THE THICK SUBCATEGORY GENERATED BY THE TRIVIAL MODULE

JON F. CARLSON^{∗}

This paper is a report of my lecture at the International Conference on the Representation Theory of Algebra which preceded the Euroconference on Infinite Length Modules. While it is true that the aim of the research is the homological algebra of finitely presented modules, a major point of the lecture was the role that infinitely generated modules could play in the investigation. The methods of idempotent modules and the theory of support varieties for infinite dimensional modules have already had a significant impact on group representation theory. It seems certain that there will be a lot more to follow. I am honored by the invitation to include the report in the conference proceedings, and I would like to thank the organizers of the conference and the workshop for the stimulating experience.

Throughout the paper G is a finite group and k is a field of characteristic p > 0. Ultimately the investigation is concerned with the structure and the behavior of the functors Ext_{kG}^{n}(M, N) for M and N finitely generated kG-modules. In particular we would like to understand how the properties of modules are reflected in their cohomology and in their extensions with other modules. Even more specifically, we ask for things like the following

Question 1. What are the properties of pairs of modules M and N that will produce the result that

Ext_{kG}^{n}(M, N) = 0.

for all n.

One of the aims of this paper is to show that the above, as well as other questions, are really questions about categories of modules. For the most part we will restrict ourselves to the study of finite dimensional kG-modules, Of course the questions that we ask have analogues for all kG-modules. However for several reasons, it seems natural to look at these issues separately.

For some notation, let mod(kG) be the category of finitely generated kG-modules and kG-module homomorphisms. We denote the corresponding category of all kGmodules by Mod(kG). Let ϕ : P ⟶ M be a projective cover of the kG-module M. Then Ω(M) is defined to be the kernel of ϕ. Thus we have an exact sequence

0 ⟶ Ω(M) ⟶ P ⟶^{ϕ} M ⟶ 0

Inductively, let Ω^{n}(M) = Ω(Ω^{n−1}(M)). Similarly let Ω^{−1}(M) be the cokernel of the inclusion M ⟶ Q of M into its injective hull. Again inductively we have Ω^{−n}(M) = Ω^{−1}(Ω^{−n+1}((M)). So for every n ∈ ℤ we have an exact sequence

Date: December 9, 1998.

Partly supported by a grant from NSF.

0 ⟶ Ω^{n+1}(M) ⟶ P roj ⟶ Ω^{n}(M) ⟶ 0.

We emphasize here that kG is a self-injective algebra. So a module is injective if and only if is projective. This hold even for infinitely generated modules.

The Natural Setting

The stable category stmod(kG) of finitely generated kG-modules is a natural setting for our investigation. Roughly speaking, it is the category we get by setting the projective modules equal to zero. Specifically we define the stable category as follows. The objects in stmod(kG) are the finite dimensional kG-modules as in mod(kG). For M and N objects, the morphisms from M to N are given by

\overline{Hom}_{kG}(M, N) = \frac{Hom_{kG}(M, N)}{PHom_{kG}(M, N)}

where PHom_{kG}(M, N) is the set of all homomorphisms that factor through a projective module.

The operators Ω and Ω^{−1} are functors on the stable category, and they are used to create a triangulated structure on stmod(kG). Basically, a triangle

⟶ L ⟶^{α} M ⟶^{β} N ⟶^{γ} Ω^{−1}(L) ⟶

in stmod(kG) corresponds to a collection of exact sequences:

0 ⟶ L ⟶^{α_{1}} M ⊕ P roj ⟶^{β_{1}} N ⟶ 0,

0 ⟶ M ⟶^{β_{2}} N ⊕ P roj ⟶^{γ_{2}} Ω^{−1}(L) ⟶ 0,

etc. Here the class modulo PHom_{kG}(L, M) of the map α_{1} is α, the class of β_{1} and β_{2} is β and the class of γ_{2} is γ. So when we pass from the module category mod(kG) to stmod(kG) we substitute “triangulated” for “abelian”.

