Module categories for finite group algebras

David J. Benson, Srikanth B. Iyengar and Henning Krause

Abstract. This survey article is intended as an introduction to the recent categorical classification theorems of the three authors, restricting to the special case of the category of modules for a finite group.

2010 Mathematics Subject Classification. Primary 20J06; Secondary 20C20, 13D45, 16E45, 18E30 Keywords. Modular representation theory, local cohomology, stable module category, idempotent modules, derived category, triangulated categories, local-global principle, stratification, costratification.

Introduction

The purpose of this paper is to explain the recent work of the three authors on the classification of localising and colocalising subcategories of triangulated categories [8, 10, 9, 11], and to put it into context. In order to be as concrete as possible, we restrict our attention to categories associated with the modular representation theory of finite groups. A more leisurely discussion, filling in requisite background from commutative algebra and triangulated categories, is given in [12].

Let G be a finite group and let k be an algebraically closed field of characteristic p > 0. With a few exceptions in characteristic two, if the Sylow p-subgroups of G are non-cyclic then the group algebra kG has wild representation type. We therefore do not hope to classify the indecomposable finitely generated kG-modules, and so we are left with several options. We can restrict the kinds of modules that we’re interested in; we can look for properties of modules that we can prove without the need for a classification; or we look for coarser classification theorems. Examples abound of all three types of approach; we shall be examining the last option.

The first indication that it might be possible to make some sort of categorical classification theorem came with Mike Hopkins’ 1987 classification [21] of the thick subcategories of the derived category D^{b}(proj(R)) of perfect complexes over a commutative Noetherian ring R. His interest was in the nilpotence theorem for the stable homotopy category, for which he regarded the derived category of perfect complexes as a toy model. The parametrisation of the thick subcategories was by specialisation closed sets of prime ideals in R.

Neeman’s 1992 paper [25] took up Hopkins’ work, clarified it and went on to classify the localising subcategories of the unbounded derived category of Rmodules D(Mod(R)). This is the appropriate big category whose compact objects are the perfect complexes D^{b}(proj(R)). The parametrisation for the localising subcategories was by all subsets of the set of prime ideals of R, not just the specialisation closed ones.

In 1997, the first author together with Carlson and Rickard [6] proved the analogue of Hopkins’ theorem for the stable category stmod(kG) of finitely generated kG-modules, over an algebraically closed field k of characteristic p. The parametrisation for the thick subcategories was by specialisation closed subsets of homogeneous primes in the cohomology ring H^{∗}(G, k), ignoring the maximal ideal of all positive degree elements.

The corresponding big category is the stable category StMod(kG) of all kGmodules; the full subcategory of compact objects in this is the finitely generated stable module category stmod(kG). It was expected by analogy with D(Mod(R)) that the classification of localising subcategories of StMod(kG) would be by all subsets of the set of non-maximal homogeneous primes in H^{∗}(G, k). But this turned out to be very hard to prove, and it was not until a couple of years ago that we managed to achieve this classification [9].

More recently, in 2009 Neeman [26] classified the colocalising subcategories of D(Mod(R)); the corresponding classification for StMod(kG) is given in [11].

Other classifications follow similar models. The work of Hovey, Palmieri and Strickland [22] and the series of papers [8, 10, 9, 11] lay the general foundations. For background on the theory of support varieties we refer to Solberg’s survey [32].

Contents.

1. The stable module category

2. Thick subcategories of stmod(kG)

3. The derived category

4. Rickard idempotent modules and functors

5. Classification of tensor ideal thick subcategories

6. Localising subcategories of StMod(kG)

7. The category KInj(kG)

8. Support for triangulated categories

9. Tensor triangulated categories

10. The local-global principle

11. Stratifying triangulated categories

12. Graded polynomial algebras

13. The BGG correspondence

14. The Koszul construction

15. Quillen stratification

16. Applications

17. Costratification

1. The stable module category

We write Mod(kG) for the module category of kG. The objects are the left kGmodules and the morphisms are the module homomorphisms. This is an abelian category. We write mod(kG) for the full subcategory of finitely generated kGmodules. We refer the reader to [2] for basic constructions and facts concerning kG-modules. The following statement is well known for finitely generated modules, but is true more generally.

Lemma 1.1. Projective and injective kG-modules coincide.

Proof. It is an easy exercise using the group basis to show that kG is self-dual as a kG-module. This implies that every projective module is injective. For the converse, we note that every kG-module M embeds in a free kG-module, via the map M → kG ⊗_{k} M sending m to ∑_{g∈G} g ⊗ g^{−1}m. Here, we make kG ⊗_{k} M into a kG-module via g(g′ ⊗ m) = gg′ ⊗ m; thus a k-basis of M gives a free kG-basis for kG ⊗_{k} M. If M is injective then this embedding splits and M is a summand of a free module, hence projective.

Projective kG-modules are well understood. Every projective is a direct sum of finitely generated projective indecomposables, and every projective indecomposable is the projective cover of a simple module.

To work “modulo projectives”, we use the stable module category StMod(kG). This has the same objects as the module category, but the morphisms are given by

\underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/ PHom_{kG}(M, N)

where PHom_{kG}(M, N) is the linear subspace of maps that factor through some projective module.

We write stmod(kG) for the full subcategory of StMod(kG) whose objects are the modules stably isomorphic to finitely generated modules. Notice that if a map between finitely generated modules factors through some projective module then it factors through a finitely generated projective module.

The subcategory stmod(kG) is distinguished categorically as the compact objects in StMod(kG).

Definition 1.2. An object M in StMod(kG) is said to be compact if given any small coproduct ⨁_{α} M_{α}, the natural map

⨁_{α} \underline{Hom}_{kG}(M, M_{α}) → \underline{Hom}_{kG}(M, ⨁_{α} M_{α})

is an isomorphism.

Although StMod(kG) and stmod(kG) are not abelian categories, they come with a natural structure of triangulated category. The “shift” is given by Ω^{−1}, the functor assigning to each module M the cokernel of an embedding of M into an injective module. At the level of Mod(kG) this is not functorial, but for StMod(kG) it is a functorial self-equivalence whose inverse is the functor Ω taking a module M to the kernel of a surjection from a projective module onto M.

The distinguished triangles

A → B → C → Ω^{−1}(A)

are by definition those isomorphic to diagrams coming from a short exact sequence

0 → A → B → C → 0

of modules as follows. Given such a short exact sequence, we embed B into an injective module I and obtain the following commutative diagram with exact rows:

[figure: 0; A; B; C; 0; 0; A; I; Ω^{−1}(A); 0.]

This gives us a map C → Ω^{−1}A which is well defined in StMod(kG).

The right hand square in this diagram gives us a short exact sequence

0 → B → C ⊕ I → Ω^{−1}(A) → 0.

Since C is isomorphic to C ⊕ I in StMod(kG), applying the construction again gives rise to the rotated triangle

B → C → Ω^{−1}(A) → Ω^{−1}(B).

Most of the axioms for a triangulated category are easy to verify. The only one that needs any comment is the octahedral axiom, which is the expression in StMod(kG) of the third isomorphism theorem in Mod(kG). More details can be found in Buchweitz [18], and in Happel [20].

2. Thick subcategories of stmod (kG)

Definition 2.1. For any triangulated category T, a thick subcategory S is a full triangulated subcategory that is closed under taking finite direct sums and summands.

It is worth spelling out what this means for the stable module category. Being a full triangulated subcategory means that the subcategory is closed under Ω and Ω^{−1}, and it has the two in three property: given a triangle in the stable module category for which two of the objects are in S, so is the third.

Pulling back to mod(kG), thick subcategories of stmod(kG) are in one to one correspondence with the subcategories 𝔠 of mod(kG) with the following properties:

(i) All projective modules are in 𝔠.

(ii) 𝔠 is closed under finite sums and summands.

(iii) If M is in 𝔠 then so are Ω(M) and Ω^{−1}(M).

(iv) If 0 → A → B → C → 0 is a short exact sequence of modules with two of A, B and C in 𝔠 then so is the third.

The clue to finding thick subcategories of stmod(kG) comes from the theory of varieties for modules, which we now briefly describe. A fuller treatment can be found in [3]. The cohomology ring H^{∗}(G, k) is defined to be Ext_{kG}^{∗}(k, k) where k denotes the trivial representation. This is a finitely generated graded commutative ring, and we write V_{G} for its maximal ideal spectrum.

