THICK SUBCATEGORIES OF THE STABLE MODULE CATEGORY

D. J. BENSON, JON F. CARLSON, AND JEREMY RICKARD

Abstract. We study the thick subcategories of the stable category of finitely generated modules for the principal block of the group algebra of a finite group G over a field of characteristic p. In case G is a p-group we obtain a complete classification of the thick subcategories. The same classification works whenever the nucleus of the cohomology variety is zero. In case the nucleus is nonzero, we describe some examples which lead us to believe that there are always infinitely many thick subcategories concentrated on each nonzero closed homogeneous subvariety of the nucleus.

1. Introduction

A subcategory of a triangulated category is said to be thick, or épaisse, if it is a triangulated subcategory and it is closed under taking direct summands. One product of the deep work of Devinatz, Hopkins and Smith [8] on stable homotopy theory was a classification of the thick subcategories of the stable homotopy category. This is described in Hopkins’ survey [11], where he also states a corresponding classification of thick subcategories of the homotopy category of bounded chain complexes of finitely generated projective Rmodules for a commutative ring R. In fact, this classification requires R to be Noetherian, as Neeman pointed out in [13], where he also gave a complete proof.

Recent work in the modular representation theory of finite groups [3, 4, 5, 18] has exploited the analogy between the stable homotopy category of algebraic topology and the stable module category of a finite group, which is also a triangulated category. Here we shall take this analogy further, and we attempt to find a classification of thick subcategories of the stable module category stmod(kG) of finitely generated representations of a finite group G over a field k. Our attempt is successful when G is a p-group, and in Theorem 3.5 we give a complete classification in terms of varieties of modules. For general groups we are not so successful, but we do get partial results in Section 5. In Section 6 we present some phenomenology in situations where our partial results do not apply, which seems to indicate that it will be difficult even to conjecture what a complete classification might be like.

Date: December 1996.

The first two authors are partly supported by grants from the NSF.

MR Classification: 20C20, 20J06.

Throughout this paper k will be a field of characteristic p > 0. For convenience, we shall assume that k is algebraically closed.

Recall that the stable module category stmod(kG) has the finitely generated kG-modules as its objects, and a map in this category between modules M and N is an equivalence class of module homomorphisms, where α, β : M ⟶ N are equivalent if the difference α − β factors through a projective module. The category StMod(kG) is defined similarly, except that all modules, not just finitely generated ones, are allowed. The space of maps between modules M and N in either of these categories is denoted by \underline{Hom}_{kG}(M, N). Readers not familiar with these categories may wish to consult [4, Section 2] and [18, Section 3].

2. Idempotent functors and varieties for infinite dimensional modules

We shall be using recent work on ‘idempotent functors’ and on varieties for infinite dimensional modules, so in this section we shall summarize the facts (from [18], [4] and [5]) that will be needed.

Let C be a thick subcategory of stmod(kG) for some finite group G, and let C^{⊕} be the smallest thick subcategory of StMod(kG) that contains C and is closed under arbitrary direct sums. As noted in the preamble to [18, Section 5], the finitely generated modules in C^{⊕} are precisely those in C.

Given a kG-module X, Section 5 of [18] describes how the techniques involved in the Brown Representability Theorem of algebraic topology can be used to construct a distinguished triangle

T_{C}(X) : E_{C}(X) ⟶ X ⟶ F_{C}(X) ⟶ Ω^{−1}(E_{C}(X))

in StMod(kG). Let us collect together the properties of this triangle that will be important to us.

Proposition 2.1. (a) The map E_{C}(X) ⟶ X is the universal map in StMod(kG) from an object of C^{⊕} to X.

(b) The map X ⟶ F_{C}(X) is the universal map in StMod(kG) from X to a C-local object of StMod(kG) (i.e., an object L for which \underline{Hom}_{kG}(C, L) = 0 for every object C of C).

(c) The triangle T_{C}(X) is natural in X. In particular E_{C} and F_{C} are functors from StMod(kG) to itself.

(d) The functors E_{C}, F_{C} : StMod(kG) ⟶ StMod(kG) are exact (i.e., they commute with Ω up to natural isomorphism and send distinguished triangles to distinguished triangles).

(e) If C and D are two thick subcategories of stmod(kG) with C ⊆ D, then the map E_{C}E_{D}(X) ⟶ E_{C}(X), obtained by applying the functor E_{C} to the first map in the triangle T_{D}(X), is an isomorphism in StMod(kG).

(f) If C and D are two thick subcategories of stmod(kG), then C = D if and only if E_{C}(X) ≅ E_{D}(X) for every kG-module X.

(g) If C has the property that C ⊗ X is in C for all objects C in C and X in stmod(kG), then for any object X in stmod(kG), T_{C}(X) is isomorphic to the triangle obtained by tensoring T_{C}(k) with X.

Proof. Parts (a) and (b) are Propositions 5.6 and 5.7 of [18], and (c) follows immediately.

Left and right adjoints to exact functors between triangulated categories are also exact (see, for example, [12, Lemma 3.9]), and so (d) follows, since E_{C} is right adjoint to the inclusion of C^{⊕} into StMod(kG) and F_{C} is left adjoint to the inclusion of the full subcategory of C-local objects into StMod(kG).

If C ⊆ D, then every D-local object is also C-local. In particular this is true of F_{D}(X), and so E_{C}F_{D}(X) ≅ 0. The third object in the triangle E_{C}(T_{D}(X)) is therefore zero, and so the first map, E_{C}E_{D}(X) ⟶ E_{C}(X), is an isomorphism. Therefore (e) is true.

If C = D, then certainly E_{C}(X) ≅ E_{D}(X) for any X. Conversely, if E_{C}(X) ≅ E_{D}(X) for every X, then

X is in C^{⊕} ⇔ X ≅ E_{C}(X)
⇔ X ≅ E_{D}(X)
⇔ X is in D^{⊕}.

Thus C^{⊕} = D^{⊕}, and so, considering the finitely generated modules in these categories, C = D. So (f) is true.

Part (g) is Proposition 5.13 of [18].

The modules E_{C}(X) and F_{C}(X) will usually not be finitely generated, even if X is, and so the use of idempotent functors forces us to consider infinitely generated modules. For this reason, we shall need to use the theory of ‘varieties’ for infinitely generated modules, which was recently developed in [4, 5]. So let us now collect together the facts that we need from this theory.

We shall assume the reader has some familiarity with the theory of varieties for finitely generated modules, where a closed homogeneous subvariety V_{G}(M) of the maximal ideal spectrum V_{G}(k) of the cohomology ring H^{∗}(G, k) is associated to each finitely generated kG-module M. A fairly comprehensive exposition can be found in Chapter 5 of [2].

In what seems to be the most useful generalization of this theory to modules that are not necessarily finitely generated, the ‘variety’ of a module M is not really a variety, but a set V_{G}(M) of closed homogeneous irreducible nonzero subvarieties of V_{G}(k) (or, equivalently, a subset of the scheme Proj H^{∗}(G, k)). We refer to [5, Definition 10.2] for the precise definition of V_{G}(M); here we just recall some of the more important properties.

Proposition 2.2. (a) The ‘variety’ V_{G}(k) of the trivial kG-module k is the set of all closed homogeneous irreducible nonzero subvarieties of V_{G}(k).

(b) A kG-module M is projective if and only if V_{G}(M) = ∅.

(c) If M is a finitely generated kG-module, then

V_{G}(M) = {V ∈ V_{G}(k) : V ⊆ V_{G}(M)}.