Likewise any exact sequence in mod(kG) defines a triangle in stmod(kG). The functor Ω^{−1} is called the translation functor for the triangulation. So a triangle consists of a triple of objects and a triple of maps and every time we go around the triangle we must apply the translation functor Ω^{−1}. There is a set of axioms that triangulated categories must satisfy. For the axioms and other details on triangulated categories see the book by Happel [13].

In a real sense, cohomology takes place in the stable category. This fact is best illustrated by the following result that is easily verified from basic principles.

Lemma 2. For M and N kG-modules, and any n > 0,

\overline{Hom}_{kG}(Ω^{n}(M), N) ≅ Ext_{kG}^{n}(M, N),

and

\overline{Hom}_{kG}(M, Ω^{−n}(N)) ≅ Ext_{kG}^{n}(M, N)

In fact if we substitute Tate cohomology Êxt for Ext then the above lemma is true for all integers n. It follows that in the stable category the familiar long exact sequence in cohomology becomes a sequence of Hom’s. Namely we have the following.

Lemma 3. Given a triangle of kG-modules

⟶ L ⟶^{α} M ⟶^{β} N ⟶^{γ} Ω^{−1}(L) ⟶

and a kG-module S we have an exact sequence

… ⟶^{γ_{∗}} \overline{Hom}_{kG}(S, Ω^{r}(L)) ⟶^{α_{∗}} \overline{Hom}_{kG}(S, Ω^{r}(M)) ⟶^{β_{∗}}

\overline{Hom}_{kG}(S, Ω^{r}(N)) ⟶^{γ_{∗}} \overline{Hom}_{kG}(S, Ω^{r−1}(L)) ⟶^{α_{∗}} …

Of course there is a similar sequence in the other variable.

Thick Subcategories

A full subcategory M of stmod(kG) is said to be thick if it is triangulated and if it is also closed under the taking of direct summands. So if M is a thick subcategory then it is closed under the operators Ω and Ω^{−1} and whenever we have a triangle in stmod(kG) (or exact sequence in mod(kG)) and two of the objects are in M then so is the third. The stable category of all modules in a particular block of kG is a thick subcategory of stmod(kG). Other examples include things like the collection of all periodic modules or the collection of all modules whose variety is contained in a fixed subvariety of V_{G}(k). More about this is presented later.

If M is a kG-module then the thick subcategory H(M) generated by M is the smallest thick subcategory of stmod(kG) containing M. It contains Ω^{n}(M) for all n. It contains any extension of one of these modules by another. It can be characterized as the full subcategory consisting of every module L that is a direct summand of a module L′ that can be filtered by a sequence of submodules such that each of the factors is isomorphic to Ω^{n}(M) for some n. With this in mind it is not difficult to derive the following fact from the long exact sequence.

Proposition 4. Suppose that M and N are kG-modules such that Êxt_{kG}^{n}(M, N) = 0 for all n. Then Êxt_{kG}^{n}(M′, N′) = 0 for all n, all M′ in H(M) and all N′ in H(N).

We should note here that if Ext_{kG}^{n}(M, N) = 0 for all n > 0 or if Êxt_{kG}^{n}(M, N) = 0 for n in a sufficiently long interval then Êxt_{kG}^{n}(M, N) = 0 for all n [4].

Proposition 5. Suppose that M and N are kG-modules and that N is in H(M), the thick subcategory generated by M. If N is not a projective module, then it must be that Ext_{kG}^{n}(M, N) ≠ 0 for infinitely many values of n.

The last result follows because if Ext_{kG}^{n}(M, N) = 0 for all n then Êxt_{kG}^{0}(N, N) = 0 and the identity homomorphism on N must factor through a projective. Thus N is a projective module.