If M is a finitely generated kG-module, we have a ring homomorphism

H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) ⟶^{M⊗_{k}−} Ext_{kG}^{∗}(M, M)

given by tensoring Yoneda extensions with M. As usual, we are tensoring over k with diagonal group action, which is exact. We write I_{M} for the kernel of this homomorphism, and V_{G}(M) for the subvariety of V_{G} determined by this ideal. Since I_{M} is a homogeneous ideal, V_{G}(M) is a closed homogeneous subvariety of the homogeneous variety V_{G}. The following theorem summarises some of the main properties of these varieties; not all of them are easy to prove.

Theorem 2.2. Let M, M_{1}, M_{2}, M_{3} be finitely generated kG-modules.

(i) V_{G}(M) = {0} if and only if M is a projective module.

(ii) More generally, the dimension of the variety V_{G}(M) determines the polynomial rate of growth of a minimal resolution of M.

(iii) V_{G}(Hom_{k}(M, k)) = V_{G}(M).

(iv) V_{G}(M_{1} ⊕ M_{2}) = V_{G}(M_{1}) ∪ V_{G}(M_{2}).

(v) V_{G}(M_{1} ⊗_{k} M_{2}) = V_{G}(M_{1}) ∩ V_{G}(M_{2}).

(vi) If 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence then for i = 1, 2, 3, V_{G}(M_{i}) is contained in the union of the varieties of the other two modules.

(vii) If 0 ≠ ζ ∈ H^{n}(G, k) is represented by a cocycle ζ̂: Ω^{n}(k) → k, we write L_{ζ} for the kernel of ζ̂. Then we have V_{G}(L_{ζ}) = V_{G}⟨ζ⟩, the variety determined by the principal ideal ⟨ζ⟩.

(viii) Given a closed homogeneous subvariety V ⊆ V_{G}, there exists a finitely generated module M with V_{G}(M) = V. Namely, if V = V_{G}⟨ζ_{1}, …, ζ_{n}⟩ then we may take M = L_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} L_{ζ_{n}} and use properties (v) and (vii).

Definition 2.3. We write V_{G} for the collection of closed homogeneous irreducible subvarieties of V_{G} (including {0}), or equivalently the spectrum of homogeneous prime ideals 𝔭 ⊆ H^{∗}(G, k) (including the maximal ideal 𝔪 of positive degree elements). We say that a subset V ⊆ V_{G} is specialisation closed if 𝔭 ∈ V, 𝔮 ∈ V_{G} and 𝔮 ⊇ 𝔭 imply 𝔮 ∈ V.

If V is a specialisation closed subset of V_{G} then we write 𝔠_{V} for the full subcategory of stmod(kG) consisting of modules M such that V_{G}(M) is a finite union of elements of V (i.e., each irreducible component of V_{G}(M) is in V).

Lemma 2.4. If V is a specialisation closed subset of V_{G} then 𝔠_{V} is a thick subcategory of stmod(kG).

Proof. This follows directly from Theorem 2.2.

We now come to an issue which will reappear in later contexts. Namely, the thick subcategories of stmod(kG) appearing in Lemma 2.4 are closed under tensor products with finitely generated kG-modules, by part (iii) of Theorem 2.2.

Lemma 2.5. If 𝔠 is a thick subcategory of stmod(kG) then the following are equivalent:

(i) 𝔠 is closed under tensor product with finitely generated kG-modules.

(ii) 𝔠 is closed under tensor product with simple kG-modules.

These conditions are automatically satisfied if G is a finite p-group.

Proof. The equivalence of (i) and (ii) follows from the fact that every finitely generated module has a finite filtration in which the filtered quotients are simple modules. If G is a finite p-group then the trivial module is the only simple kGmodule.

Definition 2.6. We say that a thick subcategory of stmod(kG) is tensor ideal, or tensor closed, if the equivalent conditions of the lemma are satisfied.

We can now state the classification theorem. This was proved in Theorem 3.4 of Benson, Carlson and Rickard [6] in case k is an algebraically closed field, and in Theorem 11.4 of Benson, Iyengar and Krause [9] for general fields k.

Theorem 2.7. There is a one to one correspondence between tensor ideal thick subcategories of stmod(kG) and non-empty specialisation closed subsets of V_{G}. Under this correspondence, V corresponds to 𝔠_{V}.

Note that the condition that V is non-empty is equivalent to the condition that V contains {0}. This subset plays no role for stmod(kG) but will make an appearance later.

If we remove the tensor ideal condition, then the classification becomes much harder. One example of a thick subcategory which is usually not tensor ideal is the full subcategory of modules in the principle block. The principal block B_{0}(kG) is the block containing the trivial module. So one might ask whether the thick subcategories of stmod(B_{0}(kG)) are all of the form 𝔠_{V} ∩ stmod(B_{0}(kG)). The obstruction to this being true is given in terms of the nucleus, defined as follows.

Definition 2.8. The nucleus Y_{G} of a finite group G is the union of the images of res_{G,H}^{∗}: V_{H} → V_{G}, as H runs over the subgroups of G such that C_{G}(H) is not p-nilpotent. Recall that a finite group is said to be p-nilpotent if it has a normal subgroup whose order is prime to p and whose index is a power of p.

Theorem 2.9 (Benson, Carlson and Robinson [7]; Benson [4]). The subvariety Y_{G} of V_{G} is equal to the union of the V_{G}(M) as M runs over the finitely generated modules M in B_{0}(kG) satisfying H^{∗}(G, M) = 0.

In particular, every non-projective module in the principal block has non-trivial cohomology if and only if the centraliser of every element of order p in G is pnilpotent.

Theorem 2.10 (Benson, Carlson and Rickard [6]). If Y_{G} is empty or equal to {0} then there is a one to one correspondence between the thick subcategories of stmod(B_{0}(kG)) and the non-empty subsets of V_{G}. Under this correspondence, V corresponds to 𝔠_{V} ∩ stmod(B_{0}(kG)).

If Y_{G} is bigger than {0} then it appears to be easy to manufacture infinite collections of thick subcategories supported on each line through the origin in Y_{G}. It appears hopeless to classify these thick subcategories. However, if we work modulo the modules supported inside the nucleus, we again obtain a classification of thick subcategories.

3. The derived category

The anomalous role of the origin in the classification of thick subcategories of stmod(kG) can be repaired by moving to a slightly bigger category, namely the derived category D^{b}(mod(kG)). Recall that the objects in this category are the finite chain complexes of finitely generated kG-modules, and the arrows are homotopy classes of maps of complexes, with the quasi-isomorphisms inverted. A quasi-isomorphism is a map of complexes that induces an isomorphism in homology. The category D^{b}(mod(kG)) is a triangulated category, in which the triangles are formed using mapping cones.

We write K^{b}(proj(kG)) for the thick subcategory of D^{b}(mod(kG)) whose objects are the perfect complexes, i.e., the complexes quasi-isomorphic to a bounded complex of finitely generated projective kG-modules. Buchweitz [18] (see also Rickard [31]) has defined a functor from D^{b}(mod(kG)) to stmod(kG) which is essentially surjective and has kernel K^{b}(proj(kG)):

K^{b}(proj(kG)) → D^{b}(mod(kG)) → stmod(kG).

The thick subcategory K^{b}(proj(kG)) is the unique minimal tensor ideal thick subcategory of D^{b}(mod(kG)). This allows us to extend the theory of support varieties to D^{b}(mod(kG)) in such a way that an object has the origin in its support if and only if it is non-zero. As before, if V is a specialisation closed subset of V_{G}, we write 𝔠_{V} for the thick subcategory of D^{b}(mod(kG)) consisting of objects whose support is a finite union of elements of V. So the following theorem is an easy consequence of Theorem 2.7.

Theorem 3.1. There is a one to one correspondence between tensor ideal thick subcategories of D^{b}(mod(kG)) and specialisation closed subsets of V_{G}. Under this correspondence, V corresponds to 𝔠_{V}.

Next, we observe that G is p-nilpotent if and only if k is the only simple module in the principal block. This allows us to extend Theorem 2.10 as follows.