(d) If

M_{1} ⟶ M_{2} ⟶ M_{3} ⟶ Ω^{−1}(M_{1})

is a distinguished triangle in StMod(kG), then

V_{G}(M_{i}) ⊆ V_{G}(M_{j}) ∪ V_{G}(M_{k})

for {i, j, k} = {1, 2, 3}.

(e) For any family {M_{i} : i ∈ I} of kG-modules,

V_{G}(⨁_{i∈I} M_{i}) = ⋃_{i∈I}(V_{G}(M_{i})).

(f) (Tensor Product Theorem) For all kG-modules M and N,

V_{G}(M ⊗_{k} N) = V_{G}(M) ∩ V_{G}(N).

(g) Let W be a closed homogeneous subvariety of V_{G}(k), and let C(W) be the thick subcategory of stmod(kG) consisting of the finitely generated modules M with V_{G}(M) ⊆ W. If a kG-module N is in C(W)^{⊕}, then

V_{G}(N) ⊆ {V ∈ V_{G}(k) : V ⊆ W }.

Proof. All of these facts follow easily from what is contained in [5].

As noted after Definition 10.2 of [5], (a) follows immediately from the definition and [5, Lemma 10.2 (ii)].

Also implicit in [5] is (b). For an elementary abelian p-group, it follows from [5, Corollary 5.6 and Theorem 10.5]. For a general group G, it then follows by Chouinard’s Theorem [7] and [5, Theorem 10.6].

Using [5, Theorem 10.6] again, the proof of (c) can be reduced to the case of an elementary abelian p-group, when it follows from [5, Remark 5.5 (ii) and Theorem 10.5].

Both (d) and (e) can be proved using [5, Lemma 10.3], which says that for any V ∈ V_{G}(k) there is a certain module κ(V) such that, for any kG-module M, V ∈ V_{G}(M) if and only if κ(V) ⊗_{k} M is not projective.

The Tensor Product Theorem (f) is Theorem 10.8 of [5].

Finally, (g) is an immediate consequence of (d) and (e).

3. Thick subcategories with ideal closure

If G is a finite group and W is a closed homogeneous subvariety of the maximal ideal spectrum V_{G}(k) of H^{∗}(G, k), then one of the more interesting thick subcategories of stmod(kG) is C(W), the full subcategory consisting of modules whose varieties are contained in W. More general examples of thick subcategories can be defined in terms of varieties as follows.

Definition 3.1. Let G be a finite group, and let X be a nonempty set of closed homogeneous subvarieties of V_{G}(k) that is closed under specialization (i.e., if W ∈ X and W′ ⊆ W, then W′ ∈ X) and finite unions. Then C(X) is the thick subcategory of stmod(kG) consisting of modules M with V_{G}(M) ∈ X.

It follows easily from standard properties of varieties for modules that C(X) is indeed a thick subcategory. If X has a maximal element W, then C(X) coincides with the thick subcategory C(W) described above. Also, if X is the set of all closed homogeneous subvarieties of V_{G}(k) of dimension less than c for some positive integer c, then C(X) is just the thick subcategory of all finitely generated modules with complexity less than c. Many familiar examples of thick subcategories are therefore subsumed by this definition.

We now consider thick subcategories of stmod(kG) which have ideal closure as defined below. It will be shown that every such thick subcategory has of the form C(X) for some X. Most importantly, all thick subcategories of stmod(kG) for G a p-group have ideal closure, and hence we have a complete classification of the thick subcategories in this case.

Definition 3.2. A thick subcategory C of stmod(kG) is said to have ideal closure if C ⊗ X is in C for any C in C and X in stmod(kG).

The following proposition is the key step in the proof of the main results of the section.

Proposition 3.3. Let M be a finitely generated kG-module, and let W = V_{G}(M). Let ⟨⟨M⟩⟩ be the thick subcategory of stmod(kP) generated by M ⊗ X for all X in stmod(kG). Then ⟨⟨M⟩⟩ is equal to C(W).

Proof. Since M is in C(W), it is clear that ⟨⟨M⟩⟩ is contained in C(W). Therefore by Proposition 2.1 the functors E_{⟨⟨M⟩⟩}E_{W} and E_{⟨⟨M⟩⟩} are isomorphic. Hence, for any kG-module N the distinguished triangle T_{⟨⟨M⟩⟩}(E_{W} (N)) is of the form

E_{⟨⟨M⟩⟩}(N) ⟶ E_{W} (N) ⟶ F_{⟨⟨M⟩⟩}E_{W} (N) ⟶ Ω^{−1}(E_{⟨⟨M⟩⟩}(N)).

We shall prove that F_{⟨⟨M⟩⟩}E_{W} (N) is projective, and so the first map in this triangle is a stable isomorphism between E_{⟨⟨M⟩⟩}(N) and E_{W} (N).

The first two terms of the triangle T_{⟨⟨M⟩⟩}(E_{W} (N)) are in C(W)^{⊕}, and therefore so is F_{⟨⟨M⟩⟩}E_{W} (N). Hence,

V_{G}(F_{⟨⟨M⟩⟩}E_{W} (N)) ⊆ {V ∈ V_{G}(k) : V ⊆ W } = V_{G}(M),

by Proposition 2.2.

Since F_{⟨⟨M⟩⟩}E_{W} (N) is ⟨⟨M⟩⟩-local and M is finitely generated,

0 = \underline{Hom}_{kG}(S ⊗ M, F_{⟨⟨M⟩⟩}E_{W} (N)) ≅ \underline{Hom}_{kG}(S, M^{∗} ⊗_{k} F_{⟨⟨M⟩⟩}E_{W} (N))

for all simple kG-modules S (and where M^{∗} denotes the k-linear dual of M). This implies that M^{∗} ⊗_{k} F_{⟨⟨M⟩⟩}E_{W} (N) is projective, and so, by the tensor product theorem for varieties (see Proposition 2.2),

∅ = V_{G}(M^{∗} ⊗_{k} F_{⟨⟨M⟩⟩}E_{W} (N))
= V_{G}(M) ∩ V_{G}(F_{⟨⟨M⟩⟩}E_{W} (N))
= V_{G}(F_{⟨⟨M⟩⟩}E_{W} (N)).

Thus F_{⟨⟨M⟩⟩}E_{W} (N) is projective, and so E_{⟨⟨M⟩⟩}(N) and E_{W} (N) are stably isomorphic. Hence C(W) and ⟨⟨M⟩⟩ are equal, by Proposition 2.1.

The classification of thick subcategories with ideal closure in stmod(kG) now follows easily.

Theorem 3.4. Suppose that C is a thick subcategory with ideal closure of stmod(kG). Then C = C(X) for some nonempty set X of closed homogeneous subvarieties of V_{G}(k) closed under specialization and finite unions.

Proof. Suppose that X is the set of subvarieties of V_{G}(k) that occur as V_{G}(M) for some object M of C. Then X is nonempty, since the zero module is in C, and is closed under finite unions because C is closed under finite direct sums. It is well-known that every closed homogeneous subvariety W′ of V_{G}(k) occurs as the variety of some finitely generated kG-module L (see, for example, [2, Corollary 5.9.2]). Therefore, if W′ is a subvariety of V_{G}(M) for some M in C, then M ⊗_{k} L has variety W′ and is in the thick subcategory ⟨⟨M⟩⟩. In particular it is in C. Hence X is closed under specialization.