So we see that the statement that Ext_{kG}^{n}(M, N) = 0 is really a statement about the subcategories that the modules M and N generate. That is, if Ext_{kG}^{n}(M, N) = 0 then the subcategories H(M) and H(N) have no nonzero object in common. As far as we know the converse of this statement is completely open.

Question 6. Suppose that M and N are kG-modules such that Ext_{kG}^{n}(M, N) ≠ 0 for some n > 0. Is it necessary that there is a nonprojective module L that is in both H(M) and H(N)?

We should notice that if M and N are indecomposable kG-modules in different blocks then Ext_{kG}^{∗}(M, N) = 0. Also we have a standard adjointness

Ext_{kG}^{n}(M, N) ≅ Ext_{kG}^{n}(k, M^{∗} ⊗ N) ≅ H^{n}(G, M^{∗} ⊗ N)

for all n. Here M^{∗} = Hom_{k}(M, k) is the k-dual and the tensor produce M^{∗} ⊗ N = M^{∗} ⊗_{k} N (≅ Hom_{k}(M, N)) is made into a G-module by letting the elements of G act diagonally. If the summand of M^{∗} ⊗N that is in the principal block is projective then there are no extensions between M and N. So it can be said that, in some sense, cohomology takes place in the principal block. Even further it measures how far the tensor product M^{∗} ⊗ N is from being in the thick subcategory generated by k.

For the remainder of the paper let K = H(k) be the thick subcategory of stmod(kG) generated by the trivial module. Let B̃_{0}(G) be the thick subcategory of modules in the principal block of kG.

Varieties and the vanishing of cohomology

The cohomology ring H^{∗}(G, k) of a finite group G is a graded-commutative, finitely generated k-algebra whose maximal ideal spectrum V_{G}(k) is a homogenious affine variety. For a finitely generated kG-module M the variety of M is defined to be the subvariety V_{G}(M) = V_{G}(J(M)) of all maximal ideals that contain the annihilator J(M) in H^{∗}(G, k) of the cohomology ring Ext_{kG}^{∗}(M, M). The varieties of modules have been studied extensively for several years now. For details on the subject see the books [1, 12]. The properties that are most of interest to us here can be summarized in the following. Note that the zero element in V_{G}(k) is the maximal ideal of all elements of positive degree.

Theorem 7. Let M be a kG-module. (1.) M is projective if and only if V_{G}(M) = {0}. (2.) If

⟶ L ⟶^{α} M ⟶^{β} N ⟶^{γ} Ω^{−1}(L) ⟶

is a triangle in stmod(kG) then V_{G}(M) ⊆ V_{G}(L) ∪ V_{G}(N). (3.) V_{G}(M) = V_{G}(Ω^{n}(M)) = V_{G}(M^{∗}) for all n. (4.) For M and N kG-modules

V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N).

It should be pointed out that the variety of a module can be determined directly from the structure of the module without any cohomological calculations whatsoever. It is done by computing the “rank variety” for the elementary abelian p-subgroups. See [1, 12] for details.

We note also that there in an extension of the notion of the variety of a module to infinitely generated modules [5], and the same properties are true in a suitably modified form. See the paper by Benson [3] in these proceedings for a synopsis of these results.

Several consequences of the Theorem are immediate to our investigation. First of all, if M and N have the property that V_{G}(M)∩V_{G}(N) = {0}, then Ext_{kG}^{∗}(M, N) = 0. Next we can see that if V is a closed subvariety of V_{G}(k) then the full subcategory M_{V} of all modules with V_{G}(M) ⊆ V is a thick subcategory of stmod(kG). The good news is that in some cases this is close to being the whole story [6].

Theorem 8. Suppose that the centralizers of p-elements in G are p-nilpotent. Then any thick subcategory C of the principal block B̃_{0}(G) is determined by varieties. That is, there is a collection V of closed subvarieties of V_{G}(k) that is closed under specialization and taking of finite unions such that C is the full subcategory M_{V} consisting of all modules M such that V_{G}(M) is an element of V.