Theorem 3.2. If Y_{G} is empty (i.e., if G is p-nilpotent) then there is a one to one correspondence between the thick subcategories of D^{b}(mod(B_{0}(kG))) and subsets of V_{G}. Under this correspondence, V corresponds to 𝔠_{V} ∩ D^{b}(mod(B_{0}(kG))).

Again, if Y_{G} is non-empty, we can work modulo objects supported inside the nucleus and obtain a classification of thick subcategories, but this returns us to quotients of stmod(B_{0}(kG)) so nothing new is gained.

4. Rickard idempotent modules and functors

The proof of Theorem 2.7 depends in an essential way on the construction of certain infinitely generated modules, and the development of a theory of support varieties in this context.

The modules in question are Rickard idempotent modules. Corresponding to any specialisation closed subset V of V_{G}, Rickard constructs two idempotent functors on StMod(kG). He writes these as E_{V} and F_{V}, but for consistency with the notation of our series of papers we shall use the notation Γ_{V} and L_{V}. More generally, given any thick subcategory 𝔠 of stmod(kG), we have functors Γ_{𝔠} and L_{𝔠}, and functorial triangles

Γ_{𝔠}(M) → M → L_{𝔠}(M).  (4.1)

The defining properties of these triangles are given in terms of localising subcategories of StMod(kG).

Definition 4.1. Let T be a triangulated category with small products and coproducts. A localising subcategory of T is a thick subcategory that is closed under coproducts, while a colocalising subcategory of T is a thick subcategory that is closed under products.

The triangle of functors (4.1) is characterised by the following properties:

(i) Γ_{𝔠}(M) is in the localising subcategory Loc(𝔠) generated by 𝔠.

(ii) If N is in 𝔠 then \underline{Hom}_{kG}(N, L_{𝔠}(M)) = 0.

The functors Γ_{𝔠} and L_{𝔠} preserve small coproducts, see for example [8, Corollary 6.5].

If 𝔠 is a tensor ideal thick subcategory then we have

Γ_{𝔠}(k) ⊗_{k} M ≅ Γ_{𝔠}(M), L_{𝔠}(k) ⊗_{k} M ≅ L_{𝔠}(M).

If 𝔠 = 𝔠_{V} then we abbreviate Γ_{𝔠_{V}} to Γ_{V} and L_{𝔠_{V}} to L_{V}, so that the Rickard triangle takes the form

Γ_{V}(M) → M → L_{V}(M).

Rickard idempotent modules allow us to develop a theory of support for modules in StMod(kG) as follows. Let 𝔭 ⊆ H^{∗}(G, k) be homogeneous prime ideal. We choose specialisation closed subsets V and W of V_{G} with the property that V ⊈ W but V ⊆ W ∪ {𝔭} (i.e., V ∖ W = {𝔭}). Then we write Γ_{𝔭} for the functor L_{W}Γ_{V} ≅ Γ_{V}L_{W}. The functor Γ_{𝔭} defined in this way is independent of the choice of V and W with the given properties, see [8, Theorem 6.2]. Note that if 𝔭 is the maximal prime 𝔪, we have Γ_{𝔪} = 0.

Definition 4.2. If M is an object in StMod(kG), then the support V_{G}(M) is the subset of V_{G} consisting of

{𝔭 ∈ V_{G} | Γ_{𝔭}(M) ≠ 0}.

The following properties of V_{G}(M) are proved in [6] for k algebraically closed, and in [9] for a general field k. This theorem should be compared with Theorem 2.2.

Theorem 4.3. Let M, M_{1}, M_{2}, M_{3} be kG-modules.

(i) V_{G}(M) = ∅ if and only if M is a projective module.

(ii) If M is finitely generated then V_{G}(M) is equal to the set of closed homogeneous irreducible subvarieties of V_{G}(M).

(iii) For a small family M_{α} of kG-modules we have

V_{G}(⨁_{α} M_{α}) = ⋃_{α} V_{G}(M_{α}).

(iv) V_{G}(M_{1} ⊗_{k} M_{2}) = V_{G}(M_{1}) ∩ V_{G}(M_{2}).

(v) If 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence then for i = 1, 2, 3, V_{G}(M_{i}) is contained in the union of the supports of the other two modules.

(vi) V_{G}(Γ_{V}(k)) = V ∖{𝔪}, V_{G}(L_{V}(k)) = V_{G} ∖V and V_{G}(Γ_{𝔭}(k)) = {𝔭} for 𝔭 ≠ 𝔪.

(vii) Given a subset V ⊆ V_{G}, there exists a module M with V_{G}(M) = V. For example we could take M = ⨁_{𝔭∈V} Γ_{𝔭}(k).

The proof of the tensor product theorem given in [6] depends on comparison with rank varieties and the following version of Dade’s lemma for modules in StMod(kG), which appeared as Theorem 5.2 of [5].

Theorem 4.4. Let E = ⟨g_{1}, …, g_{r}⟩ ≅ (Z/p)^{r}, let k be an algebraically closed field of characteristic p, and let K be an algebraically closed extension field of k of transcendence degree at least r − 1. Then a kE-module M is projective if and only if for every choice of (λ_{1}, …, λ_{r}) ∈ K^{r} with not all the λ_{i} equal to zero, the restriction of K ⊗_{k} M to the cyclic subgroup of KE of order p generated by

1 + λ_{1}(g_{1} − 1) + ⋯ + λ_{r}(g_{r} − 1)

is projective.

The role of this version of Dade’s lemma in the development given in [5, 6] is what necessitates the requirement that k is algebraically closed. Later we shall describe another proof avoiding rank varieties and avoiding any version of Dade’s lemma, and which works for a general field k.

Remark 4.5 (Maximal versus prime ideals). The reader will have observed that when we were dealing with the stable category stmod(kG) of finitely generated modules, we used V_{G}, the spectrum of maximal ideals in H^{∗}(G, k), whereas for the stable category StMod(kG) of all modules we used V_{G}, the spectrum of homogeneous prime ideals in H^{∗}(G, k). In the case of a finitely generated module, V_{G}(M) is determined by V_{G}(M), see Theorem 4.3 (ii), whereas for an infinitely generated module it is not, see part (vii) of the same theorem.

It would have been possible to use V_{G} consistently throughout, but we chose not to, partly for historical reasons. The origin of the use of V_{G} is Quillen’s work [27, 28], and much of the literature on finitely generated modules has been written in this context.

A theorem of Hilbert states that for the maximal ideal spectrum and the homogeneous prime ideal spectrum of a finitely generated commutative algebras over a field, each determines the other. A graded commutative ring is commutative modulo its nil radical, so this applies here.

The homogeneous prime spectrum of a nonstandardly graded ring can be quite confusing. For example a polynomial ring can give a spectrum with singularities even though it is regular in the commutative algebra sense. An explicit example of this is the ring k[x, y, z] with |x| = 2, |y| = 4, |z| = 6. The affine open patch corresponding to y ≠ 0 has coordinate ring generated by α = x^{2}/y, β = xz/y^{2}, γ = z^{2}/y^{3}, with the single relation αγ = β^{2}. This has a singularity at the origin. This example arises as H^{∗}(G, k) modulo its nil radical with G = (Z/p)^{3} ⋊ Σ_{3} for p ≥ 5, and k a field of characteristic p.

Finally, in the Quillen stratification theorem as described in [27, 28] (see also Section 15 of this survey), part of the statement is that N_{G}(E)/C_{G}(E) acts freely on the stratum corresponding to E. This is true for inhomogeneous maximal, but not for homogeneous prime ideals. Fortunately, our use of Quillen stratification does not involve this feature.

5. Classification of tensor ideal thick subcategories

The key step in the classification of tensor ideal thick subcategories of stmod(kG) (Theorem 2.7) is the following theorem.

Theorem 5.1. Let M be a finitely generated kG-module with k algebraically closed, and let V be the collection of closed homogeneous irreducible subvarieties of V_{G}(M). Then the tensor ideal thick subcategory generated by M is equal to 𝔠_{V}.

Proof. Write 𝔠 for the tensor ideal thick subcategory generated by M. It is clear that 𝔠 is contained in 𝔠_{V}. So the natural transformation Γ_{𝔠}Γ_{V} → Γ_{𝔠} is an isomorphism. So if N is any kG-module then we obtain a triangle

Γ_{𝔠}(N) → Γ_{V}(N) → L_{𝔠}Γ_{V}(N).