Clearly C is contained in C(X). But also, for each object N of C(X), there is an object N′ of C with V_{G}(N′) = V_{G}(N). By Proposition 3.3, N and N′ generate the same thick subcategory, and so N must be in C. So C = C(X).

Corollary 3.5. Let P be a p-group, and let C be a thick subcategory of stmod(kP). Then C = C(X) for some nonempty set X of closed homogeneous subvarieties of V_{P} (k) closed under specialization and finite unions.

Proof. The important thing to note here is that stmod(kP) = ⟨k⟩, the thick subcategory generated by the trivial module k. Also if C is an object in C and X is in ⟨k⟩ then C ⊗ X is in C. So C has ideal closure.

4. The nucleus and modules with no cohomology

For a general finite group G, the modules in the principal block form a proper thick subcategory of stmod(kG). More interestingly, perhaps, the modules M in the principal block for which the Tate cohomology Ĥ^{∗}(G, M) vanishes in all degrees form a thick subcategory that is not usually trivial. Such modules were studied by in [6], where several conjectures were proposed that were settled recently in [3] using some of the theory of idempotent functors described in Section 2. In the next section we shall need to use this work, so let us give a brief summary of some of the important points.

One of the important theorems is a classification of the finite groups G for which Ĥ^{∗}(G, M) ≠ 0 for all nonprojective modules M in the principal block. In [6, Section 2], the term trivial homology module is introduced. It follows immediately from what is proved in [6, Sections 2 and 3] about these modules that they are precisely the modules that are in the triangulated subcategory of stmod(kG) generated by the trivial module. In [3, Theorem 1.4] the following result was proved, which had previously been established in odd characteristic in [6].

Proposition 4.1. Let G be a finite group. Then the following are equivalent.

(a) The full subcategory of stmod(kG) consisting of modules in the principal block is generated, as a triangulated category, by the trivial module k.

(b) If M is a finitely generated kG-module for which Ĥ^{∗}(G, M) = 0, then no nonprojective direct summand of M belongs to the principal block.

(c) Every element of order p in G has a p-nilpotent centralizer.

In fact, [3, Theorem 1.4] gives twelve equivalent conditions.

To say something about groups that do not satisfy the conditions of Proposition 4.1, Benson, Carlson and Robinson introduced a closed homogeneous subvariety Y_{G} of V_{G}(k) called the nucleus. Let us recall the definition.

Definition 4.2 ([6]). Let G be a finite group. The nucleus Y_{G} is the union of the images of the restriction maps

res_{G,H}^{∗} : V_{H}(k) ⟶ V_{G}(k),

as H runs over the set of subgroups of G for which the centralizer C_{G}(H) is not p-nilpotent. (Except that if G is p-nilpotent, then Y_{G} should be taken to be {0} rather than ∅.)

Note that Y_{G} = {0} if and only if the equivalent conditions of Proposition 4.1 hold.

The nucleus was introduced in order to state the following result, conjectured in [6] and proved in [3].

Proposition 4.3. Let G be a finite group. Every finitely generated module in the principal block B_{0}(kG) is contained in ⟨k, C(Y_{G})⟩, the thick subcategory of stmod(kG) generated by the trivial module k together with all the modules whose variety is contained in the nucleus.

The thick subcategory ⟨k, C(Y_{G})⟩ consists of the direct summands of what are called nuclear homology modules in [6, 3], so this is just a restatement of [3, Theorem 1.2].

Proposition 4.4. Let G be a finite group. If M is a module in the principal block such that Ĥ^{∗}(G, M) = 0, then M is in C(Y_{G})^{⊕}.

If M is finitely generated, then the conclusion is just that V_{G}(M) ⊆ Y_{G}, and this was shown in [6, Corollary 10.12] to be a consequence of Proposition 4.3. We shall give a proof using idempotent functors that does not require M to be finitely generated.

Proof. Consider the distinguished triangle T_{C(Y_{G})}(M). By [18, Proposition 5.13], this is isomorphic to the triangle obtained by taking the tensor product of M with T_{C(Y_{G})}(k). Also, the construction of T_{C(Y_{G})}(k) in [18, Section 6] makes it clear that E_{C(Y_{G})}(k) and F_{C(Y_{G})}(k) are in ⟨k⟩^{⊕}. It follows that E_{C(Y_{G})}(M) and F_{C(Y_{G})}(M) are in ⟨M⟩^{⊕}, and in particular are in the category of ⟨k⟩-local modules of the principal block.

Since F_{C(Y_{G})}(M) is also C(Y_{G})-local, it follows from Proposition 4.3 that it must be projective, and so

M ≅ E_{C(Y_{G})}(M),

which is in C(Y_{G})^{⊕}.

We shall also need the following consequence of Proposition 4.3.

Proposition 4.5. Let G be a finite group, and let M be a finitely generated kG-module in the principal block for which

V_{G}(M) ∩ Y_{G} = {0}.

Then M is in the thick subcategory ⟨k⟩ of stmod(kG) generated by the trivial module k.

Proof. Consider the distinguished triangle

T_{⟨k⟩}(M) : E_{⟨k⟩}(M) ⟶ M ⟶ F_{⟨k⟩}(M) ⟶ Ω^{−1}(E_{⟨k⟩}(M)).

The third term, F_{⟨k⟩}(M), is ⟨k⟩-local, and so, by Propositions 4.4 and 2.2,

V_{G}(F_{⟨k⟩}(M)) ⊆ {W ∈ V_{G}(k) : W ⊆ Y_{G}}.

Since M is finitely generated, it follows from the Tensor Product Theorem of Proposition 2.2 that

V_{G}(Hom_{k}(M, F_{⟨k⟩}(M))) = ∅,

and so

\underline{Hom}_{kG}(M, F_{⟨k⟩}(M)) = 0.

By the universal property of the second map in the triangle T_{⟨k⟩}(M), it follows that F_{⟨k⟩}(M) is projective, and so

M ≅ E_{⟨k⟩}(M),

which is in ⟨k⟩^{⊕}. Since M is finitely generated, it must be in ⟨k⟩.

5. Thick subcategories in the principal block

In this section, we shall give some generalizations of Corollary 3.5, the classification of thick subcategories of stmod(kP) for a p-group P, to more general groups. If G is a group for which kG has more than one block, then the modules belonging to a particular block form a thick subcategory. We shall restrict our attention to modules that belong to the principal block B_{0}(kG), although there are doubtless things that could be said about other blocks. Even with this restriction, the thick subcategories cannot, in general, be classified in terms of varieties in the same way as they could for pgroups: for example, the thick subcategory ⟨k⟩ generated by the trivial module contains modules with all possible varieties, but it does not coincide with the thick subcategory of all modules in the principal block unless the nucleus Y_{G} is trivial. What we shall show is that, in a sense, the nucleus is the only obstruction to a full generalization: if we restrict our attention to thick subcategories containing no nonprojective modules whose variety is in the nucleus, or containing all modules with variety in the nucleus, then we do get a satisfactory classification. In particular, Corollary 3.5 generalizes to groups which have a trivial nucleus.

Since we shall be considering only modules in the principal block, let us introduce the following notation for various thick subcategories.

Definition 5.1. Let G be a finite group.

(a) We shall denote by stmod_{0}(kG) the thick subcategory of stmod(kG) consisting of the finitely generated kG-modules that belong to the principal block B_{0}(kG). Similarly, StMod_{0}(kG) is the thick subcategory of StMod(kG) consisting of all modules that belong to the principal block.

