LOCALISING SUBCATEGORIES FOR COCHAINS ON THE CLASSIFYING SPACE OF A FINITE GROUP

DAVE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE

Abstract. The localising subcategories of the derived category of the cochains on the classifying space of a finite group are classified. They are in one to one correspondence with the subsets of the set of homogeneous prime ideals of the cohomology ring H^{∗}(G, k).

1. Introduction

Let G be a finite group and k a field of characteristic p. Let C^{∗}(BG; k) be the cochains on the classifying space BG. Using the machinery of Elmendorf, Kříž, Mandell and May [8], one can regard C^{∗}(BG; k) as a strictly commutative S-algebra over the field k. The derived category D(C^{∗}(BG; k)) has thus a structure of a tensor triangulated category via the left derived tensor product − ⊗_{C^{∗}(BG;k)}^{L} −. The unit for the tensor product is C^{∗}(BG; k).

In this paper we apply techniques and results from [3, 4, 5, 6] to classify the localising subcategories of D(C^{∗}(BG; k)). More precisely, there is a notion of stratification for triangulated categories via the action of a graded commutative ring which implies that the localising subcategories are parameterised by sets of homogeneous prime ideals [4]. For D(C^{∗}(BG; k)) we use the natural action of the endomorphism ring of the tensor identity which is isomorphic to the cohomology algebra H^{∗}(G, k) of the group G.

Theorem 1.1. The derived category D(C^{∗}(BG; k)) is stratified by the ring H^{∗}(G, k). This yields a one to one correspondence between the localising subcategories of D(C^{∗}(BG; k)) and subsets of the set of homogeneous prime ideals of H^{∗}(G, k).

It is proved in [6] that there is an equivalence of tensor triangulated categories between D(C^{∗}(BG; k)) and the localising subcategory of K(Inj kG) generated by the tensor identity. Here, K(Inj kG) is the homotopy category of complexes of injective (= projective) kG-modules, studied in [6, 9].

The main theorem of [5] states that K(Inj kG) is stratified as a tensor triangulated category by H^{∗}(G, k). Theorem 1.1 is a consequence of a more general result concerning tensor triangulated categories, which is described below.

Let (T, ⊗, 𝟙) be a compactly generated tensor triangulated category, as described in [3, §8], and R a graded commutative noetherian ring acting on T via a homomorphism R → End_{T}^{∗}(𝟙). In this case, for each homogeneous prime ideal 𝔭 of R there exists a local cohomology functor Γ_{𝔭}: T → T; see [3]. The support of an object X in T is then defined to be

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

The research of the second author was undertaken during a visit to the University of Bielefeld, supported by a research prize from the Humboldt Foundation, and by NSF grant DMS 0903493.

The condition that T is stratified by the action of R means that assigning a subcategory S of T to its support

supp_{R} S = ⋃_{X∈S} supp_{R} X

yields a bijection between tensor ideal localising subcategories of T and subsets of the homogeneous prime ideal spectrum Spec R contained in supp_{R} T; see [4, Theorem 4.2]. Theorem 1.1 is thus a special case of the result below that relates tensor ideal localising subcategories of T and the localising subcategories of Loc_{T}(𝟙), the localising subcategory of T generated by the tensor unit. We note that Loc_{T}(𝟙) is a compactly generated tensor triangulated category in its own right and that R acts on it as well.

Theorem 1.2. Suppose that the Krull dimension of R is finite. If T is stratified by R as a tensor triangulated category, then so is Loc_{T}(𝟙), and there is a bijection

\{ \begin{matrix}Tensor ideal localising \\ subcategories of T \end{matrix} \} ⟶^{∼} \{ \begin{matrix}Localising subcategories \\ of Loc_{T}(𝟙) \end{matrix} \} .

It assigns each tensor ideal localising subcategory S of T to S ∩ Loc_{T}(𝟙).

Remark 1.3. The theorem is not true without the assumption that T is stratified by R. For example, let T be the derived category of quasi-coherent sheaves on the projective line ℙ_{k}^{1}. The tensor unit is O. In this example there are no proper localising subcategories of Loc_{T}(O) since End_{T}^{∗}(O) = k, while there are many tensor ideal localising subcategories of T.

Remark 1.4. The assumption that the Krull dimension of R is finite is artificial, and is used only to ensure that for each X ∈ T and 𝔭 ∈ Spec R the object Γ_{𝔭}X belongs to Loc_{T}(X). One can replace this condition by, for instance, the assumption that T arises as the homotopy category of a Quillen model category [10, §6].

2. Localising subcategories of Loc_{T}(𝟙)

In this section T is a triangulated category with set-indexed coproducts and the tensor product ⊗ provides a symmetric monoidal structure with unit 𝟙 on T, which is exact in each variable and preserves set-indexed coproducts.

The proof of Theorem 1.2 is based on a sequence of elementary lemmas. The first one describes the tensor ideal localising subcategory of T which is generated by a class C of objects; we denote this by Loc_{T}^{⊗}(C).