Of course, if G is a p-group then B̃_{0}(G) = stmod(kG) and we have a complete classification of the thick subcategories of stmod(kG). The proof in this case uses the idempotent modules of Rickard, which we say more about later, and the extended notion of varieties for infinitely generated modules. When G is not a p-group then the proof of the Theorem requires also a deep result of Benson [2]. In any event it rests on the following facts, the first of which is proved in [4] for p > 2 and in [2] for the general case.

Theorem 9. Suppose that the centralizers of p-elements in G are p-nilpotent. Then B̃_{0}(G) = K.

A thick subcategory M is tensor ideal closed if whenever M is in M and N is another module then M ⊗ N is in M. The fact that we need for the proof of the theorem is that any tensor ideal closed subcategory is determined by varieties [6]. It is not difficult to verify that any thick subcategory of K is tensor ideal closed in K.

All of this tells us that, if we are in the case where the centralizers of p-elements are p-nilpotent, then two modules M and N in B̃_{0}(G) generate orthogonal subcategories (Ext_{kG}^{∗}(M, N) = 0) if and only if V_{G}(M) ∩ V_{G}(N) = {0}. This holds more generally for any two modules in K.

Before going further we should mention that there is another fortunate occurence in the case that B̃_{0}(G) = K. Namely we have some strong control on the selfequivalences. Every stable self-equivalence F : K ⟶ K must take the trivial module k to an endotrivial module F(k). A module M is endotrivial if the kendomorphism ring Hom_{k}(M, M) ≅ M^{∗} ⊗ M is isomorphic to k ⊕ P where P is a projective module. That is, it is isomorphic to k in the stable category stmod(kG)

[11]. This has some importance in the investigation of questions such as Broué’s conjecture on derived equivalences of block algebras with abelian defect groups [7].

The nucleus and other thick subcategories

But what happens in other cases? What if the centralizers of p-elements are not pnilpotent – if B̃_{0}(G) ≠ K? For the principal block there is a theorem of Benson that gives us a reasonably satisfactory conclusion, at least to the question of vanishing cohomology and the relation to varieties.

To state the theorem we need the following definition. Note that for H a subgroup of G we have a restriction map res_{G,H} : H^{∗}(G, k) ⟶ H^{∗}(H, k) and a corresponding map on varieties res_{G,H}^{∗} : V_{H}(k) ⟶ V_{G}(k). We should remember Quillen’s theorem that V_{G}(k) = ⋃ res_{G,E}^{∗}(V_{E}(k)) where E runs through the elementary abelian psubgroups of G [1, 12].

Definition 10. The nucleus of G is the subvariety of V_{G}(k) defined by

N = ⋃ res_{G,E}^{∗}(V_{E}(k))

where E runs through the set of all elementary abelian p-subgroups such that C_{G}(E) is not p-nilpotent.

Then Benson’s Theorem can be stated as follows.

Theorem 11. [2] Suppose that M is a module in B̃_{0}(G) and that

H^{∗}(G, M) = Ext_{kG}^{∗}(k, M) = 0.

Then V_{G}(M) ⊆ N.

Moreover the theorem can be extended to cover varieties defined by cohomology modules [9]. To understand this let I(M) denote the annihilator in H^{∗}(G, k) of the cohomology H^{∗}(G, M) for a kG-module M, and let W_{G}(M) = V_{G}(I(M)) be the subvariety of V_{G}(k) consisting of all maximal ideals that contain I(M). It is not difficult to show that W_{G}(M) ⊆ V_{G}(M) because H^{∗}(G, M) is a module over Ext_{kG}^{∗}(M, M). So the theorem says that for M in B̃_{0}(G)

V_{G}(M) ∖ W_{G}(M) ⊆ N .