(b) If W is a closed homogeneous subvariety of V_{G}(k), then C_{0}(W) is the thick subcategory of stmod(kG) consisting of modules in C(W) that belong to the principal block.

(c) If X is a nonempty family of closed homogeneous subvarieties of V_{G}(k) that is closed under specialization and finite unions, then C_{0}(X) is the thick subcategory of stmod(kG) consisting of modules in C(X) that belong to the principal block.

Let us start with some easy lemmas regarding idempotent functors and tensor products as applied to modules in the principal block.

Lemma 5.2. Let G be a finite group, let C be a thick subcategory of the category stmod_{0}(kG), and let N be a kG-module. Then E_{C}(N) is in StMod_{0}(kG).

If N is in StMod_{0}(kG), then so is F_{C}(N).

Proof. Since StMod_{0}(kG) contains C and is closed under arbitrary direct sums, it contains C^{⊕}, of which E_{C}(N) is an object.

The first term of the distinguished triangle

E_{C}(N) ⟶ N ⟶ F_{C}(N) ⟶ Ω^{−1}(E_{C}(N))

is in StMod_{0}(kG), and so if either N or F_{C}(N) is in StMod_{0}(kG), so is the other.

Lemma 5.3. Let G be a finite group, and let C be a thick subcategory of StMod_{0}(kG). Let N be a module in C, and let M be a finitely generated kG-module. If M is in the thick subcategory ⟨k⟩ generated by the trivial module, then M ⊗_{k} N and Hom_{k}(M, N) are both in C. In particular, if M is in stmod_{0}(kG) and V_{G}(M) ∩ Y_{G} = {0}, then M ⊗_{k} N and Hom_{k}(M, N) are both in C.

Proof. Let D be the class of finitely generated kG-modules X for which X ⊗_{k} N and Hom_{k}(X, N) are both in C. Then D forms a thick subcategory of stmod(kG) which contains the trivial module k, and so it must contain all of ⟨k⟩. The final statement follows because if M is in stmod_{0}(kG) and V_{G}(M) ∩ Y_{G} = {0}, then M is in ⟨k⟩, by Proposition 4.5.

Let us start by studying thick subcategories of stmod_{0}(kG) consisting entirely of modules whose variety intersects the nucleus Y_{G} trivially. As in the case of p-groups, the key step in the classification of such subcategories consists of identifying the thick subcategory generated by a single module. We have the following analogue of Proposition 3.3.

Proposition 5.4. Let G be a finite group, and let M be a kG-module in stmod_{0}(kG) whose variety V_{G}(M) = W satisfies W ∩ Y_{G} = {0}. Then the thick subcategories ⟨M⟩ and C_{0}(W) coincide.

Proof. If N is any kG-module, the distinguished triangle T_{⟨M⟩}(E_{C_{0}(W)}(N)) has the form

E_{⟨M⟩}(N) ⟶ E_{C_{0}(W)}(N) ⟶ F_{⟨M⟩}E_{C_{0}(W)}(N) ⟶ Ω^{−1}(E_{⟨M⟩}(N)).

As in the proof of Proposition 3.3, we shall show that E_{⟨M⟩}(N) and E_{C_{0}(W)}(N) are stably isomorphic (and so ⟨M⟩ = C_{0}(W), by Proposition 2.1) by showing that F_{⟨M⟩}E_{C_{0}(W)}(N) is projective.

Since the first two terms of the triangle are in C_{0}(W)^{⊕}, so is the third. Therefore F_{⟨M⟩}E_{C_{0}(W)}(N) is in StMod_{0}(kG) and

V_{G}(F_{⟨M⟩}E_{C_{0}(W)}(N)) ⊆ {V ∈ V_{G}(k) : V ⊆ W } = V_{G}(M).

Because F_{⟨M⟩}E_{C_{0}(W)}(N) is ⟨M⟩-local, we have, for every n ∈ ℤ, that

0 = \underline{Hom}_{kG}(Ω^{n}M, F_{⟨M⟩}E_{C_{0}(W)}(N)) ≅ Ĥ^{n}(G, M^{∗} ⊗_{k} F_{⟨M⟩}E_{C_{0}(W)}(N)).

But, by Lemma 5.3, M^{∗} ⊗_{k} F_{⟨M⟩}E_{C_{0}(W)}(N) is in StMod_{0}(kG), its variety is contained in V_{G}(M) by Proposition 2.2(f), and so, by Proposition 4.5, it must be projective. Hence,

∅ = V_{G}(M^{∗} ⊗_{k} F_{⟨M⟩}E_{C_{0}(W)}(N))
= V_{G}(M) ∩ V_{G}(F_{⟨M⟩}E_{C_{0}(W)}(N))
= V_{G}(F_{⟨M⟩}E_{C_{0}(W)}(N)).

Thus F_{⟨M⟩}E_{C_{0}(W)}(N) is projective, as required.

The following partial classification of thick subcategories for a general group G follows from the previous result in the same way that Theorem 3.4 and Corollary 3.5 followed from Proposition 3.3.

Theorem 5.5. Let G be a finite group, and let C be a thick subcategory of stmod_{0}(kG) containing only modules M for which V_{G}(M)∩Y_{G} = {0}. Then C = C_{0}(X) for some nonempty set X of closed homogeneous subvarieties of V_{G}(k), closed under specialization and finite unions, such that

W ∈ X ⇒ W ∩ Y_{G} = {0}.

Using the classification of groups G for which the nucleus Y_{G} is trivial, the following corollary follows immediately.

Corollary 5.6. Let G be a finite group in which the centralizers of all elements of order p are p-nilpotent. Then every thick subcategory of stmod_{0}(kG) is of the form C_{0}(X) for some nonempty set X of closed homogeneous subvarieties of V_{G}(k), closed under specialization and finite unions.

In a slightly different direction, we can use the same methods to classify thick subcategories of stmod_{0}(kG) that contain C_{0}(Y_{G}). Of course, this is equivalent to classifying thick subcategories of the quotient category stmod_{0}(kG)/C_{0}(Y_{G}). The analogue of Proposition 3.3 that we need is as follows.

Proposition 5.7. Let G be a finite group, let M be a kG-module in the category stmod_{0}(kG), and let W = V_{G}(M). Then Y_{0}(M), the thick subcategory of stmod_{0}(kG) generated by C_{0}(Y_{G}) together with M, is equal to C_{0}(Y_{G} ∪W).

Proof. Let us start by choosing some module Y in stmod_{0}(kG) with V_{G}(Y) = Y_{G}.

For any kG-module N, the distinguished triangle T_{Y_{0}(M)}(E_{C_{0}(Y_{G}∪W)}(N)) has the form

E_{Y_{0}(M)}(N) ⟶ E_{C_{0}(Y_{G}∪W)}(N) ⟶ F_{Y_{0}(M)}E_{C_{0}(Y_{G}∪W)}(N) ⟶ Ω^{−1}(E_{Y_{0}(M)}(N)).

As in the proofs of Propositions 3.3 and 5.4, we just have to show that the third term of this triangle is projective.

Certainly V_{G}(F_{Y_{0}(M)}E_{C_{0}(Y_{G}∪W)}(N)) is contained in V_{G}(M ⊕ Y), since this is true of the first two terms of the triangle.