The first two terms in this triangle are in Loc(𝔠_{V}), and hence so is the third. So

V_{G}(L_{𝔠}Γ_{V}(N)) ⊆ V.

If S is any simple kG-module then

\underline{Hom}_{kG}(S, M^{∗} ⊗_{k} L_{𝔠}Γ_{V}(N)) ≅ \underline{Hom}_{kG}(S ⊗_{k} M, L_{𝔠}Γ_{V}(N)) = 0.

Thus M^{∗} ⊗_{k} L_{𝔠}Γ_{V}(N) is projective, and hence

∅ = V_{G}(M^{∗} ⊗_{k} L_{𝔠}Γ_{V}(N)) = V_{G}(M) ∩ V_{G}(L_{𝔠}Γ_{V}(N)) = V_{G}(L_{𝔠}Γ_{V}(N)).

It follows that L_{𝔠}Γ_{V}(N) is projective, and so Γ_{𝔠}(N) → Γ_{V}(N) is a stable isomorphism. Thus 𝔠 = 𝔠_{V}.

6. Localising subcategories of StMod (kG)

First we describe the goal. As with the thick subcategories of stmod(kG), we only hope to classify the tensor ideal localising subcategories of StMod(kG); this is all localising subcategories only in the case where G is a p-group.

There are a lot more tensor ideal localising subcategories of StMod(kG) than tensor ideal thick subcategories of stmod(kG), as we can see by comparing Theorems 2.2 and 4.3. For any subset V, not just for a specialisation closed one, let C_{V} be the full subcategory of StMod(kG) consisting of those modules M with V_{G}(M) ⊆ V. The following lemma is the analogue of Lemma 2.4.

Lemma 6.1. If V is a subset of V_{G} then C_{V} is a tensor ideal localising subcategory of StMod(kG).

The statement of the classification theorem is as follows.

Theorem 6.2 (Benson, Iyengar and Krause [9]). There is a one to one correspondence between tensor ideal localising subcategories of StMod(kG) and subsets of V_{G} ∖ {0}. Under this correspondence, V corresponds to C_{V}.

The proof of this classification theorem is much harder than the corresponding proof for tensor ideal thick subcategories of stmod(kG). If one tries to mimic the arguments of Neeman [25], there is a basic obstruction, which is the lack of appropriate field objects. Specifically, given a prime 𝔭 ⊆ H^{∗}(G, k), a field object for 𝔭 would be a module M such that Ĥ^{∗}(G, M) = \underline{Hom}_{kG}^{∗}(k, M) is isomorphic to the graded field of fractions of H^{∗}(G, k)/𝔭. Such an object does not always exist, as can be seen using the obstruction theory of Benson, Krause and Schwede [14, 15].

There is one case where there are field objects, and that is the case of an elementary abelian 2-group E. The point here is that the group algebra of an elementary abelian 2-group is the same as an exterior algebra in characteristic two. The basic property of the cohomology in this case is that it is “formal”, in the sense that the cochains and the cohomology are equivalent—there is no higher order information. This is made precise by the Bernstein–Gelfand–Gelfand (BGG) correspondence [16], which gives a correspondence between appropriate categories of modules for the exterior algebra and for a polynomial algebra, where the polynomial algebra is regarded as the cohomology of the exterior algebra and vice versa. In this case, one can perform the classification for the graded polynomial ring H^{∗}(E, k) and then use the BGG correspondence to prove the classification for StMod(kE); see [12] for details.

For a general finite group in characteristic two, the classification can be achieved using the Quillen stratification theorem and Chouinard’s theorem to reduce to elementary abelian subgroups. We shall describe these later.

If E is an elementary abelian p-group for p odd, the problem is that the cochains are not formal. However, there is a device coming from commutative algebra that allows us to reduce to the formal case. Namely, we regard the group algebra kE as a complete intersection, and then use a suitable Koszul complex. We shall describe this construction later.

Before addressing these points, we wish to set up a slightly cleaner version of the module category, where the origin {0} ∈ V_{G} is not excluded, as it is in Theorems 2.7 and 6.2. This is the category KInj(kG) described in the next section. It bears the same relation to StMod(kG) as the bounded derived category D^{b}(mod(kG)) does to stmod(kG). It might be thought that D(Mod(kG)) would play this role, but the compact objects in D(Mod(kG)) are just the perfect complexes, namely the complexes quasiisomorphic to finite complexes of finitely generated projective modules. So we would get K^{b}(proj(kG)) rather than the desired D^{b}(mod(kG)).

7. The category KInj(kG)

The objects of KInj(kG) are the complexes of injective (or equivalently projective, see Lemma 1.1) kG-modules. We should emphasise that this means unbounded complexes of not necessarily finitely generated injective modules. The arrows are homotopy classes of degree preserving maps of complexes. This is a triangulated category in which the triangles come from the mapping cone construction. This category is investigated in detail in Krause [24], Benson and Krause [13].

Let K_{ac}Inj(kG) be the full subcategory of KInj(kG) whose objects are the acyclic complexes. These objects can be described as Tate resolutions of modules.

Definition 7.1. If M is a kG-module then a Tate resolution of M is formed by splicing together an injective resolution and a projective resolution of M:

\xymatrix{⋯ \ar[r] & P_{1} \ar[r] & P_{0} \ar[rr] \ar[dr] &  & I_{0} \ar[r] & I_{1} \ar[r] & ⋯ \\  &  &  & M \ar@{-->}[ur] \ar[dr] \\  &  & 0 \ar@{-->}[ur] &  & 0}

Every acyclic complex of injectives

⋯ → P_{1} → P_{0} → P_{−1} → P_{−2} → ⋯

can be regarded as a Tate resolution of the image of P_{0} → P_{−1}. Furthermore, given a module homomorphism M → M′, it can be extended to a map of Tate resolutions. Such extensions are homotopic if and only if the homomorphisms differ by an element of PHom_{kG}(M, M′). It follows that Tate resolutions give an equivalence of categories between StMod(kG) and K_{ac}Inj(kG).

We write tk for a Tate resolution of the trivial module k, as an object in KInj(kG), ik for an injective resolution, and pk for a projective resolution. So there is a triangle in KInj(kG) of the form

pk → ik → tk

expressing tk as the mapping cone of the map pk → ik. We can make projective, injective and Tate resolutions of any module by tensoring:

M ⊗_{k} pk → M ⊗_{k} ik → M ⊗_{k} tk.

Tensor products in KInj(kG) are taken to be tensor products over k of complexes of modules. Note that ik is the tensor identity of KInj(kG).

Now K_{ac}Inj(kG) is a localising subcategory of KInj(kG), and the quotient category KInj(kG)/ K_{ac}Inj(kG) is the unbounded derived category D(Mod(kG)). The inclusion

K_{ac}Inj(kG) → KInj(kG)

and the quotient functor

KInj(kG) → D(Mod(kG))

each have both a left and a right adjoint, so that we get a diagram of categories and functors

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

The compact objects in these categories are only preserved by the left adjoints, and give us back Rickard’s sequence

stmod(kG) ← D^{b}(mod(kG)) ← K^{b}(proj(kG)).

Lemma 7.2. The only tensor ideal localising subcategories of the unbounded derived category D(Mod(kG)) are zero and the whole category.

Proof. Let C be a non-zero tensor ideal localising subcategory. If X is a non-zero object in C then X has non-zero homology. The homology of kG ⊗_{k} X is thus non-zero and free. Any summand isomorphic to kG of the homology splits off as a summand of kG ⊗_{k} X, and so kG is in C. But kG generates the whole of D(Mod(kG)).

It follows from this lemma that corresponding to any tensor ideal localising subcategory of StMod(kG), there are two tensor ideal localising subcategories of KInj(kG), one of which is contained in K_{ac}Inj(kG) and the other of which is generated by this together with the image of D(Mod(kG)) under − ⊗_{k} pk.

Correspondingly, we have a notion of support varieties for objects in KInj(kG). The definitions involve exactly the same definitions of functors Γ_{𝔠} and L_{𝔠} as for StMod(kG), and the only difference is that if X is an object in KInj(kG) then the origin {0} is either in V_{G}(X) or not according as the complex X has homology or not.