For a pair of modules M and N in the principal block, we can say even more. Notice that I(M^{∗} ⊗ N) is the annihilator of Ext_{kG}^{∗}(M, N) ≅ H^{∗}(G, M^{∗} ⊗ N). Hence

W_{G}(M^{∗} ⊗ N) ⊆ V_{G}(M^{∗} ⊗ N) = V_{G}(M) ∩ V_{G}(N).

Again we have that

V_{G}(M) ∩ V_{G}(N) ∖ W_{G}(M^{∗} ⊗ N) ⊆ N .

For a block B other than the principal block we can define a block nucleus N_{B} that has some of the properties of the principal block nucleus [9]. But this time the definition has some serious problems with it, as pointed out in the paper [9]. Perhaps a better definition would come from a more specialized treatment of the block algebra such as that in [13]. The approach there has the advantage of alleviating the difficulty of not having any module to play the role of the trivial module in an arbitrary block. On the other hand it has the disadvantage of not being defined in the same context as the varieties of modules in the principal block. There are a lot of open questions here.

However, even in the case of the principal block, if N ≠ {0}, then there is still no direct and certain way to predict the orthogonality of the subcategories generated by the modules M and N (Ext_{kG}^{∗}(M, N) = 0) by looking only at the module structure. The current state of affairs with regards to the thick subcategories of B̃_{0}(G) is just as muddled. We might ask the question:

Question 12. Suppose that N ≠ {0}. Is it possible to get a complete classification of the thick subcategories of B̃_{0}(G)?

In [6], we looked at this and related question in some detail. In particular we considered several seemingly simple examples. In not even one case were we able to answer the above question. Indeed, the situation appears to be very chaotic. In every example that we considered there is an infinite number of (mutually orthogonal) thick subcategories of the “no cohomology” category U. This the full subcategory of all modules M in B̃_{0}(G) such that H^{∗}(G, M) = {0}. It seems to be a complete mystery. Another question we ask is the following.

Question 13. Suppose that N ≠ {0}. Let W be the full subcategory of all modules M in B̃_{0}(G) such that Ext_{kG}^{∗}(N, M) = 0 for all N in U. Is W = K?

Again, in no case, not even the easiest of examples, were we able to answer this question in [6]. However some recent progress in the area is encouraging. Now we can at least say what happens in some of the very special examples discussed in [6]. The rest of this work will focus on the few new results that we have in this direction. To get started, we need a new techniques as given below.

Idempotent modules and functors

It is here that we begin to call on the methods of infinitely generated modules. A few years ago Jeremy Rickard introduced a very useful tool in the form of idempotent modules and functors corresponding to a subcategory. As we have already said, the method was used in the original solution of the theorem of Benson (Theorem 11) as well as in the classification of the thick subcategories in the case that N = {0}. More details as well as other applications of the functors can be found in the survey of Benson [3]. See [8] for another leisurely treatment of the background. The statement goes as follows.

Theorem 14. [14] Suppose that C is a thick subcategory of stmod(kG). Then there exist functors E_{C} and F_{C} such that for any module M there is a triangle

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

with the two properties (1). E_{C}(M) ∈ C^{⊕} and (2). F_{C}(M) is C-local.

Moreover the two properties characterize the triangle and the modules E_{C}(M) and F_{C}(M).

Here C^{⊕} is the subcategory of the stable category StMod(kG) of all kG-modules consisting of all modules that are filtered colimits of objects in C. It is the triangulated subcategory generated by the closure of C in StMod(kG) under the operation of taking arbitrary direct sums. A kG-module N in Mod(kG) is said to be C-local if \overline{Hom}_{kG}(C, N) = 0 for all C ∈ C.

In the case that the subcategory C is tensor ideal closed, the modules E_{C}(k) and F_{C}(k) are idempotent in the stronger sense that E_{C}(k) ⊗ E_{C}(k) ≅ E_{C}(k) and F_{C}(k) ⊗ F_{C}(k) ≅ F_{C}(k) and E_{C}(k) ⊗ F_{C}(k) ≅ {0} in the stable category. That is, as modules

E_{C}(k) ⊗ E_{C}(k) ≅ E_{C}(k) ⊕ P

for P a projective module. In almost all cases the modules are infinite dimensional. Indeed, it can be shown that any finite dimensional nonprojective module M with the property that M ⊗M ≅ M in the stable category must have the form M ≅ k ⊕proj [8].