But, since M ⊕ Y is in Y_{0}(M), F_{Y_{0}(M)}E_{C_{0}(Y_{G}∪W)}(N) is (M ⊕ Y)-local, and so, for any n ∈ ℤ,

0 = Hom(Ω^{n}(M ⊕ Y), F_{Y_{0}(M)}E_{C_{0}(Y_{G}∪W)}(N))
≅ Ĥ^{n}(G, (M ⊕ Y)^{∗} ⊗_{k} F_{Y_{0}(M)}E_{C_{0}(Y_{G}∪W)}(N)).

So

V_{G}(F_{Y_{0}(M)}E_{C_{0}(Y_{G}∪W)}(N)) = V_{G}(M ⊕ Y) ∩ V_{G}(F_{Y_{0}(M)}E_{C_{0}(Y_{G}∪W)}(N))
= V_{G}((M ⊕ Y)^{∗} ⊗_{k} F_{Y_{0}(M)}E_{C_{0}(Y_{G}∪W)}(N))
⊆ {V ∈ V_{G}(k) : V ⊆ Y_{G}},

where the final inclusion follows from Proposition 4.4. By Proposition 2.2, then, the module F_{Y_{0}(M)}E_{C_{0}(Y_{G}∪W)}(N) is in C_{0}(Y_{G})^{⊕}, but it is also C_{0}(Y_{G})- local, and so must be projective.

As before, we easily deduce the following classification.

Theorem 5.8. Let G be a finite group, and let C be a thick subcategory of stmod_{0}(kG) that contains C_{0}(Y_{G}). Then C = C_{0}(X) for some set X of closed homogeneous subvarieties of V_{G}(k) that is closed under specialization and finite unions and includes Y_{G}.

6. Inside the nucleus

In this section we shall examine some examples of thick subcategories of stmod_{0}(kG) that are not covered by the theorems of Section 5. In particular, we shall consider thick subcategories of C_{0}(Y_{G}) for several groups with nontrivial nuclei.

In preparation for studying these examples, we begin with a theorem which applies in some of the cases we shall consider. For notation, let N denote the thick subcategory of StMod_{0}(kG) consisting of all modules M with Ĥ^{n}(G, M) = 0 for all n.

Theorem 6.1. Suppose that G has a normal subgroup N, and let k be a field of characteristic p.

(i) Let M be a kG-module with the property that M ↓_{N} is projective and

Hom_{kN}(k, M ↓_{N}) = 0.

Then H^{∗}(G, M) = 0, and hence M is in N.

(ii) Suppose that G/N is a p-group. Let M be an indecomposable and nonprojective module in the principal block, and suppose that for every nontrivial p-subgroup Q ⊆ N, C_{N}(Q) is p-nilpotent. If M is in N then M ↓_{N} is projective and Hom_{kN}(k, M ↓_{N}) = 0.

Proof. (i) The hypotheses imply that the E_{2} page of the Lyndon-Hochschild-Serre spectral sequence

H^{∗}(G/N, H^{∗}(N, M)) ⇒ H^{∗}(G, M)

is zero, and so H^{∗}(G, M) = 0.

(ii) Since G/N is a p-group, M ↓_{N}↑^{G}≅ k_{N} ↑^{G} ⊗M is in the thick subcategory generated by M, and so

Ĥ^{∗}(N, M ↓_{N}) ≅ Ĥ^{∗}(G, M ↓_{N}↑^{G}) = 0

Hence, by Proposition 4.4, M ↓_{N} is projective. Since M is indecomposable and nonprojective, Ĥ^{0}(G, M) = 0 implies Hom_{kG}(k, M) = 0, and hence Hom_{kN}(k, M ↓_{N})^{G/N} = 0. Since G/N is a p-group, this implies that Hom_{kN}(k, M ↓_{N}) = 0.

Example 1 (C_{2} × A_{4} in characteristic two). Let G = C_{2} × A_{4}, and let k be an algebraically closed field of characteristic two. Then kG has only one block, the principal block, and the only nontrivial subgroup of G that does not have a 2-nilpotent centralizer is C_{2} × {1}. The nucleus Y_{G} therefore consists of a single line in V_{G}(k), and so every nonprojective module in C(Y_{G}) has Y_{G} as its variety. By Theorem 6.1 a kG-module is in N if and only if its restriction to {1} × A_{4} is projective and has no direct summand isomorphic to the projective cover of the trivial module.

Let M be the kG-module which, as a module for A_{4}, is the projective cover P_{k} of the trivial module, and on which C_{2} acts trivially. In other words, M is isomorphic to the permutation module Ind_{C_{2}×C_{3}}^{G}(k). Then M ≅ ΩM.

Now consider the element

γ = ∑_{x∈A_{4}} x

of kG, which is a central element whose square is zero. For any λ ∈ k, there is an automorphism θ_{λ} of kG that fixes A_{4} and sends the generator g of C_{2} to g + λγ.

We can now form an infinite family {M_{λ} : λ ∈ k} of modules, where M_{λ} is just the module M with the kG-action twisted by the automorphism θ_{λ}. That is, M = U/V where U = kG.u is the projective left ideal generated by u = 1+w+w^{2} for w ∈ A_{4} an element of order three, and V = kG((g −1)u− λγ). In particular M_{0} = M. Since M is periodic of period one, the same is true of all the M_{λ}. Also, as M_{λ} is an indecomposable periodic module, V_{G}(M_{λ}) must be a single line through the origin, which must in fact be the nucleus Y_{G}, since M_{λ} is not projective on restriction to C_{2} × {1}.

As modules for A_{4}, all of these modules are isomorphic to P_{k}, but instead of acting trivially, the generator g of C_{2} acts as id + λϕ, where ϕ is a fixed nonzero kA_{4}-module homomorphism obtained as a composition

P_{k} ⟶ P_{k}/ rad(P_{k}) ⟶ soc(P_{k}) ⟶ P_{k}.

Proposition 6.2. For different values of λ, the modules M_{λ} for C_{2} × A_{4} (in characteristic two) described above generate thick subcategories of C(Y_{G}) which intersect trivially.

Proof. Since the automorphisms of kG involved in the definition of the modules satisfy

θ_{λ}θ_{μ} = θ_{λ+μ}

for all λ, μ ∈ k, it is sufficient to show that, for λ ≠ 0, the thick subcategory generated by M_{λ} intersects trivially with the thick subcategory generated by M.

By Frobenius Reciprocity,

\underline{Hom}_{kG}(M, M_{λ}) ≅ \underline{Hom}_{k(C_{2}×C_{3})}(k, M_{λ} ↓_{C_{2}×C_{3}}),

which is zero (if λ ≠ 0) since it is easy to check that there is only one indecomposable summand of M_{λ} ↓_{C_{2}×C_{3}} that has nonzero fixed points, and that this summand is projective.

Since M ≅ ΩM, it follows that \underline{Hom}_{kG}(X, Y) = 0 for every X in the thick subcategory of stmod(kG) generated by M and Y in the thick subcategory generated by M_{λ}. So these thick subcategories must intersect trivially.

We can produce two more families of examples by simply taking the tensor product of the modules M_{λ} with each of the two nontrivial simple kGmodules, which are both one dimensional. The subcategories produced in this way have the property that they are contained in N.