This allows us to formulate the following version of the classification theorem for KInj(kG), where the origin has lost its special role.

Theorem 7.3. There is a one to one correspondence between tensor ideal localising subcategories of KInj(kG) and subsets of V_{G}. Under this correspondence, V corresponds to C_{V}.

This is the theorem whose proof we shall outline in the sections to follow.

8. Support for triangulated categories

At this stage, it is appropriate to describe the general setup introduced in [8] for discussing support for objects in triangulated categories. This setup allows us to move from one category to another without too much effort. Then the game plan, which is inspired by the work of Avramov, Buchweitz, Iyengar, and Miller [1], is as follows:

(i) Reduce from a finite group to its elementary abelian subgroups.

(ii) Use a Koszul construction to move from complexes of modules over an elementary abelian group to differential graded modules over a graded exterior algebra.

(iii) Use a version of the BGG correspondence to move from a graded exterior algebra to a graded polynomial algebra.

(iv) Use a version of Neeman’s classification [25] to deal directly with differential graded modules over graded polynomial algebras.

Definition 8.1. Let T be a compactly generated triangulated category with small coproducts. We write Σ for the shift in T and T^{c} for the full subcategory of compact objects in T.

We write Z^{∗}(T) for the graded centre of T. Namely, Z^{∗}(T) is the graded ring whose degree n component Z^{n}(T) is the set of natural transformations

η: Id_{T} → Σ^{n}

satisfying

ηΣ = (−1)^{n}Ση.

Then Z^{∗}(T) is a graded commutative ring, in the sense that for x, y ∈ Z^{∗}(T) we have

yx = (−1)^{|x||y|}xy.

Let R be a graded commutative Noetherian ring. We say that T is an R-linear triangulated category if we are given a homomorphism of graded commutative rings ϕ: R → Z^{∗}(T). This amounts to giving, for each object X in T, a homomorphism of graded rings

ϕ_{X}: R → End_{T}^{∗}(X),

such that for X and Y objects in T, the two induced actions of an element r ∈ R on α ∈ Hom_{T}^{∗}(X, Y) via ϕ_{X} and ϕ_{Y} are related by

ϕ_{Y} (r)α = (−1)^{|r||α|}αϕ_{X}(r).

Example 8.2. If T = StMod(kG) then T^{c} = stmod(kG) and Σ = Ω^{−1}. In this case, we take R = H^{∗}(G, k). Although it might seem more natural to use Tate cohomology Ĥ^{∗}(G, k) = \underline{Hom}_{kG}^{∗}(k, k), the problem is that this ring is usually not Noetherian. This is related to the awkward role of the origin in the theory of the support varieties for StMod(kG).

If T = KInj(kG) then T^{c} = D^{b}(mod(kG)) and Σ is the usual shift. In this case, we also take R = H^{∗}(G, k). In this case, R is just the graded endomorphism ring of the tensor identity ik, and the origin no longer has an awkward role.

We write Spec R for the set of homogeneous prime ideals in R. For each prime 𝔭 ∈ Spec R and each graded R-module M, we denote by M_{𝔭} the graded localisation of M at 𝔭.

If V is a specialisation closed subset of Spec R, we set

T_{V} = {X ∈ T | Hom_{T}^{∗}(C, X)_{𝔭} = 0 for all C ∈ T^{c}, 𝔭 ∈ Spec R ∖ V}.

This is a localising subcategory of T, and there is a localisation functor L_{V}: T → T such that L_{V}(X) = 0 if and only if X is in T_{V}. We then define Γ_{V}(X) by completing X → L_{V}(X) to a triangle:

Γ_{V}(X) → X → L_{V}(X).

Example 8.3. In the case where T is either StMod(kG) or KInj(kG), and R = H^{∗}(G, k), this is exactly Rickard’s triangle for the subset V of Spec H^{∗}(G, k).

As in Section 4, if 𝔭 ∈ Spec R we choose specialisation closed subsets V and W of Spec R satisfying V ⊈ W and V ⊆ W ∪ 𝔭 and define

Γ_{𝔭} = L_{W}Γ_{V} = Γ_{V}L_{W}.

Again, this is independent of choice of V and W with the given properties. Definition 8.4. If X is an object in T then the support of X is the subset

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

This is a subset of

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

Example 8.5. Continuing Example 8.3, this notion of support agrees with V_{G}(X) as defined in Section 4. We have

supp_{H^{∗}(G,k)}(StMod(kG)) = V_{G} ∖ {0},
supp_{H^{∗}(G,k)}(KInj(kG)) = V_{G}.

The following theorem summarises the properties of support for R-linear triangulated categories; confer Theorems 2.2 and 4.3. Proofs can be found in [8]. Theorem 8.6. Let X, Y and Z be objects in T. Then

(i) X = 0 if and only if supp_{R}(X) = ∅. (ii) supp_{R}(ΣX) = supp_{R}(X). (iii) For a small family of objects X_{α} we have

supp_{R}(⨁_{α} X_{α}) = ⋃_{α} supp_{R}(X_{α}).

(iv) If X → Y → Z is a triangle in T then

supp_{R}(Y) ⊆ supp_{R}(X) ∪ supp_{R}(Z).

(v) For V ⊆ Spec R we have

supp_{R}(Γ_{V}(X)) = V ∩ supp_{R}(X)
supp_{R}(L_{V}(X)) = (Spec R ∖ V) ∩ supp_{R}(X).

(vi) If cl(supp_{R}(X)) ∩ supp_{R}(Y) = ∅ then Hom_{T}(X, Y) = 0. Here, cl(V) denotes the specialisation closure of a subset V.

9. Tensor triangulated categories

In this section we consider compactly generated triangulated categories with some extra structure. Namely, we want to consider an internal tensor product

⊗: T × T → T,

exact in each variable, preserving small coproducts, and with a unit 𝟙. It then follows (using the Brown representability theorem) that there are function objects Hom(X, Y) in T with natural isomorphisms

Hom_{T}(X ⊗ Y, Z) ≅ Hom_{T}(X, Hom(Y, Z)).

We define the dual of an object X in T to be

X^{∨} = Hom(X, 𝟙).

Definition 9.1. We say that (T, ⊗, 𝟙) is a tensor triangulated category if the following hold:

(i) T is a compactly generated triangulated category with small coproducts.

(ii) The tensor product ⊗ and unit 𝟙 make T a symmetric monoidal category.

(iii) The tensor product is exact in each variable and preserves small coproducts.

(iv) The unit 𝟙 is compact.

(v) Compact objects are strongly dualisable in the sense that if C is compact and X is any object in T then the canonical map

C^{∨} ⊗ X → Hom(C, X)

is an isomorphism.

See [8, 11] for further discussion of this structure.

Example 9.2. Both StMod(kG) and KInj(kG) are tensor triangulated categories.

The symmetric monoidal structure ensures that the endomorphism ring of the tensor identity End_{T}^{∗}(𝟙) is a graded commutative ring. Given any object X, this ring acts on End_{T}^{∗}(X) via

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

So if R is a graded commutative Noetherian ring, a homomorphism R → End_{T}^{∗}(𝟙) gives T a structure of an R-linear category.

Definition 9.3. We say that the action of R on T is canonical if it arises from a homomorphism R → End_{T}^{∗}(𝟙) as described in the previous paragraph.

Proposition 9.4. If the action of R on T is canonical then for each specialisation closed subset V ⊆ Spec R and each prime 𝔭 ∈ Spec R there are natural isomorphisms

Γ_{V}(X) ≅ X ⊗ Γ_{V}(𝟙),  L_{V}(X) ≅ X ⊗ L_{V}(𝟙),  Γ_{𝔭}(X) ≅ X ⊗ Γ_{𝔭}(𝟙).

10. The local-global principle

If X is an object or collection of objects in a triangulated category T, we write Loc_{T}(X) for the localising subcategory of T generated by X.

Definition 10.1. Let T be an R-linear triangulated category with small coproducts. We say that the local-global principle holds for T if for each object X in T we have

Loc_{T}(X) = Loc_{T}({Γ_{𝔭}(X) | 𝔭 ∈ supp_{R}(T)}).

This condition is usually satisfied, because of the following theorem [10, Corollary 3.5]:

Theorem 10.2. The local-global principle holds provided R has finite Krull dimension.