Also for subcategories defined by varieties there is a straightforward construction of the modules E_{C}(k) and F_{C}(k) by direct limits (see [8] or the original paper [14]). For the subcategory K or any subcategory generated by a single object there is also a straight forward construction of E_{K}(M) and F_{K}(M) arising from the cohomology of M. In particular the modules are a direct limit of finitely generated modules and we can get some control of the situation. To illustrate what happens we elaborate on one example from the list of examples in [6].

The example C_{3} × S_{3} in characteristic 3

Suppose that G ≅ C_{3} × S_{3} ≅ ⟨z⟩ × ⟨y, x⟩ where z^{3} = y^{3} = x^{2} = 1 and xyx = y^{2}. Let k be a field of characteristic 3. Then there are two simple kG-modules: the trivial module k and a module ϵ of dimension one on which the element x of order 2 acts by multiplication by −1. As a left kG-module

kG ≅ P_{k} ⊕ P_{ϵ},

where P_{k} and P_{ϵ} are the projective covers of k and ϵ respectively. Both of these modules have dimension 9 and their restrictions to the Sylow 3-subgroup S = ⟨z, y⟩ of G is a copy of kS. Let Y = y − y^{2} ∈ kG, so that xY x = −Y. Let Z = z − 1. So the radical of kS is generated by Y and Z. Now notice that L = P_{k}/(Y · P_{k}) has dimension 3 and has a filtration

{0} = Z^{3} · L ⊆ Z^{2} · L ⊆ Z · L ⊆ L

where (Z^{i} · L)/(Z^{i+1} · L) ≅ k. So this module is in K. Also L ≅ Y^{2} · P_{k} is in K. So (Y · P_{k})/(Y^{2} · P_{k}) ≅ P_{ϵ}/(Y · P_{ϵ}) is in K.

On the other hand look at M = P_{ϵ}/(Z · P_{ϵ}). It can be checked that this is a periodic module (Ω(M) = Z · P_{ϵ} ⊆ P_{ϵ} and Ω^{2}(M) ≅ M). Moreover there exists no nonzero homomorphism k ⟶ Ω^{n}(M). Therefore M is in the category U of no-cohomology modules. (Note that the principal block is the only block.) That is, Ext_{kG}^{∗}(k, M) = Ext_{kG}^{∗}(M, k) = 0 only because the socle of M and Ω(M) is isomorphic to ϵ and hence \overline{Hom}(k, M) = \overline{Hom}(k, Ω(M)) = 0.

With all of this in mind we want to find the structure of the modules in the triangle

⟶ E_{K}(U) ⟶ U ⟶ F_{K}(U) ⟶ Ω^{−1}(E_{K}(U)) ⟶

where U = k ⊕ ϵ. Now notice that E_{K}(k ⊕ ϵ) ≅ k ⊕ E_{K}(ϵ) and that F_{K}(k ⊕ ϵ) ≅ F_{K}(ϵ) since F_{K}(k) = 0. The point of this is that U is a generator for the category stmod(kG) since it is the direct sum of the only two irreducible modules. Thus, stmod(kG) = H(U). Consequently for any module M, we have that Ext_{kG}^{∗}(U, M) = 0 or Ext_{kG}^{∗}(M, U) = 0 if and only if M is projective (injective).

The point of what is to follow is that the generators of the category have a lot to say about the module structure. Before going further we should remark that the converse to the above statement about the generator for the category is a completely open question. That is, we can ask the following question.