Remark . One might suspect that any thick subcategory with support in the nucleus is generated by

{M_{λ} | λ ∈ X} ∪ {M_{λ} ⊗ ω | λ ∈ Y } ∪ {M_{λ} ⊗ ω̄ | λ ∈ Z}

for suitable subsets X, Y and Z of k. Here, ω and ω̄ denote the two nontrivial simple kG-modules. If this were true, then any self-equivalence of the stable module category would permute the minimal elements of this collection of thick subcategories, namely those generated by a single M or M ⊗ω or M ⊗ω̄. However, there is a self-equivalence of the derived category of A_{4} that takes the projective cover P_{k} of the trivial module to a complex

⋯ → 0 → P_{ω} ⊕ Pω̄ → P_{ω} ⊕ Pω̄ → P_{k} → 0 → ⋯

where P_{ω} and Pω̄ are the other indecomposable projectives. This induces a self-equivalence of the stable module category of C_{2} × A_{4} that takes M_{0} to a 20-dimensional indecomposable module whose restriction to A_{4} is P_{k} ⊕ 2P_{ω} ⊕ 2Pω̄. The reader will find more details of some examples of thick subcategories given by self-equivalences of the derived category in Example 4.

Example 2 (C_{2} × A_{5} in characteristic two). Since the principal blocks of A_{4} and A_{5} are derived equivalent [17, Section 3] (i.e., the bounded derived categories of finitely generated modules for the two blocks are equivalent as triangulated categories), it follows from Theorem 2.1 of [16] that the principal blocks of C_{2} × A_{4} and C_{2} × A_{5} are also derived equivalent, and hence are stably equivalent by Corollary 2.2 of [15].

An infinite collection of thick subcategories that intersect trivially and have varieties contained in the nucleus can be obtained by taking the modules N_{λ} that we get by applying this stable equivalence to the modules M_{λ} given in the last example.

Example 3. (Janko’s sporadic group J_{1} in characteristic two) Let G be the sporadic group J_{1} of Janko and let k be an algebraically closed field of characteristic two. Then G has just one conjugacy class of involutions, and the centralizer H is isomorphic to C_{2} × A_{5}. The nucleus Y_{G} is equal to the image of Y_{H} → V_{G}(k), and is a single line through the origin in the three dimensional variety V_{G}(k).

By Theorem 4.2 of [3], induction gives an equivalence of categories between C(Y_{H}) and C(Y_{G}). It follows that the modules obtained by inducing N_{λ} from C_{2} ×A_{5} to J_{1} generate thick subcategories of C(Y_{G}) which intersect trivially.

Proposition 6.3. For different values of λ, the modules N_{λ} for C_{2} ×A_{5} (in characteristic two) described above induce up to J_{1} to give modules which generate thick subcategories of C(Y_{G}) intersecting trivially.

Example 4 (C_{3} × S_{3} in characteristic three). Now let

G = C_{3} × S_{3} = ⟨x, y, z | x^{3} = y^{3} = z^{2} = 1, xy = yx, xz = zx, yz = zy^{2}⟩

and let k be an algebraically closed field of characteristic three. As in the previous example, kG has only one block, and there is only one nontrivial subgroup (namely C_{3} × {1}) whose centralizer is not 3-nilpotent, and so again the nucleus Y_{G} is a single line through the origin. There are two simple kG-modules, k and ϵ, both one-dimensional.

The following construction produces an infinite collection of thick subcategories of C(Y_{G}). Set X = x − 1 and Y = y^{2} − y, so that zX = Xz, zY = −Y z, X^{3} = 0, XY = Y X, and Y^{3} = 0. For 0 ≠ λ ∈ k, we define M_{λ} to be the kG-module generated by elements a and b, satisfying the following relations: za = −a, zb = −b, Y^{2}a = X^{2}b, λXa = Y^{2}b. Then M_{λ} has a k-basis consisting of the elements a, Y a, Y^{2}a = X^{2}b, b, Y b, Y^{2}b = λXa, Xb, XY b and XY^{2}b = λX^{2}a.

The projective cover of M_{λ} is a direct sum of two copies of P_{ϵ} generated by elements â and b̂ mapping to a and b. The kernel is generated by a′ = Y^{2}â − X^{2}b̂ and b′ = λXâ − Y^{2}b̂, which satisfy the same relations as a and b. Counting dimensions, it follows that ΩM_{λ} ≅ M_{λ}, so that M_{λ} is periodic with period one. In fact it can be seen that M_{λ} is projective on restriction to S_{3}, and that M_{λ}^{G} = 0 and Hom_{kG}(M_{λ}, k) = 0. So by Theorem 6.1, every M_{λ} is in N.

If M_{μ} is another such module, with corresponding generators c and d, we calculate Hom_{kG}(M_{λ}, M_{μ}) as follows. The images of a and b must be in the −1 eigenspace of z, which has a k-basis c, Y^{2}c = X^{2}d, d, Y^{2}d = μXc, Xd and XY^{2}d = μX^{2}c. Examining the relations in M_{λ} and M_{μ}, we obtain a system of linear equations which must be satisfied by these images. For λ ≠ μ, these imply that Hom_{kG}(M_{λ}, M_{μ}) has a k-basis consisting of the following homomorphisms:

θ_{1} : a ↦ Y^{2}c = X^{2}d  b ↦ 0
θ_{2} : a ↦ μX^{2}c = XY^{2}d  b ↦ 0
θ_{3} : a ↦ 0  b ↦ (μ − λ)Y^{2}c
θ_{4} : a ↦ 0  b ↦ (λ − μ)Xc
θ_{5} : a ↦ 0  b ↦ μX^{2}c = XY^{2}d
θ_{6} : a ↦ μXc = Y^{2}d  b ↦ λXd.

Each of these lifts to a map from M_{λ} to the projective cover of M_{μ} as follows:

θ̂_{1} : a ↦ X^{2}d̂  b ↦ 0
θ̂_{2} : a ↦ XY^{2}d̂  b ↦ 0
θ̂_{3} : a ↦ 0  b ↦ μY^{2}ĉ − λX^{2}d̂
θ̂_{4} : a ↦ μX^{2}ĉ − XY^{2}d̂  b ↦ λXĉ − Y^{2}d̂
θ̂_{5} : a ↦ 0  b ↦ XY^{2}d̂
θ̂_{6} : a ↦ Y^{2}d̂  b ↦ λXd̂.

It follows that \underline{Hom}_{kG}(M_{λ}, M_{μ}) = 0 for λ ≠ μ. This, together with the fact that these modules are periodic of period one, implies that the modules M_{λ} again generate thick subcategories which intersect trivially.

Proposition 6.4. For different values of λ, the modules M_{λ} for C_{3} ×S_{3} (in characteristic three) described above generate thick subcategories of C(Y_{G}) which intersect trivially.

Here are some more isolated examples of thick subcategories of stmod(kG).

(a) The intersection of C(Y_{G}) with ⟨k⟩ or with ⟨ϵ⟩.

(b) Of course, N is a thick subcategory of C(Y_{G}). As noted before all of the modules M_{λ} are in N. However, there are many more modules in N and may be many more thick subcategories of N.

(c) Let L be the projective cover of the trivial module for S_{3}, regarded as a kG-module via inflation. Then L has Y_{G} as its variety, and so it generates a thick subcategory of C(Y_{G}). Since ⟨L, k⟩ contains ϵ, and must therefore be the whole of stmod(kG), L cannot be in ⟨k⟩. Also \underline{Hom}_{kG}(k, L) ≠ 0, so L is not in either of the categories (a) or (b).

(d) The modules X of C(Y_{G}) for which \underline{Hom}_{kG}(Ω^{n}L, X) = 0 for all n ∈ ℤ also form a nontrivial thick subcategory. One such module has the following Loewy structure.