Lemma 2.1. Let C be a class of objects of T. Then

Loc_{T}^{⊗}(C) = Loc_{T}({X ⊗ Y | X ∈ C, Y ∈ T}).

Proof. Set S = Loc_{T}({X ⊗ Y | X ∈ C, Y ∈ T}). It suffices to show that S is tensor ideal. This means that F S ⊆ S for each tensor functor F = − ⊗ Y, which is an immediate consequence of Lemma 2.2 below. □

Lemma 2.2. Let F: U → V be an exact functor between triangulated categories that preserves set-indexed coproducts. If C is a class of objects of U, then

F Loc_{U}(C) ⊆ Loc_{V}(F C).

Proof. The preimage F^{−1} Loc_{V}(F C) is a localising subcategory of U containing C. Thus it contains Loc_{U}(C), and one gets

F Loc_{U}(C) ⊆ F F^{−1} Loc_{V}(F C) ⊆ Loc_{V}(F C).  □

LOCALISING SUBCATEGORIES FOR COCHAINS ON BG 3

Lemma 2.3. Let Γ: T → T be a colocalization functor which preserves set-indexed coproducts. Then for any X ∈ T and Y ∈ Loc_{T}(𝟙), there is a natural isomorphism

Γ X ⊗ Y ⟶^{∼} Γ (X ⊗ Y).

Remark 2.4. There is an analogous result for a localization functor L: T → T which preserves set-indexed coproducts: For any X ∈ T and Y ∈ Loc_{T}(𝟙), there is a natural isomorphism L(X ⊗ Y) ⟶^{∼} LX ⊗ Y.

Proof. A colocalisation functor Γ comes with a natural morphism Γ X → X. Tensoring this with an object Y ∈ Loc_{T}(𝟙) gives a morphism Γ X⊗Y → X⊗Y that factors through the natural morphism Γ (X⊗Y) → X⊗Y. Here, one uses that Γ X⊗Y belongs to Loc_{T}(𝟙), since the objects Y′ ∈ T with Γ X ⊗Y′ ∈ Loc_{T}(𝟙) form a localising subcategory containing 𝟙. The induced morphism ϕ_{Y} : Γ X ⊗ Y → Γ (X ⊗ Y) is an isomorphism. To see this, observe that the objects Y′ ∈ T such that ϕ_{Y′} is an isomorphism form a localising subcategory containing 𝟙. □

Proposition 2.5. Suppose that the unit 𝟙 is compact in T and let Γ: T → Loc_{T}(𝟙) denote the right adjoint of the inclusion Loc_{T}(𝟙) → T. If S is a localising subcategory of Loc_{T}(𝟙), then

Loc_{T}^{⊗}(S) ∩ Loc_{T}(𝟙) = Γ (Loc_{T}^{⊗}(S)) = S.

Proof. We verify each of the following inclusions

S ⊆ Loc_{T}^{⊗}(S) ∩ Loc_{T}(𝟙) ⊆ Γ (Loc_{T}^{⊗}(S)) ⊆ S.

The first one is clear. Composing the functor Γ with the inclusion Loc_{T}(𝟙) → T yields a colocalisation functor which preserves set-indexed coproducts, since 𝟙 is compact. For an object X in Loc_{T}^{⊗}(S) ∩ Loc_{T}(𝟙), we have Γ X ≅ X. This gives the second inclusion. Applying Lemma 2.3 together with the description of Loc_{T}^{⊗}(S) from Lemma 2.1 yields the third inclusion. □

Corollary 2.6. Suppose that the unit 𝟙 is a compact object in T. Assigning each localising subcategory S of Loc_{T}(𝟙) to Loc_{T}^{⊗}(S) gives a bijection

\{ \begin{matrix}Localising subcategories \\ of Loc_{T}(𝟙) \end{matrix} \} ⟶^{∼} \{ \begin{matrix}Tensor ideal localising subcategories of \\ T generated by objects from Loc_{T}(𝟙) \end{matrix} \} .

Proof. The inverse map sends U ⊆ T to U ∩ Loc_{T}(𝟙). □

We are now ready to prove Theorem 1.2. Note that in this T is a compactly generated tensor triangulated category, which entails a host of additional requirements; see [3, §8] for a list.

Proof of Theorem 1.2. It follows from Proposition 2.5 that the assignment

S ⟼ Loc_{T}^{⊗}(S)

is an injective map from the localising subcategories of Loc_{T}(𝟙) to the tensor ideal localising subcategories of T. In general, it is not bijective, as the example of Remark 1.3 shows. However, since T is stratified by R as a tensor triangulated category, it follows from [4, §7] that each tensor ideal localising subcategory is generated by a set of objects of the form Γ_{𝔭}𝟙. Since R has finite Krull dimension, [4, Theorem 3.4] yields that Γ_{𝔭}𝟙 is in Loc_{T}(𝟙). Therefore, given a tensor ideal localising subcategory U of T, the localising subcategory

U′ = Loc_{T}({Γ_{𝔭}𝟙 | 𝔭 ∈ Supp_{R} U}) ⊆ Loc_{T}(𝟙)

satisfies Loc_{T}^{⊗}(U′) = U. This proves the surjectivity of the assignment. Moreover, we have shown that each localising subcategory of Loc_{T}(𝟙) is generated by objects of the form Γ_{𝔭}𝟙, so Loc_{T}(𝟙) is stratified by the action of R; see [4, Theorem 4.2]. □

3. The cohomological nucleus

Let (T, ⊗, 𝟙) be a compactly generated tensor triangulated category and let R be a graded commutative noetherian ring acting on T via a homomorphism R → End_{T}^{∗}(𝟙). Suppose in addition that R has finite Krull dimension.

