategory corresponding to V is the full subcategory of objects X in D^{b}(mod kG) such that V_{G}(X) ⊆ V.

Proof. Let C be a tensor ideal thick subcategory of D^{b}(mod kG), and let C′ be the tensor ideal localising subcategory of K(Inj kG) generated by C. It follows using the arguments described in [32, §5] that C′ ∩ D^{b}(mod kG) = C. The supports of C and C′ coincide, so the map sending C to its support is injective by Theorem 10.1.

It follows from Lemma 11.3 that the support of a tensor ideal thick subcategory is specialisation closed. On the other hand, if V is a specialisation closed subset of V_{G}, then the support of the tensor ideal thick subcategory generated by {ik//𝔭 | 𝔭 ∈ V} equals V, by Proposition 3.5(2). So every specialisation closed subset of V_{G} occurs as the support of some tensor ideal thick subcategory C of D^{b}(mod kG). □

Localising subcategories closed under products and duality.

Let k be a field and (T, ⊗, 𝟙) a k-linear tensor triangulated category. There are two notions of duality in T. The Spanier-Whitehead dual X^{∨} of an object X is defined as the function object Hom(X, 𝟙) as in [6, §8] by the adjunction

Hom_{T}(− ⊗ X, 𝟙) ≅ Hom_{T}(−, Hom(X, 𝟙)).

The Brown-Comenetz dual X^{∗} of an object X is defined by the isomorphism

Hom_{k}(Hom_{T}(𝟙, − ⊗ X), k) ≅ Hom_{T}(−, X^{∗}).

The language and notation is borrowed from stable homotopy theory. The commutativity of the tensor product implies that both dualities are self adjoint. Thus we have a natural isomorphism Hom_{T}(X, Y^{∗}) ≅ Hom_{T}(Y, X^{∗}) which gives rise to a natural biduality morphism X → X^{∗∗}.

Lemma 11.5. The following statements hold for each X in T.

(1) There is a natural isomorphism X^{∗} ≅ Hom(X, 𝟙^{∗}).

(2) For each compact object C, applying Hom_{T}(C, −) to the morphism X → X^{∗∗} yields the biduality map

Hom_{T}(C, X) → Hom_{k}(Hom_{k}(Hom_{T}(C, X), k), k).

(3) If X^{∗} = 0, then X = 0.

Proof. The first claim is a consequence of the isomorphisms

Hom_{T}(−, X^{∗}) ≅ Hom_{T}(− ⊗ 𝟙, X^{∗})
≅ Hom_{T}(−, Hom(𝟙, X^{∗}))
≅ Hom_{T}(−, Hom(X, 𝟙^{∗}))

where the last one is immediate from the definition of the Brown-Comenetz dual.

(2) Set (−)^{†} = Hom_{k}(−, k). There is a natural isomorphism η: Hom_{T}(C, X)^{†} ⟶^{∼} Hom_{T}(C^{∨}, X^{∗}) which is compatible with the adjunction isomorphisms for (−)^{∗} and (−)^{†}. This follows from the defining isomorphism of the Brown-Comenetz dual and the fact that C is a strongly dualising object. Thus the following diagram commutes.

\xymatrix{Hom_{T}(X^{∗}, X^{∗}) \ar[dd]_{∼} \ar[r]^{Hom_{T}(C^{∨},−)} & Hom_{k}(Hom_{T}(C^{∨}, X^{∗}), Hom_{T}(C^{∨}, X^{∗})) \ar[d]_{∼}^{Hom_{k}(η,η^{−1})} \\  & Hom_{k}(Hom_{T}(C, X)^{†}, Hom_{T}(C, X)^{†}) \ar[d]_{∼} \\ Hom_{T}(X, X^{∗∗}) \ar[r]^{Hom_{T}(C,−)} & Hom_{k}(Hom_{T}(C, X), Hom_{T}(C, X)^{††})}

This justifies the claim.