Question 15. Suppose that M is an object in a thick subcategory M of stmod(kG) and that Ext_{kG}^{∗}(M, V) ≠ 0 for all V in the subcategory. Must it be true that M is a generator for the category in the sense that H(M) = M?

In the case that we are considering we can actually determine the module F_{K}(U) ≅ F_{K}(ϵ) in terms of generators and relations. Let F be the direct sum of an infinite number of copies of P_{ϵ} generated by the elements {a_{i} : i = 1, 2, … }. That is, each a_{i} represents the idempotent (x − 1)/2 ∈ kG that generates P_{ϵ} as a direct summand of kG. Thus x · a_{i} = −a_{i}. Let R be the submodule generated by the elements

Z · a_{2i−1} − Y^{2} · a_{2i} and Z^{2} · a_{2i} − Y^{2} · a_{2i+1}

for i = 1, 2, . . ..

We claim that F = F/R ≅ F_{K}(ϵ). First note that the distinguished map θ : ϵ ⟶ F is given by θ(m) = Y^{2} · a_{1} + R where m is a generator for ϵ. If we let M_{i} be the submodule of F/θ(ϵ) generated by the classes of the elements a_{1}, …, a_{i} then each quotient M_{i}/M_{i−1} is in K. That is, each is a quotient of P_{ϵ}/(Y^{2} ·P_{ϵ}) by a submodule all of whose composition factors are isomorphic to k. So F/θ(ϵ) is the direct limit of the system

M_{1} ⟶ M_{2} ⟶ M_{3} ⟶ …

of objects in K. The fact that F is K-local is also easy because the socle and top (F/Rad(F)) of F consist of direct sums of copies of ϵ and F is periodic of period 2 (Ω^{2}(F) ≅ F). Consequently, by the criteria of Theorem 14, we have our claim.

We want to look at \overline{End}_{kG}(F), the endomorphism ring of F_{K}(U) = F_{K}(ϵ) in the stable category. It can be shown in this case that the stable endomorphism ring is isomorphic to k[α, β]/(β^{2}). Here α is the homomorphism that takes each a_{i} to a_{i+2} for all i, and β is represented by the homomorphism given by the images β(a_{2i−1}) = Z · a_{2i} and β(a_{2i}) = a_{2i+1}. It can be calculated that β^{2} factors through a projective module and hence is zero in the stable category.

Now suppose that N is a module in the subcategory U of B̃_{0}(G) consisting of all modules M with H^{∗}(G, M) = 0. Then for the generator U = k ⊕ ϵ of the example we have that Ext_{kG}^{∗}(N, E_{K}(U)) = 0 and by the long exact sequence, Ext_{kG}^{∗}(N, U) = Ext_{kG}^{∗}(N, F_{K}(U)) which has finite dimension over k. Because U is a generator for B̃_{0}(G) we are assured that Ext_{kG}^{∗}(N, U) ≠ 0. But for each n, Ext_{kG}^{n}(N, F_{K}(U)) is a module over the stable endomorphism ring \overline{End}_{kG}(F_{K}(U)). We emphasize that it must be a finite dimensional module. It is this fact that we need to use to answer Question 13. Even more can be proved.

The double perp of K

Something very similar to the situation that we have seen in the above example occurs in several other cases including all of the examples in [6]. It does not seem to be unversal or even generic, but it is still encouraging that we can answer some of these questions in even very special circumstance. We finish this paper by giving a brief sketch of how the information on the stable endomorphism ring \overline{End}_{kG}(F_{K}(U)) for a generator U of B̃_{0}(G) can be employed effectively. Basically we define a new sort of support variety for modules in the “no cohomology” subcategory U. The reader should be warned that we are skipping over several very sticky details. These will be dealt with fully in a future paper [10]. We hope that the account presented here will at least give the flavor of the subject.