We define the cohomological nucleus of T as the set of homogeneous prime ideals 𝔭 of R such that there exists an object X ∈ T satisfying Hom_{T}^{∗}(𝟙, X) = 0 and Γ_{𝔭}X ≠ 0. This definition is motivated by work of Benson, Carlson, and Robinson in the context of modular group representations [2].

For 𝔭 in Spec R consider the tensor ideal localising subcategory

Γ_{𝔭}T = {Y ∈ T | Y ≅ Γ_{𝔭}X for some X ∈ T}.

Note that an object X ∈ T belongs to Γ_{𝔭}T if and only if Hom_{T}^{∗}(C, X) is 𝔭-local and 𝔭-torsion for every compact C ∈ T, by [3, Corollary 4.10]. The result below gives a local description of the cohomological nucleus.

Proposition 3.1. Let 𝔭 be a homogeneous prime ideal of R. The following conditions are equivalent:

(1) Every object X in T with Hom_{T}^{∗}(𝟙, X) = 0 satisfies Γ_{𝔭}X = 0.

(2) Every localising subcategory of Γ_{𝔭}T is a tensor ideal of T.

Proof. The Krull dimension of R is finite, so Γ_{𝔭}X is in Loc_{T}(X) for each X in T, by [4, Theorem 3.4]. This fact is used without further comment.

(1) ⇒ (2): Set S = Loc_{T}(Γ_{𝔭}𝟙). Note that S ⊆ Γ_{𝔭}T; we claim that equality holds. Indeed, S ⊆ Loc_{T}(𝟙) and also Loc_{T}^{⊗}(S) = Γ_{𝔭}T, since Γ_{𝔭} = Γ_{𝔭}𝟙 ⊗ −. Thus, for any X in Γ_{𝔭}T from Proposition 2.5 one gets an exact triangle Γ X → X → X′ → with Γ X ∈ S and Hom_{T}^{∗}(𝟙, X) = 0. Then (1) implies X′ = 0 and hence X ∈ S.

Now for a localising subcategory R of Γ_{𝔭}T one has Loc_{T}^{⊗}(R) ⊆ Loc_{T}(𝟙), and it follows, again from Proposition 2.5, that R is a tensor ideal of T.

(2) ⇒ (1): Assume Hom_{T}^{∗}(𝟙, X) = 0; then Hom_{T}^{∗}(𝟙, Γ_{𝔭}X) = 0, as 𝟙 is compact. Condition (2) implies that Loc_{T}(Γ_{𝔭}𝟙) = Γ_{𝔭}T. Thus Γ_{𝔭}X belongs to Loc_{T}(Γ_{𝔭}𝟙) and therefore also to Loc_{T}(𝟙). So one obtains Hom_{T}^{∗}(Γ_{𝔭}X, Γ_{𝔭}X) = 0, which implies Γ_{𝔭}X = 0. □

Consider as an example for T the stable module category StMod kG of a finite group G with the canonical action of R = H^{∗}(G, k). We refer to [1, 2] for the discussion of two variations of the nucleus, namely the group theoretic and the representation theoretic nucleus. There it is shown that Loc_{T}(𝟙) = T if and only if the centraliser of every element of order p in G is p-nilpotent, where p denotes the characteristic of the field k.

It is convenient to define for any class C of objects of T

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

Now let S = Loc_{T}(𝟙). The representation theoretic nucleus is by definition

⋃_{X∈S^{⊥}∩T^{c}} supp_{R} X.

Clearly, this is contained in the cohomological nucleus. It is a remarkable fact that the representation theoretic nucleus is non-empty if S^{⊥} ≠ 0; this is proved in [1, 2]. Moreover, Question 13 of [7] asks whether S = ^{⊥}(S^{⊥} ∩ T^{c}). Note that S = ^{⊥}(S^{⊥}) follows from general principles.

Acknowledgments. It is a pleasure to thank Greg Stevenson for helpful comments on this work.

References

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[8] A. D. Elmendorf, I. Kříž, M. A. Mandell, and J. P. May, Rings, modules and algebras in stable homotopy theory, Surveys and Monographs, vol. 47, American Math. Society, 1996.

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[10] G. Stevenson, Support theory via actions of tensor triangulated categories, arXiv:1105.4692v1.

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

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

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