(3) When X^{∗} = 0 it follows from (2) that Hom_{T}(C, X) = 0 for any compact object C, since k is a field. Thus X = 0 as claimed. □

In StMod kG the function object is Hom_{k}(M, N), with diagonal action. The Spanier-Whitehead dual and the Brown-Comenetz dual of kG-modules are closely related. As usual for a kG-module N we write ΩN and Ω^{−1}N for the kernel of a projective cover of N and the cokernel of an injective envelope of N, respectively.

Proposition 11.6. In StMod(kG) for each kG-module M there are isomorphisms

M^{∨} ≅ Hom_{k}(M, k) and M^{∗} ≅ Ω Hom_{k}(M, k).

Hence M^{∨} = 0 if and only if M^{∗} = 0 if and only if M is projective.

Proof. The expression for M^{∨} is by definition. Set T = StMod(kG). For each kGmodule N, Tate duality [12, Chapter XII, Theorem 6.4] gives the third isomorphism below:

Hom_{T}(N, Ω Hom_{k}(M, k)) ≅ Hom_{T}(Ω^{−1}N, Hom_{k}(M, k))
≅ Hom_{T}(Ω^{−1}N ⊗_{k} M, k)
≅ Hom_{k}(Hom_{T}(k, Ω((Ω^{−1}N) ⊗_{k} M)), k)
≅ Hom_{k}(Hom_{T}(k, N ⊗_{k} M), k).

The other isomorphisms are standard. Thus M^{∗} ≅ Ω Hom_{k}(M, k).

In StMod(kG) one has ΩN = 0 if and only if N = 0. Therefore the last claim follows from Lemma 11.5(3). □

The situation in K(Inj kG) is more complicated. Observe that in this category the function object of complexes X and Y is the complex Hom_{k}(X, Y) of injective kG-modules with diagonal action: (gϕ)(x) = g(ϕ(g^{−1}x)).

Proposition 11.7. For each X in K(Inj kG) there are isomorphisms

X^{∨} = Hom_{k}(X, ik) and X^{∗} ≅ Hom_{k}(X, ik^{∗}) ≅ Hom_{k}(X, pk).

Hence X^{∨} is semi-injective, and X^{∨} = 0 if and only if X is acyclic. Furthermore X^{∗} = 0 if and only if X = 0.

Proof. The expression for X^{∨} is by definition. For X^{∗} use Lemma 11.5(1) and an isomorphism ik^{∗} ≅ pk, which is a variant of Tate duality:

Hom_{K(kG)}(−, ik^{∗}) ≅ Hom_{k}(Hom_{K(kG)}(ik, −), k)
≅ Hom_{k}(Hom_{K(kG)}(k, −), k)
≅ Hom_{K(kG)}(−, k)
≅ Hom_{K(kG)}(−, pk).

The adjunction isomorphism Hom_{T}(−, X^{∨}) ≅ Hom_{T}(X ⊗_{k} −, ik) implies that X^{∨} is semi-injective, since ik is semi-injective. Given this it is clear that X^{∨} = 0 precisely when X is acyclic. The statement about X^{∗} is part of Lemma 11.5. □

Given a subcategory C of a triangulated category T we define full subcategories

⊥C = {X ∈ T | Hom_{T}^{∗}(X, Y) = 0 for all Y in C}
C^{⊥} = {X ∈ T | Hom_{T}^{∗}(Y, X) = 0 for all Y in C}.

Evidently, ^{⊥}C is a localising subcategory, and C^{⊥} is a colocalising subcategory, i.e., a thick subcategory closed under products.

Theorem 11.8. For any tensor ideal localising subcategory C of K(Inj kG) the following are equivalent:

(a) The subcategory C is closed under products.

(b) The complement of the support of C in Spec H^{∗}(G, k) is specialisation closed.

(c) The subcategory C is equal to D^{⊥} with D a subcategory of compact objects.

(d) The Brown-Comenetz dual of any object in C is also in C. The analogous statement where K(Inj kG) is replaced by StMod(kG) and the set Spec H^{∗}(G, k) is replaced by Proj H^{∗}(G, k) also holds.