For notation let S = k[α] ⊆ \overline{End}_{kG}(F_{K}(U)) be the polynomial subring. As we notice, if N is in U then Ext_{kG}^{n}(N, F_{K}(U)) is a finite dimensional module over S. Because of the periodicity of the cohomology it must be that the annihilator of Ext_{kG}^{∗}(N, F_{K}(U)) in S is a nonzero ideal which we denote J (N). Then it has a variety X(N) in the maximal ideal spectrum X ≅ k of S.

The first thing we prove is that for any triangle in U

⟶ N_{1} ⟶ N_{2} ⟶ N_{3} ⟶ Ω^{−1}(N_{1}) ⟶

we have J (N_{1}) · J (N_{3}) ⊆ J (N_{2}). Therefore if Y ⊆ X is a subvariety and if U_{Y} is the collection of all modules N in U with X(N) ⊆ Y then U_{Y} is a thick subcategory of U.

Next note that the homomorphism α is injective though not surjective. The same is true for any f(α) where f is a nonconstant polynomial. So for any such endomorphism we have an exact sequence in Mod(kG) (triangle in StMod(kG))

0 ⟶ F_{K}(U) ⟶^{f(α)} F_{K}(U) ⟶ N_{f} ⟶ 0.

where N_{f} is finitely generated. We can prove at this point that f^{2} ∈ J (N_{f}) so that X(N_{f}) = X(f) the zero set of f. So the subcategory U_{X(f)} is not empty. Therefore we have the following.

Fact 1. The category U has an infinite number of thick subcategories. Moreover the modules all have the same variety.

The thick subcategories correspond to the subvarieties of X. If N is a module in U then the variety V_{G}(N) of N is precisely the nucleus N which is a line in V_{G}(k). Note also that any module in U is in a proper subcategory. Thus we have the following.

Fact 2. No object in U is a generator of the category U.

Finally, we can get an answer to Question 13 in this particular case. We end the paper by presenting a brief sketch of the proof.

Fact 3. If M is a finitely generated kG-module with Ext_{kG}^{∗}(M, N) = {0} for all n in U then M is in K.

The first thing to notice is that, in the above notation, Ext_{kG}^{∗}(M, N_{f}) = {0} for all polynomials f(α) ∈ S. Thus we have that multiplication by f(α)

f(α) : Ext_{kG}^{∗}(M, F_{K}(U)) ⟶ Ext_{kG}^{∗}(M, F_{K}(U))

is an injection for all f. But Ext_{kG}^{∗}(M, F_{K}(U)) is a finitely generated module over the polynomial ring S. So the only way that this can happen is if Ext_{kG}^{∗}(M, F_{K}(U)) = {0}. But U is a generator for the category so we can create any module out of U by a series of extensions of translations by Ω or taking of direct summands. Any such process will preserve the property listed above, because F_{K} is a functor. Hence we must have that

Ext_{kG}^{∗}(M, F_{K}(M)) = {0}.

The last statement leads to our final contradiction because in the triangle

⟶ E_{K}(M) ⟶ M ⟶ F_{K}(M) ⟶ Ω^{−1}(E_{K}(M)) ⟶

the map from M to F_{K}(M) is the zero map. Hence the identity map from M to M factors through an element of K^{⊕} and M must be in K.

References

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[3] D. J. Benson, Infinite dimensional modules for finite groups, These proceedings.

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[9] J. F. Carlson, Varieties for cohomology with twisted coefficients, Acta Acad. Sinica (to appear).

[10] J. F. Carlson, Orthogonal subcategories of modules over group algebras, (in preparation).

[11] J. F. Carlson and R. Rouquier, Self-equivalences of stable module categories, Math. Zeit. (to appear).

[12] L. Evens, “The Cohomology of Groups”, Oxford University Press, 1991.

[13] D. Happel, “Triangulated categories in the Representation Theory of Finite Dimensional Algebras”, London Math. Soc. Lecture Notes No. 119, Cambridge Univ. Press, 1988.

[13] M. Linckelmann, Varieties in block theory, (preprint).

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

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

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