ϵ  k  k  ϵ
ϵ  k  ϵ  k  ϵ  k
k  ϵ  k  ϵ  k  ϵ
ϵ  k

(e) One way of producing new examples of thick subcategories is to apply self-equivalences of stmod(kG) to old examples. By [15, Theorem 2.1], a selfequivalence of the derived category D^{b}(mod(kG)) induces a self-equivalence of the stable module category, and so the main theorem of [14] can be used to give examples.

For instance, if we denote by Q_{k} and Q_{ϵ} the projective covers of the two simple kS_{3}-modules, then let T be the direct sum of the complex

… ⟶ 0 ⟶ Q_{ϵ} ⟶ 0 ⟶ 0 ⟶ …,

concentrated in degree 1, and the complex

… ⟶ 0 ⟶ Q_{ϵ} ⟶ Q_{k} ⟶ 0 ⟶ …,

with a non-zero differential and concentrated in degrees 1 and 0. It is easy to check that T is a tilting complex [14, Definition 6.5] for kS_{3} whose endomorphism ring is isomorphic to kS_{3}. Thus, by [14, Theorem 6.4], it induces a self-equivalence of D^{b}(mod(kS_{3})). The induced self-equivalence of stmod(kS_{3}) is actually isomorphic to the identity functor. However, by [16, Theorem 2.1], the tensor product kC_{3} ⊗ T is a tilting complex for kG whose endomorphism ring is isomorphic to kG. In this case, the induced functor

S : stmod(kG) ⟶ stmod(kG)

generates an infinite cyclic group of nontrivial self-equivalences of the category stmod(kG).

We can use this self-equivalence to produce more examples of thick subcategories of C(Y_{G}). For example, if L is the module described in (c), then the modules L, S(L) and S^{2}(L) have Loewy structures of the form

\begin{aligned}k \\ & ϵ \\ & & k \end{aligned} ,  \begin{aligned}& ϵ & & k \\ ϵ & & k & & ϵ \\ & k & & ϵ & & k \\ & & ϵ \end{aligned} ,  \begin{aligned}ϵ & & & & ϵ & & k \\ & k & & ϵ & & k & & ϵ \\ & & ϵ & & k & & ϵ & & k \\ & & & & & ϵ \end{aligned} ,

the structure of S^{3}(L) is

\begin{aligned}& ϵ \\ ϵ & & k & & ϵ & & & & ϵ & & k \\ & k & & ϵ & & k & & ϵ & & k & & ϵ \\ & & ϵ & & & & ϵ & & k & & ϵ & & k \\ & & & & & & & & & ϵ \end{aligned} ,

and so on. Also, S^{−n}(L) is the dual of S^{n}(L).

Each of these modules generates a thick subcategory of C(Y_{G}). Since \underline{Hom}_{kG}(Ω^{n}ϵ, L) = 0 for all n ∈ ℤ, but \underline{Hom}_{kG}(ϵ, S^{m}L) is non-zero for m ≠ 0, all of these subcategories are different. Similarly, we can apply powers of S to the thick subcategory described in (d).

(f) More examples can be constructed from those described above by taking the tensor product with ϵ.

Example 5 (SL(2, 𝔽_{3}) in characteristic two). Let

G = SL(2, 𝔽_{3}) ≅ Q_{8}⋊ C_{3}
= ⟨x, y, z | zy = y^{−1}x = yx^{−1}, z^{3} = 1, xz = zy, yz = zxy⟩

and let k be an algebraically closed field of characteristic two. Then kG has only one block, the principal block, and the only nontrivial subgroup of G that does not have a 2-nilpotent centralizer is Z(G) ≅ C_{2}. The nucleus Y_{G} is the whole of V_{G}(k), which is a single line through the origin. So every nonprojective module has Y_{G} as its variety.

The following construction provides an infinite family of thick subcategories of C(Y_{G}) = stmod(kG). Let ω, ω̄ be the primitive cube roots of unity in k, so that ω + ω̄ = 1 = ωω̄, and set X = x + ωy + ω̄xy, Y = x^{−1} + ω̄y^{−1} + ω(xy)^{−1} in kG. Then X^{2} = Y XY, Y^{2} = XY X, X^{2}Y = XY^{2} = Y^{2}X = Y X^{2} = 0, zX = ωXz and zY = ω̄Y z. The group algebra kQ_{8} of the subgroup Q_{8} = ⟨x, y⟩ has basis 1, X, Y, XY, Y X, XY X = Y^{2}, Y XY = X^{2}, XY XY = Y XY X = X^{3} = Y^{3}. A basis for kG can be obtained by multiplying these basis elements by 1, z and z^{2}. The advantage of this basis is that it is well adapted to the study of the simple modules, their extensions, the Loewy series of the group algebra, and so on.

For 0 ≠ λ ∈ k, let M_{λ} be the kG-module with a single generator a, satisfying za = a and XY a = λY Xa. Thus M_{λ} is four dimensional over k, with basis a, Xa, Y a and XY a. We have that X^{2}a = Y XY a = λY^{2}Xa = 0 and that Y^{2}a = XY Xa = λ^{−1}X^{2}Y a = 0. The action of z is given by za = a, zXa = ωXa, zY a = ω̄Y a and zXY a = XY a. The projective cover of M_{λ} is the eight dimensional kG-module generated by a single element â mapping to a and satisfying zâ = â. The kernel is generated by a′ = (XY + λY X)â, and is also four dimensional. Since

(XY + λ^{−1}Y X)a′ = (XY XY + λXY^{2}X + λ^{−1}Y X^{2}Y + Y XY X)â = 0,

we have that Ω(M_{λ}) ≅ M_{λ^{−1}}.

If M_{μ} is another such module with corresponding generator b, and μ ∉ {λ, λ^{−1}}, then Hom_{kG}(M_{λ}, M_{μ}) is one dimensional, generated by the homomorphism

a ↦ (λ + μ^{−1})XY b = (λXY + Y X)b.

This lifts to the homomorphism a ↦ (λXY + Y X)b̂ to the projective cover of M_{μ}, and so \underline{Hom}_{kG}(M_{λ}, M_{μ}) = 0. It follows from this, together with the fact that Ω(M_{λ}) ≅ M_{λ^{−1}}, that the pairs {M_{λ}, M_{λ^{−1}}} generate thick subcategories which intersect trivially, for different pairs {λ, λ^{−1}}.

Proposition 6.5. For different pairs {λ, λ^{−1}}, the pairs {M_{λ}, M_{λ^{−1}}} for SL(2, 𝔽_{3}) (in characteristic two) generate thick subcategories of C(Y_{G}) which intersect trivially.

Example 6 (GL(2, 𝔽_{3}) in characteristic two). Let G be the group

G = GL(2, 𝔽_{3}) ≅ Q_{8}⋊ S_{3},

and let H be the subgroup SL(2, 𝔽_{3}) discussed above. Let k be an algebraically closed field of characteristic two. Then kG again has only one block, and the only nontrivial subgroup of G that does not have a 2nilpotent centralizer is Z(G) = Z(H) ≅ C_{2}. So the nucleus Y_{G} is equal to Im(V_{H}(k) → V_{G}(k)), a single line through the origin in the affine plane V_{G}(k).