The proof of this result relies on a construction of objects T (I) in K(Inj kG) which we recall from [8, §11]. Given an injective H^{∗}(G, k)-module I, the object T (I) is defined in terms of the following natural isomorphism

(11.9)  Hom_{H^{∗}(G,k)}(H^{∗}(G, −), I) ≅ Hom_{K(kG)}(−, T (I)).

For each prime ideal 𝔭 of H^{∗}(G, k) let I(𝔭) be the injective envelope of H^{∗}(G, k)/𝔭.

Lemma 11.10. Let 𝔭 and 𝔮 be prime ideals in H^{∗}(G, k) with 𝔮 ⊆ 𝔭, and let I be an injective H^{∗}(G, k)-module.

(1) V_{G}(T (I(𝔭))) = {𝔭}.

(2) T (I(𝔮)) is a direct summand of a direct product of shifts of T (I(𝔭)).

(3) The natural morphism T (I) → T (I)^{∗∗} is a split monomorphism.

Proof. (1) Let C be a compact object of K(Inj kG). Then H^{∗}(G, C) is finitely generated as a module over H^{∗}(G, k). It follows that Hom_{K(kG)}^{∗}(C, T (I(𝔭))) is 𝔭local and 𝔭-torsion as a H^{∗}(G, k)-module, for it is isomorphic to

Hom_{H^{∗}(G,k)}^{∗}(H^{∗}(G, C), I(𝔭)).

Now apply [6, Corollary 5.9].

(2) The module I(𝔮) is 𝔭-local and the shifted copies of I(𝔭) form a set of injective cogenerators for the category of 𝔭-local modules. Thus I(𝔮) is a direct summand of a product of shifted copies of I(𝔭). Now apply the functor T and observe that T preserves products.

(3) The induced map H^{∗}(G, T (I)) → H^{∗}(G, T (I)^{∗∗}) is a monomorphism, by Lemma 11.5(2). Applying Hom_{H^{∗}(G,k)}(−, I) to this map and using (11.9), one gets an epimorphism

Hom_{K(kG)}(T (I)^{∗∗}, T (I)) → Hom_{K(kG)}(T (I), T (I))

which provides an inverse for T (I) → T (I)^{∗∗}. □

Proof of Theorem 11.8. First we prove the theorem for K(Inj kG).

(a) ⇒ (b): Let 𝔭 be a prime ideal in the support of C. Theorem 10.1 implies that T (I(𝔭)) is in C, since its support is {𝔭}, by Lemma 11.10(1). Therefore T (I(𝔮)) is in C for every 𝔮 ⊆ 𝔭 by Lemma 11.10, since C is closed under products. Thus the complement of the support of C is specialisation closed.

(b) ⇒ (c): Let V denote the complement of the support of C. Since it is specialisation closed the localising subcategory K_{V} of K(Inj kG) corresponding to V is generated by compact objects, by Proposition 3.5. Therefore K_{V}^{⊥} = (K_{V}^{c})^{⊥} where K_{V}^{c} denotes the full subcategory consisting of the compact objects in K_{V}. On the other hand, K_{V}^{⊥} is the localising subcategory consisting of all objects with support contained in the complement of V, by [6, Corollary 5.7]. Thus Theorem 10.1 implies C = (K_{V}^{c})^{⊥}.

(c) ⇒ (a): This implication is clear.

(c) ⇒ (d): Suppose that C = D^{⊥} with D a subcategory of compact objects. Fix objects D in D and X in C. We need to show that Hom_{K(kG)}(D, X^{∗}) = 0. For any compact object C there are isomorphisms

Hom_{K(kG)}(D ⊗_{k} C, X) ≅ Hom_{K(kG)}(D, Hom_{k}(C, X))
≅ Hom_{K(kG)}(D, C^{∨} ⊗_{k} X) = 0

where the last one holds because C is tensor ideal. Hence D ⊗_{k} C is in ^{⊥}C for any compact object C. Thus D^{∨} ⊗_{k} C is in ^{⊥}C as well, since D^{∨} is a direct summand of D^{∨} ⊗_{k} D ⊗_{k} D^{∨}. A similar argument now yields

Hom_{K(kG)}(C, D ⊗_{k} X) ≅ Hom_{K(kG)}(C ⊗_{k} D^{∨}, X) = 0

for any compact object C. Therefore D ⊗_{k} X = 0, so that

Hom_{K(kG)}(D, X^{∗}) ≅ Hom_{K(kG)}(D, Hom_{k}(X, ik^{∗}))
≅ Hom_{K(kG)}(D ⊗_{k} X, ik^{∗}) = 0.