For a tensor triangulated category T, we modify the definition slightly. If X is an object or collection of objects in T, we write Loc_{T}^{⊗}(X) for the tensor ideal localising subcategory of T generated by X. The following theorem says that for a tensor triangulated category the appropriate analogue of the local-global principle always holds [10, Theorem 7.2]:

Theorem 10.3. Let T be an R-linear tensor triangulated category with canonical R-action. Then for each object X in T the following holds:

Loc_{T}^{⊗}(X) = Loc_{T}^{⊗}({Γ_{𝔭}(X) | 𝔭 ∈ Spec R}).

When the local-global principle holds, the classification of localising subcategories can be achieved one prime at a time. The way to express this is via the following maps:

\{ \begin{matrix}Localising \\ subcategories of T \end{matrix} \} [figure: σ; τ]\{ \begin{matrix}Families (S(𝔭))_{𝔭∈supp_{R}(T)} with S(𝔭) a \\ localising subcategory of Γ_{𝔭}(T) \end{matrix} \}

where σ(S) = (S∩Γ_{𝔭}(T)) and τ(S(𝔭)) = Loc_{T}(S(𝔭) | 𝔭 ∈ supp_{R}(T)). The following is [10, Proposition 3.6].

Theorem 10.4. If the local-global principle holds then the maps σ and τ are mutually inverse bijections.

In other words, specifying a localising subcategory of T is equivalent to specifying a localising subcategory of Γ_{𝔭}(T) for each prime 𝔭 ∈ supp_{R}(T).

Since the tensor ideal version of the local-global principle always holds in a tensor triangulated category, we have the following theorem.

Theorem 10.5. Let T be an R-linear tensor triangulated category with canonical R-action. Then the maps σ and τ above induce mutually inverse bijections between the tensor ideal localising subcategories of T and families (S(𝔭))_{𝔭∈supp_{R}(T)} with S(𝔭) a tensor ideal localising subcategory of Γ_{𝔭}(T).

Remark 10.6. If the tensor identity 𝟙 generates T then every localising subcategory of T is tensor ideal. In the case of StMod(kG) and KInj(kG) this holds if and only if G is a finite p-group.

11. Stratifying triangulated categories

In the last section, we showed how to classify localising subcategories of an R-linear triangulated category T one prime at a time. The easiest case is where each Γ_{𝔭}(T) with 𝔭 ∈ supp_{R}(T), is a minimal subcategory.

Definition 11.1. We say that a localising subcategory of T is minimal if it is non-zero and has no proper non-zero localising subcategories.

We say that T is stratified by R if the local-global principle holds, and each Γ_{𝔭}(T) with 𝔭 ∈ supp_{R} T is a minimal localising subcategory.

Suppose that T is stratified by R, and consider the maps σ and τ of Section 10. To name a family (S(𝔭))_{𝔭∈supp_{R}(T)} with S(𝔭) a localising subcategory of Γ_{𝔭}(T), we just have to name the set of 𝔭 ∈ supp_{R}(T) for which S(𝔭) = Γ_{𝔭}(T), since the remaining primes 𝔭 will satisfy S(𝔭) = 0. So the following is a direct consequence of Theorem 10.4.

Theorem 11.2. Let T be an R-linear triangulated category. If T is stratified by R then the maps σ and τ establish a bijection between the localising subcategories of T and the subsets of supp_{R}(T).

In the case of an R-linear tensor triangulated category T, because the corresponding local-global principle is automatic (Theorem 10.3), we say that T is stratified by R if for each 𝔭 ∈ supp_{R}(T), Γ_{𝔭}(T) is minimal as a tensor ideal localising subcategory.

Theorem 11.3. Let T be an R-linear tensor triangulated category. If T is stratified by R then the maps σ and τ establish a bijection between the tensor ideal localising subcategories of T and the subsets of supp_{R}(T).

In order to prove stratification in a given situation, we need a criterion for minimality of localising subcategories. The following can be found in Lemma 4.1 of [10] and Lemma 3.9 of [9].

Lemma 11.4. (i) Let T be an R-linear triangulated category. A non-zero localising subcategory S of T is minimal if and only if for all non-zero objects X and Y in S we have Hom_{T}^{∗}(X, Y) ≠ 0.

(ii) Let T be an R-linear tensor triangulated category. A non-zero tensor ideal localising subcategory S of T is minimal if and only if for all non-zero objects X and Y in S there exists an object Z in T such that Hom_{T}^{∗}(X ⊗ Z, Y) ≠ 0. The object Z may be taken to be a compact generator for T.

We can now restate Theorems 6.2 and 7.3.

Theorem 11.5. The tensor triangulated categories StMod(kG) and KInj(kG) are stratified by H^{∗}(G, k).

In the next few sections, we outline the proof of this theorem. A lot of details will be swept under the carpet here, but are spelt out in [9].

12. Graded polynomial algebras

The first step in proving Theorem 11.5 is to stratify the derived category of differential graded modules a polynomial ring. Let S be a graded polynomial algebra over the field k on a finite number of indeterminates. If k does not have characteristic two, we assume that the degrees of the indeterminates are even, so that S is graded commutative.

We view S as a differential graded (abbreviated to dg) algebra with zero differential, and we write D(S) for the derived category of dg S-modules. The objects of this category are the dg S-modules, and the morphisms are homotopy classes of degree preserving chain maps with quasi-isomorphisms inverted. See for example Keller [23] for further details.

The category D(S) is a tensor triangulated category in which the tensor product is the derived tensor product over S, X ⊗_{S}^{L} Y and the tensor identity is S, which is a compact generator for D(S). In particular, every localising subcategory is tensor ideal. The canonical action of S on D(S) makes D(S) into an S-linear tensor triangulated category. The following theorem is a dg analogue of the theorem of Neeman [25], and is proved in Theorem 5.2 of [9]. The existence of field objects plays a crucial role in the proof.

Theorem 12.1. The category D(S) is stratified by the canonical S-action. So the maps σ and τ of Section 10 are mutually inverse bijections between the localising subcategories of D(S) and subsets of Spec S.

13. A BGG correspondence

The second step in the proof of Theorem 11.5 is to use a version of the BGG correspondence to transfer the stratification from polynomial rings to exterior algebras.

Let k be a field and let Λ be an exterior algebra on indeterminates ξ_{1}, …, ξ_{c} of negative odd degrees. We view Λ as a dg algebra with zero differential. Let S be a graded polynomial algebra on variables x_{1}, …, x_{c} with |x_{i}| = −|ξ_{i}| + 1. Let J be the dg Λ ⊗_{k} S-module with

J^{♮} = Hom_{k}(Λ, k) ⊗_{k} S

and

d(f ⊗ s) = ∑_{i} ξ_{i}f ⊗ x_{i}s.

If M is a dg Λ-module then Hom_{Λ}(J, M) is in a natural way a dg S-module.

In general, for a dg algebra A, we write A^{♮} for the underlying graded algebra of A, forgetting the differential. If M is a dg A-module, we write M^{♮} for the underlying graded A^{♮}-module. We say that a dg A-module I is graded-injective if I^{♮} is injective in the category of graded A^{♮}-module. Finally, we write KInj(A) for the homotopy category of graded-injective dg A-modules.

The following version of the BGG correspondence [16] extends the one from [1].

Theorem 13.1. The functor

Hom_{Λ}(J, −): KInj(Λ) → D(S)

is an equivalence of categories.

We give Λ a comultiplication

Δ(ξ_{i}) = ξ_{i} ⊗ 1 + 1 ⊗ ξ_{i}.

This allows us to define a tensor product of dg Λ-modules, and this makes KInj(Λ) into a tensor triangulated category. Its tensor identity is ik, which is a compact generator. We have

S ≅ Ext_{Λ}^{∗}(k, k) ≅ Hom_{KInj(Λ)}(J, J)

which makes KInj(Λ) into an S-linear tensor triangulated category with canonical S-action. The equivalence described in Theorem 13.1 allows us to stratify KInj(Λ).

Theorem 13.2. The category KInj(Λ) is stratified by the canonical action of S. So the maps σ and τ of Section 10 give a bijection between localising subcategories of KInj(Λ) and subsets of Spec S.