We write G as H⋊ ⟨t⟩, with relations t^{2} = 1, xt = tx^{−1}, yt = t(xy)^{−1}, zt = tz^{−1}. Thus we have Xt = tY and Y t = tX. It is easily seen that conjugation by t swaps M_{λ} with M_{λ^{−1}}. So M̂_{λ} = M_{λ} ↑^{G}≅ M_{λ^{−1}} ↑^{G} is an indecomposable kG-module whose restriction to H is isomorphic to M_{λ} ⊕ M_{λ^{−1}}. It follows from Frobenius reciprocity that for μ ∉ {λ, λ^{−1}}, \underline{Hom}_{kG}(M̂_{λ}, M̂_{μ}) = 0. Furthermore, Ω(M̂_{λ}) ≅ M̂_{λ}. So again, the modules M̂_{λ} generate thick subcategories of stmod(kG) which intersect trivially for different pairs {λ, λ^{−1}}.

Proposition 6.6. For different pairs {λ, λ^{−1}}, the modules M̂_{λ} for the group GL(2, 𝔽_{3}) (in characteristic two) generate thick subcategories of C(Y_{G}) which intersect trivially.

Example 7 (M_{11} and SL(3, 𝔽_{3}) in characteristic two). Let G be one of the groups M_{11} or SL(3, 𝔽_{3}), and let k be an algebraically closed field of characteristic two. It is well known that the principal blocks of these two groups are Morita equivalent^{1}, so we can discuss them simultaneously. The group G has a single conjugacy class of involutions, and the centralizer H is isomorphic to the group GL(2, 𝔽_{3}) discussed above. The nucleus Y_{G} is equal to Im(Y_{H} → V_{G}(k)), and is again a single line through the origin in the affine plane V_{G}(k).

By Theorem 4.2 of [3], induction gives an equivalence of categories between C(Y_{H}) and C(Y_{G}). It follows that the modules M̂_{λ} ↑^{G} generate thick subcategories of C(Y_{G}) which intersect trivially.

Proposition 6.7. For different pairs {λ, λ^{−1}}, the modules M̂_{λ} ↑^{G} for M_{11} or SL(3, 𝔽_{3}) (in characteristic two) generate thick subcategories of C(Y_{G}) which intersect trivially.

7. Some open questions

We shall finish with some questions that remain unanswered. The most important, of course, is:

Question 7.1. Is there a classification of the thick subcategories of the category stmod kG for a general finite group G? Or, less ambitiously, what about the thick subcategories in the principal block?

In this paper, we have not looked at nonprincipal blocks, but perhaps the results of Section 5 could be generalized.

Question 7.2. Is there an object associated to a nonprincipal block that plays the role that the nucleus Y_{G} does for the principal block? If so, are there analogues of Theorems 5.5 and 5.8?

Returning to principal blocks, there are some more specific questions.

1The structure of the projective indecomposable modules for M_{11} is given in Schneider’s thesis [19]. Blocks with semidihedral defect groups are classified in Erdmann [9, 10], but there is a misprint on page 162 of [10], where it states that the principal block of M_{11} is in family IV. In fact it is in family III along with the groups P SL(3, 𝔽_{q}) for q ≡ 3 modulo 4.

Question 7.3. For a group G with a nontrivial nucleus Y_{G}, and a closed homogeneous subvariety V of Y_{G}, is there always an infinite collection of thick subcategories of C(V) intersecting trivially, and each containing modules with variety V?

Given a thick subcategory C of stmod(kG), there is another thick subcategory whose objects are the modules Y such that \underline{Hom}_{kG}(X, Y) = 0 for all X in C. Let us call this the perpendicular category C^{⊥} of C. For example, ⟨k⟩^{⊥} is the category N of modules M for which Ĥ^{∗}(G, M) = 0. By ‘Tate duality’ [1, Proposition 4.12.9] between \underline{Hom}(M, N) and \underline{Hom}(N, ΩM), there is no difference between the ‘left perpendicular’ and ‘right perpendicular’ categories of a thick subcategory.

In the case that G is a p-group and X is a collection of subvarieties closed under finite unions and specializations, C(X)^{⊥} is just C(X′), where

X′ = {V ∈ V_{G}(k) : V ∩ W = {0} for all W ∈ X }.

It follows that C(X)^{⊥⊥} = C(X) if and only if X is closed under arbitrary unions in the sense that if a subvariety V is a union of elements of X then it is itself in X.

Question 7.4. If Y_{G} is not trivial, for which thick subcategories C of the category stmod(kG) does C = C^{⊥⊥}? In particular, is it always true that ⟨k⟩ = ⟨k⟩^{⊥⊥}?

Even in the specific examples of Section 6, we do not know whether ⟨k⟩ = ⟨k⟩^{⊥⊥}.

References

[1] D. J. Benson Representations and cohomology I. Cambridge studies in advanced mathematics, vol. 30, Camb. Univ. Press (1990).

[2] D. J. Benson Representations and cohomology II. Cambridge studies in advanced mathematics, vol. 31, Camb. Univ. Press (1991).

[3] D. J. Benson Cohomology of modules in the principal block of a finite group. New York Journal of Mathematics 1 (1995), 196–205.

[4] D. J. Benson, J. F. Carlson and J. Rickard Complexity and varieties for infinitely generated modules. Math. Proc. Camb. Phil. Soc. 118 (1995) 223–243.

[5] D. J. Benson, J. F. Carlson and J. Rickard Complexity and varieties for infinitely generated modules, II. Math. Proc. Camb. Phil. Soc., to appear.

[6] D. J. Benson, J. F. Carlson and G. R. Robinson On the vanishing of group cohomology. J. Algebra 131 (1990) 40–73.

[7] L. Chouinard Projectivity and relative projectivity over group rings. J. Pure Appl. Algebra 7 (1976) 278–302.

[8] E. S. Devinatz, M. J. Hopkins and J. H. Smith Nilpotence and stable homotopy theory, I. Ann. of Math. (2) 128 (1988) 207–241.

[9] K. Erdmann. Algebras and semidihedral defect groups I. Proc. London Math. Soc. 57 (1988), 109–150.

[10] K. Erdmann. Algebras and semidihedral defect groups II. Proc. London Math. Soc. 60 (1990), 123–165.

[11] M. J. Hopkins Global methods in homotopy theory. In: Homotopy theory — Proc. Durham Symp. 1985. CUP, 1987.

[12] A. Neeman Stable homotopy as a triangulated functor. Invent. Math. 109 (1992) 17–40.

[13] A. Neeman The chromatic tower for D(R). Topology 31 (1992) 519–532.

[14] J. Rickard Morita theory for derived categories. J. London Math. Soc. (2) 39 (1989) 436–456.

[15] J. Rickard Derived categories and stable equivalence. J. Pure Appl. Algebra 61 (1989) 303–317.

[16] J. Rickard Derived equivalences as derived functors. J. London Math. Soc. (2) 43 (1991) 37–48.

[17] J. Rickard Splendid equivalences: derived categories and permutation modules. Proc. London Math. Soc. (3) 72 (1996) 331–358.

[18] J. Rickard Idempotent modules in the stable category. J. London Math. Soc., to appear.

[19] G. Schneider. Die 2-modularen Darstellungen der Mathieu-Gruppe M_{12}. Doctoral dissertation, Essen, 1981.

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

  E-mail address: djb@byrd.math.uga.edu

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

  E-mail address: jfc@sloth.math.uga.edu

University of Bristol, School of Mathematics, University Walk, Bristol BS8 1TW, ENGLAND

  E-mail address: J.Rickard@bristol.ac.uk