(d) ⇒ (b): Suppose that C is closed under Brown-Comenetz duality, and let 𝔭 be a prime in the support of C. We apply Lemma 11.10 several times. First notice that T (I(𝔭)) is in C, by Theorem 10.1. Given any set {X_{α}} where each X_{α} is a shift of T (I(𝔭)), it follows that the complex

∏_{α} X_{α}^{∗∗} = (⨁_{α} X_{α}^{∗})^{∗}

is in C. The natural map ∏_{α} X_{α} → ∏_{α} X_{α}^{∗∗} is a split monomorphism, and hence ∏_{α} X_{α} is in C. If 𝔮 ⊆ 𝔭 then T (I(𝔮)) is a direct summand of a direct product of shifts of T (I(𝔭)). It follows that 𝔮 is also in the support of C. Thus the complement of this set is specialisation closed.

This completes the proof of the theorem for K(Inj kG). It remains to consider the category StMod(kG). We use the same arguments as before for the implications (b) ⇒ (c) ⇒ (d) and (c) ⇒ (a), because they are formal, given Theorem 10.3. For the other implications, we identify StMod(kG) with the category K_{ac}(Inj kG) of acyclic complexes and view each tensor ideal localising subcategory of StMod(kG) as a tensor ideal localising subcategory of K(Inj kG). Then we use the fact that the inclusion into K(Inj kG) preserves products and Brown-Comenetz duals. Moreover, the support of K_{ac}(Inj kG) equals Proj H^{∗}(G, k), by Proposition 9.6. □

The telescope conjecture for StMod(kG) and K(Inj kG).

A localising subcategory C of a triangulated category T is strictly localising if the inclusion has a right adjoint. This is equivalent to the statement that there is a localisation functor L: T → T such that an object X of T is in C if and only if LX = 0; see for example [23, Lemma 3.5]. The obstruction to constructing the right adjoint is that the collections of morphisms in the Verdier quotient may be too large to be sets.

Lemma 11.11. Tensor ideal localising subcategories of K(Inj kG) and StMod(kG) are strictly localising.

Proof. Let C be a tensor ideal localising subcategory of K(Inj kG), and let V be its support, which is a subset of Spec H^{∗}(G, k). Fix a compact generator C of K(Inj kG) and consider the functor from K(Inj kG) to the category of graded abelian groups sending an object X to ⨁_{𝔭∉V} Hom_{K(kG)}^{∗}(C, Γ_{𝔭}X). Theorem 10.1 implies that its kernel is C. The functor is cohomological and preserves coproducts. Thus there exists a localisation functor K(Inj kG) → K(Inj kG) with kernel C; see for example [6, Proposition 3.6].

An analogous argument works for StMod(kG). □

A strictly localising subcategory C of a triangulated category T is smashing if the localisation functor T → T with kernel C preserves coproducts. If T is tensor triangulated and generated by its tensor unit 𝟙, then a localisation functor L: T → T preserves coproducts if and only if the natural morphism L𝟙⊗X → LX is an isomorphism for all X in T. This fact explains the term “smashing”, because in algebraic topology the smash product plays the role of the tensor product.

Next we discuss the telescope conjecture which is due to Bousfield and Ravenel [9, 30]. In its general form, the conjecture asserts for a compactly generated triangulated category T that every smashing localising subcategory is generated by objects that are compact in T; see [26]. The following result confirms this conjecture for K(Inj kG) and StMod(kG), at least for all smashing subcategories that are tensor ideal.

Theorem 11.12. Let C be a tensor ideal localising subcategory of K(Inj kG). The following conditions are equivalent:

(a) The localising subcategory C is smashing.