14. The Koszul construction

If E is an elementary abelian 2-group and k has characteristic two, then kE is an exterior algebra on generators of degree zero. So we can use Theorem 13.2 to stratify KInj(kE). This proves Theorem 11.5 in this case; see [12] for details. Elementary abelian p-groups with p odd cannot be treated this way. In this section, we show how to use a Koszul construction to deal with this case.

So let p be a prime, k be a field of characteristic p, and

E = ⟨g_{1}, …, g_{r}⟩ ≅ (Z/p)^{r}

be an elementary abelian p-group. Let z_{i} = g_{i} − 1 ∈ kE, so that

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

We regard kE as a complete intersection, and form the Koszul construction as follows. Let A be the dg algebra

A = kE⟨y_{1}, …, y_{r}⟩

where kE is in degree zero, and y_{1}, …, y_{r} are exterior generators of degree −1 with

y_{i}^{2} = 0,  y_{i}y_{j} = −y_{j}y_{i},  d(y_{i}) = z_{i},  d(z_{i}) = 0.

Let Λ be an exterior algebra on generators ξ_{1}, …, ξ_{r} of degree −1, regarded as a dg algebra with zero differential, and S be a polynomial algebra on generators x_{1}, …, x_{r} of degree 2, again regarded as dg algebra with zero differential. The following is Lemma 7.1 of [9].

Lemma 14.1. The map ϕ: Λ → A defined by

ϕ(ξ_{i}) = z_{i}^{p−1}y_{i}

is a quasi-isomorphism of dg algebras. In particular,

Ext_{A}^{∗}(k, k) ≅ Ext_{Λ}^{∗}(k, k) ≅ S.

We give A a comultiplication

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

This gives KInj(A) the structure of a tensor triangulated category with a canonical action of S. The following is a special case of Proposition 4.6 of [9].

Theorem 14.2. The map ϕ: Λ → A induces an equivalence of tensor triangulated categories

Hom_{Λ}(A, −): KInj(Λ) → KInj(A).

As a consequence, we can stratify KInj(A). The following is Theorem 7.2 of [9].

Theorem 14.3. The S-linear tensor triangulated category KInj(A) is stratified by the canonical action of S. So the maps σ and τ of Section 10 give a bijection between localising subcategories of KInj(A) and subsets of Spec S.

Now the inclusion map kE → A induces a restriction map

Ext_{A}^{∗}(k, k) → Ext_{kE}^{∗}(k, k).

The structure of Ext_{kE}^{∗}(k, k) is as follows. If p is odd then it is a tensor product of an exterior algebra and a polynomial algebra

Ext_{kE}^{∗}(k, k) ≅ Λ(u_{1}, …, u_{r}) ⊗ k[x_{1}, …, x_{r}]

with |u_{i}| = 1 and |x_{i}| = 2. The elements x_{i} are the restrictions of the elements of the same name in Ext_{A}^{∗}(k, k) ≅ S. If p = 2 then

Ext_{kE}^{∗}(k, k) = k[u_{1}, …, u_{r}]

with |u_{i}| = 1. The elements x_{i} ∈ Ext_{A}^{∗}(k, k) restrict to u_{i}^{2}.

In both cases, this allows us to regard S ≅ Ext_{A}^{∗}(k, k) as a subring of H^{∗}(E, k) over which it is finitely generated as a module. The restriction map from Ext_{A}^{∗}(k, k) to H^{∗}(E, k) = Ext_{kE}^{∗}(k, k) induces a bijection

Spec H^{∗}(E, k) → Spec S.

Using the induction and restriction functors, the criterion of Lemma 11.4 is essential in deducing the following theorem from Theorem 14.3. See Theorem 4.11 of [9] for details.

Theorem 14.4. The tensor triangulated category KInj(kE) is stratified by the canonical action of H^{∗}(E, k), or equivalently of S ⊆ H^{∗}(E, k). So the maps σ and τ of Section 10 give a bijection between localising subcategories of KInj(kE) and subsets of Spec H^{∗}(E, k).

There is an issue here with the tensor structure. The usual diagonal map Δ(g_{i}) = g_{i} ⊗ g_{i} on kE is not compatible with the inclusion kE → A. There is another diagonal map on kE given by Δ(z_{i}) = z_{i} ⊗ 1 + 1 ⊗ z_{i} coming from regarding kE as a restricted universal enveloping algebra, and this diagonal map is compatible with the inclusion. Each diagonal map gives rise to a canonical action of H^{∗}(E, k) on KInj(kE), but the two actions are not the same. Lemma 3.10 of [9] helps us out at this point:

Lemma 14.5. Let T be a triangulated category admitting two tensor triangulated structures with the same unit 𝟙, and assume that 𝟙 generates T. Let

ϕ, ϕ′: R → Z^{∗}(T)

be two actions of R on T. If the maps R → End_{T}^{∗}(𝟙) induced by ϕ and ϕ′ agree then T is stratified by R through ϕ if and only if it is stratified by R through ϕ′.

15. Quillen stratification

There is a general machine for understanding cohomological properties of a general finite group from its elementary abelian subgroups, called Quillen stratification. In this section, we explain how to use this machine to compare localising subcategories of KInj(kE) and KInj(kG).

Let V_{G} = Spec H^{∗}(G, k). If H is a subgroup of G then the restriction map H^{∗}(G, k) → H^{∗}(H, k) induces a map

res_{G,H}^{∗}: V_{H} → V_{G}.

Quillen [27, 28] proved the following. Given a prime 𝔭 ∈ V_{G} we say that 𝔭 originates in an elementary abelian p-subgroup E ≤ G if 𝔭 is in the image of res_{G,E}^{∗} but not of res_{G,E′}^{∗} for E′ a proper subgroup of E.

Theorem 15.1. Given a prime 𝔭 ∈ V_{G}, there exists a pair (E, 𝔮) such that 𝔭 originates in E and res_{G,E}^{∗}(𝔮) = 𝔭, and all such pairs are G-conjugate. This sets up a bijection between primes 𝔭 ∈ V_{G} and conjugacy classes of pairs (E, 𝔮) where 𝔮 ∈ V_{E} originates in E.

In order to be able to use this, we first need a version of the “subgroup theorem” for elementary abelian p-groups. The following is Theorem 9.5 of [9], and its proof is a fairly straightforward consequence of the Stratification Theorem 14.4 for elementary abelian p-groups.

Theorem 15.2. Let E′ ≤ E be elementary abelian p-groups. If X is an object in KInj(kE) then

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

Next, we need a version of Chouinard’s theorem [19] for KInj(kG), see Proposition 9.6 of [9]:

Theorem 15.3. An object X in KInj(kG) is zero if and only if the restriction of X to every elementary abelian p-subgroup of G is zero.

We are now ready to outline the proof of Theorem 11.5. This amounts to showing that for 𝔭 ∈ V_{G}, the tensor ideal localising subcategory Γ_{𝔭} KInj(kG) is minimal. For this purpose, we use the criterion of Lemma 11.4. Let X and Y be non-zero objects in Γ_{𝔭} KInj(kG). By Theorem 15.3 there exists an elementary abelian subgroup E_{0} such that X↓_{E_{0}} is non-zero. Choose a prime 𝔮_{0} ∈ V_{E_{0}}(X↓_{E_{0}}). Using standard properties of support under induction and restriction, we obtain res_{G,E_{0}}^{∗}(𝔮_{0}) = 𝔭. So we can choose a pair (E, 𝔮) with E_{0} ≥ E and 𝔮_{0} = res_{E_{0},E}^{∗}(𝔮), so that the conjugacy class of (E, 𝔮) corresponds to 𝔭 under the bijection of Theorem 15.1. By Theorem 15.2 we have Γ_{𝔮}(X↓_{E}) ≠ 0. Since the pair (E, 𝔮) is determined up to conjugacy by 𝔭, the object Y ∈ Γ_{𝔭} KInj(kG) also has Γ_{𝔮}(Y ↓_{E}) ≠ 0.

Let Z be an injective resolution of the permutation module k(G/E), as an object in KInj(kG). This has the property that

X ⊗_{k} Z ≅ X ⊗_{k} k(G/E) ≅ X↓_{E}↑^{G}.

Thus using Frobenius reciprocity we have

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

Since Γ_{𝔮} KInj(kE) is minimal, using the first part of Lemma 11.4 in one direction shows that the right hand side is non-zero. Using the other part in the other direction then shows that Γ_{𝔭} KInj(kG) is minimal. Thus KInj(kG) is stratified as a tensor triangulated category by H^{∗}(G, k).

16. Applications

In this section, we give some applications of the classification of localising subcategories of StMod(kG) and KInj(kG) (Theorem 11.5). To illustrate the methods, we give the proof in the case of the tensor product theorem. The remaining proofs can be found in Section 11 of [9].

The tensor product theorem. The tensor product theorem states that if X and Y are objects in StMod(kG), or more generally in the larger category KInj(kG), then

V_{G}(X ⊗_{k} Y) = V_{G}(X) ∩ V_{G}(Y).

This was first proved by Benson, Carlson and Rickard [5, Theorem 10.8] for StMod(kG) in the case where k is algebraically closed. The method of proof was to reduce to elementary abelian subgroups and then use the version of Dade’s lemma given in Theorem 4.4.

The proof of the Stratification Theorem 11.5 does not involve any form of Dade’s lemma, and so we get a new proof of the tensor product theorem as follows. Since

Γ_{𝔭}(X ⊗_{k} Y) ≅ Γ_{𝔭}(𝟙) ⊗_{k} X ⊗_{k} Y ≅ Γ_{𝔭}(X) ⊗_{k} Y ≅ X ⊗ Γ_{𝔭}(Y),

if either Γ_{𝔭}(X) or Γ_{𝔭}(Y) is zero then so is Γ_{𝔭}(X ⊗_{k} Y). This shows that

V_{G}(X ⊗_{k} Y) ⊆ V_{G}(X) ∩ V_{G}(Y).

For the reverse containment, suppose that 𝔭 ∈ V_{G}(X) ∩ V_{G}(Y). Thus Γ_{𝔭}(X) ≠ 0 and Γ_{𝔭}(Y) ≠ 0. It follows from Theorem 11.5 and Γ_{𝔭}(X) ≠ 0 that Γ_{𝔭}(𝟙) is in Loc^{⊗}(Γ_{𝔭}(X)), and hence that Γ_{𝔭}(Y) is in Loc^{⊗}(Γ_{𝔭}(X ⊗_{k} Y)). Since Γ_{𝔭}(Y) ≠ 0, this implies that Γ_{𝔭}(X ⊗_{k} Y) ≠ 0.

The subgroup theorem. The Subgroup Theorem 15.2 for elementary abelian groups was proved using the stratification theorem in that context, and was used in order to prove the stratification theorem for general finite groups. The following general version of the subgroup theorem follows in the same way from the stratification theorem for finite groups.

Theorem 16.1. Let H ≤ G be finite groups. If X is an object in KInj(kG) then

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

Thick subcategories. Theorem 11.5 also gives a new proof for the classification of tensor ideal thick subcategories of stmod(kG) (Theorem 2.7) and D^{b}(mod(kG)), avoiding the use of the version of Dade’s lemma given in Theorem 4.4.

Localising subcategories closed under products and duality. We state the following theorem for StMod(kG). A similar statement holds for KInj(kG).

Theorem 16.2. Under the bijection given in Theorem 6.2, the following properties of a tensor ideal localising subcategory C_{V} of StMod(kG) are equivalent.

(i) C_{V} is closed under products.

(ii) V_{G} ∖ V is specialisation closed.

(iii) There exists a set X of finitely generated kG-modules such that C_{V} is the full subcategory of modules M satisfying \underline{Hom}_{kG}(N, M) for all N in X.

(iv) When a kG-module M is in C_{V} so is Hom_{k}(M, k).

The telescope conjecture. Our final application is a statement about certain localising such categories which are defined as follows.

Definition 16.3. A localising subcategory C of a triangulated category T is said to be strictly localising if there is a localisation functor L: T → T such that an object X is in C if and only if L(X) = 0.

A localising subcategory C is said to be smashing if it is strictly localising, and the localisation functor L preserves coproducts.

The following is the analogue for StMod(kG) and KInj(kG) of the telescope conjecture of algebraic topology (Bousfield [17], Ravenel [29, 30]).

Theorem 16.4. A tensor ideal localising subcategory of StMod(kG) or of KInj(kG) is smashing if and only if it is generated by compact objects.

17. Costratification

Let T be a compactly generated triangulated category with small products and coproducts. Recall that a colocalising subcategory of T is a thick subcategory that is closed under products.

If T is tensor triangulated then such a subcategory C is closed under tensor products with simple modules if and only if it is Hom closed, in the sense that for all X in T and all Y in C, the function object (see Section 9) Hom(X, Y) is in C.

The main theorem of [11] classifies the Hom closed colocalising subcategories of StMod(kG) and of KInj(kG).

Theorem 17.1. There is a one to one correspondence between Hom closed colocalising subcategories D of StMod(kG) (respectively KInj(kG)) and tensor ideal localising subcategories C of StMod(kG) (respectively KInj(kG)) given by C = ^{⊥}D, the full subcategory of objects X satisfying Hom^{∗}(X, Y) = 0 for all Y in D.

The proof of this theorem goes via the notions of cosupport and costratification. We explain these concepts and fix to this end an R-linear tensor triangulated category T, as in Section 9. In addition to the axioms listed there, we assume also Hom(?, Y) is exact for each object Y ∈ T; all tensor triangulated categories encountered in this work have this property.

For each prime 𝔭 ∈ Spec R denote by Λ^{𝔭} the right adjoint of the functor Γ_{𝔭} which exists by the Brown representability theorem.

Definition 17.2. If X is an object in T then the cosupport of X is the subset

cosupp_{R}(X) = {𝔭 ∈ Spec R | Λ^{𝔭}(X) ≠ 0}.

Note that Λ^{𝔭} and Γ_{𝔭} provide mutually inverse equivalences between Γ_{𝔭}(T) and Λ^{𝔭}(T). Thus cosupp_{R}(X) is a subset of supp_{R}(T).

We say that the tensor triangulated category T is costratified by R if for each 𝔭 ∈ Spec R the category Λ^{𝔭}(T) is either zero or minimal among all Hom closed colocalising subcategories of T.

As for localising subcategories, the classification of colocalising subcategories can be achieved one prime at a time. The way to express this is via the following maps:

\{ \begin{matrix}Colocalising \\ subcategories of T \end{matrix} \} [figure: σ; τ]\{ \begin{matrix}Families (S(𝔭))_{𝔭∈supp_{R}(T)} with S(𝔭) a \\ colocalising subcategory of Λ^{𝔭}(T) \end{matrix} \}

where σ(S) = (S ∩ Λ^{𝔭}(T)) and τ(S(𝔭)) = Coloc_{T}(S(𝔭) | 𝔭 ∈ supp_{T} (T)). The following is [11, Corollary 9.2].

Theorem 17.3. Let T be an R-linear tensor triangulated category. If T is costratified by R then the maps σ and τ establish a bijection between the Hom closed colocalising subcategories of T and the subsets of supp_{R}(T).

From this result one deduces Theorem 17.1 by proving costratification first for KInj(kG) (see [11, Theorem 11.10]) and then for StMod(kG) (see [11, Theorem 11.13]).

Let us include an application which justifies the study of support and cosupport; it is [11, Corollary 9.6].

Theorem 17.4. Suppose the tensor triangulated category T is generated by its unit. Then T is stratified by R if and only if for all objects X and Y in T one has

Hom_{T}^{∗}(X, Y) = 0 ⟺ supp_{R}(X) ∩ cosupp_{R}(Y) = ∅.

Note that StMod(kG) (respectively KInj(kG)) is generated by its unit if and only if G is a p-group. We refer to [11] for more general results which do not depend on the fact that T is generated by its unit.

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David J. Benson, Institute of Mathematics, Fraser Noble Building, University of Aberdeen, Aberdeen AB24 3UE, UK E-mail: bensondj@maths.abdn.ac.uk

Srikanth B. Iyengar, Department of Mathematics, University of Nebraska, Lincoln NE 68588-0130, USA E-mail: iyengar@math.unl.edu

Henning Krause, Universität Bielefeld, Fakultät für Mathematik, Postfach 100 131, D-33 501 Bielefeld, Germany E-mail: hkrause@math.uni-bielefeld.de