(b) The localising subcategory C is generated by objects compact in K(Inj kG).

(c) The support of C is a specialisation closed subset of Spec H^{∗}(G, k).

A similar result holds for StMod(kG) with Spec H^{∗}(G, k) replaced by Proj H^{∗}(G, k).

Proof. We prove the theorem for K(Inj kG); an analogous argument works for the stable module category. Let L: K(Inj kG) → K(Inj kG) be a localisation functor with kernel C. For each object X in K(Inj kG), there exists a localisation triangle Γ X → X → LX → with Γ X in C and LX in C^{⊥}. Using this exact triangle one shows that C is smashing if and only if C^{⊥} is closed under coproducts.

(a) ⇒ (b): If C is smashing then C^{⊥} is a localising subcategory closed under products. Thus C^{⊥} = D^{⊥} for some category D consisting of compact objects by Theorem 11.8. It follows that C is generated by D.

(b) ⇒ (c): Let D be a subcategory of compact objects such that C = Loc(D). Then C and D have same support. Now one uses that the support of each compact object is specialisation closed by Lemma 11.3.

(c) ⇒ (a): Let V be the support of C. Then C consists of all objects with support contained in V, by Theorem 10.1. This implies that C^{⊥} is the localising subcategory consisting of all objects with support disjoint from V, since V is specialisation closed; see [6, Corollary 5.7]. In particular, C^{⊥} is closed under coproducts, and therefore C is smashing. □

Left perpendicular categories.

If V is a subset of Spec H^{∗}(G, k), we write cl(V) for the specialisation closure of V, namely the smallest specialisation closed subset containing it.

Theorem 11.13. For X and Y in K(Inj kG) the following are equivalent:

(a) Hom_{K(kG)}^{∗}(X, Y′) = 0 for all Y′ ∈ Loc^{⊗}(Y).

(b) cl(V_{G}(X)) ∩ V_{G}(Y) = ∅

Proof. The implication (b) ⇒ (a) is part of [6, Corollary 5.8]. Assume (a) holds. Choose primes 𝔮 ∈ V_{G}(Y) and 𝔭 ∈ V_{G}(X), and a compact object C in K(Inj kG) with Hom_{k}(C, Γ_{𝔭}X) ≠ 0. Since V_{G}(T (I(𝔮))) = {𝔮}, by Lemma 11.10, Theorem 10.1 yields C ⊗_{k} T (I(𝔮)) ∈ Loc^{⊗}(Y). Since Γ_{𝔭}X = Γ_{𝔭}ik ⊗_{k} X ∈ Loc^{⊗}(X) holds, one has

Hom_{H^{∗}(G,k)}^{∗}(H^{∗}(G, Hom_{k}(C, Γ_{𝔭}X)), I(𝔮)) ≅ Hom_{K(kG)}^{∗}(Hom_{k}(C, Γ_{𝔭}X), T (I(𝔮)))
≅ Hom_{K(kG)}^{∗}(Γ_{𝔭}X, C ⊗_{k} T (I(𝔮))) = 0.

The H^{∗}(G, k)-module H^{∗}(G, Hom_{k}(C, Γ_{𝔭}X)) is non-zero and 𝔭-local, so it follows that 𝔭 ⊈ 𝔮 as required. □

If V is a subset of V_{G}, we write ^{⊥}V for the set of primes 𝔮 ∈ V_{G} such that for all 𝔭 ∈ V we have 𝔮 ⊉ 𝔭. In other words, ^{⊥}V is the largest specialisation closed subset of V_{G} that has trivial intersection with V. The statement of the result below was suggested by a question of Jeremy Rickard.

Corollary 11.14. Let C be a tensor ideal localising subcategory of K(Inj kG). If V is the support of C, then ^{⊥}V is the support of ^{⊥}C.

Proof. This follows from Theorem 11.13. □

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Dave Benson, Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland, U.K.

Srikanth B. Iyengar, Department of Mathematics, University of Nebraska, Lincoln NE 68588, U.S.A.

Henning Krause, Institut für Mathematik, Universität Paderborn, 33095 Paderborn, Germany

Current address: Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany.