COHOMOLOGY OF MODULES IN THE PRINCIPAL BLOCK OF A FINITE GROUP

D. J. BENSON

1. Introduction

Recent developments in modular representation theory of finite groups have involved a re-evaluation of the role of infinitely generated modules. In particular, Rickard [5] has introduced some infinitely generated modules which are idempotent in the stable category, in the sense that the tensor square is isomorphic to the original module plus a projective. This work, together with a version of Dade’s lemma for infinitely generated modules, has allowed Benson, Carlson and Rickard [1, 2] to formulate and prove a generalization of the usual theory of complexity and varieties for modules to the infinitely generated situation.

In this paper, we shall demonstrate how the recent work described above can be used to address some older questions about finitely generated modules. In particular, we shall prove the conjectures formulated in the paper of Benson, Carlson and Robinson [3]. Before stating our main theorem, we state a more easily understood consequence, which provides an affirmative solution to Conjecture 1.2 of that paper. The proof may be found in Section 5.

Theorem 1.1. Let k be a field of characteristic p, and let G be a finite group. Then the centralizer of every element of order p in G is p-nilpotent, if and only if for every nonprojective module M in the principal block B_{0}(kG), H^{n}(G, M) ≠ 0 for some n ≠ 0.

We remark that in general, for a kG-module M, H^{n}(G, M) ≠ 0 for some n ≠ 0 if and only if Ĥ^{n}(G, M) ≠ 0 for infinitely many values of n both positive and negative, cf. Theorem 1.1 of [3]. We also remark that in the statement of the above theorem, it does not matter whether we restrict our attention to finitely generated kG-modules.

In the language of [3], our main theorem is the following. This almost provides an affirmative answer to Conjecture 10.10 of that paper, which does not mention passing down to summands. Terminology used in this introduction is explained in Section 2, and the proof may be found in Section 5.

Theorem 1.2. Let k be an algebraically closed field of characteristic p, and let G be a finite group. Then every finitely generated kG-module in the principal block is a direct summand of a nuclear homology module.

It follows from this theorem that every kG-module in the principal block is a filtered colimit of nuclear homology modules. In order to prove that every finitely generated module in the principal block actually is a nuclear homology module, it would suffice to show that the characters of nuclear homology modules span the principal block, and then the argument would run as in the proof of Proposition 4.4 of [3]. It does not seem immediately clear that this character theoretic statement is true.

Partially supported by a grant from the NSF.

As pointed out in Corollary 10.12 of [3] (passage to direct summands does not affect this), it follows from this theorem that the nucleus Y_{G} (which is the union of the images in the cohomology variety V_{G}, of the elementary abelian p-subgroups whose centralizers are not p-nilpotent) coincides with the representation theoretic nucleus Θ_{G} (which is the union of the varieties of the finitely generated modules in the principal block having no cohomology).

Corollary 1.3. For any finite group G, we have Y_{G} = Θ_{G}.

In the case where Y_{G} = {0}, this implies Theorem 1.1. More precisely, we prove the following strengthened form of Theorem 1.4 of [3]:

Theorem 1.4. Suppose that G is a finite group and k is a field of characteristic p. Then the following are equivalent:

(A) Every finitely generated module in the principal block B_{0}(kG) is a trivial homology module.

(A′) Every simple module in B_{0}(kG) is a direct summand of a trivial homology module. (A′′) Every (finitely generated) trivial source module in B_{0}(kG) is a direct summand of a

trivial homology module.

(A′′′) Every module (finitely or infinitely generated) in B_{0}(kG) is a filtered colimit of trivial homology modules.

(B) For every finitely generated non-projective module M in B_{0}(kG), we have H^{n}(G, M) ≠ 0 for some n > 0.

(B′) For every finitely generated non-projective periodic module M in B_{0}(kG), we have H^{n}(G, M) ≠ 0 for some n > 0.

(B′′) For every (finitely or infinitely generated) non-projective module M in B_{0}(kG), we have H^{n}(G, M) ≠ 0 for some n > 0.

(B′′′) For every (finitely or infinitely generated) module M of complexity one in B_{0}(kG), we have H^{n}(G, M) ≠ 0 for some n > 0.

(C) For every non-projective (finitely generated) trivial source module M in B_{0}(kG), we have H^{n}(G, M) ≠ 0 for some n > 0.

(D) The centralizer of every element of order p in G is p-nilpotent.

(D′) The centralizer of every nontrivial p-subgroup of G is p-nilpotent.

(E) For every non-projective finitely generated indecomposable module M in B_{0}(kG), with vertex R and Green correspondent f(M), we have H^{n}(N_{G}(R), f(M)) ≠ 0 for some n > 0.

On the way to proving these theorems, we prove a remarkable property of modules of complexity one. In general, such a module decomposes as a direct sum of modules whose variety consists of a single line through the origin in the cohomology variety V_{G}(k). The following theorem is proved at the end of Section 3.

Theorem 1.5. Suppose that M is a kG-module whose variety V_{G}(M) consists of a single line L through the origin in V_{G}(k). Let E be an elementary abelian p-subgroup of G, minimal with respect to the property that L is contained in the image of the map res_{G,E}^{∗} : V_{E}(k) → V_{G}(k) induced by restriction from G to E in cohomology. Let L = res_{G,E}^{∗}(ℓ_{0}) with ℓ_{0} a line through the origin in V_{E}(k), and let D be the subgroup of N_{G}(E) consisting of the elements which stabilize ℓ_{0} setwise. Then the direct sum of M with some projective kG-module is induced from D.

We remark that this paper makes Sections 8 and 9 of [3] obsolete, and there is no longer anything special about odd primes in our proofs. We also remark that we have not been able to tackle the original question which motivated [3], namely whether every simple module in the principal block necessarily has nonvanishing cohomology. One possible approach to this might be to try to prove that the variety of a simple module in the principal block cannot be contained in the nucleus. Since the property of being simple is not easy to work with, it may be better to consider modules whose endomorphism rings, modulo traces from suitable subgroups, are isomorphic to the field.

I would like to thank Jon Carlson and Jeremy Rickard for conversations which inspired this work, and I would also like to thank Jon Carlson for pointing out a serious error in an earlier version of this paper.

2. Terminology and background material

Let G be a finite group and k an algebraically closed field of characteristic p. Let stmod(kG) be the stable category of finitely generated kG-modules, considered as a triangulated category. The homomorphisms \underline{Hom}_{kG}(M, N) in this category are homomorphisms in the usual module category, modulo those that factor through some projective module. The triangles in stmod(kG) come from the short exact sequences in mod(kG) in the normal way. Similarly, StMod(kG) is the stable category of all (not necessarily finitely generated) kG-modules, which is again a triangulated category.

We write V_{G}(k) for the maximal ideal spectrum of H^{∗}(G, k). Note that for p odd, elements of odd degree square to zero, so that H^{∗}(G, k) modulo its nil radical is commutative. Thus V_{G}(k) is a homogeneous affine variety. Associated to any (not necessarily finitely generated) kG-module M, there is a collection V_{G}(M) of closed homogeneous irreducible subvarieties of V_{G}(k) (see [2] for details). These varieties have good properties with respect to tensor products, and M is projective if and only if V_{G}(M) = ∅.

We next remark that there is a mistake in the definition of Y_{G} given in Section 10 of [3]. The nucleus Y_{G} should be defined as the subvariety of V_{G}(k) given as the union of the images of the maps res_{G,H}^{∗} : V_{H}(k) → V_{G}(k) induced by res_{G,H} : H^{∗}(G, k) → H^{∗}(H, k), as H runs over the set of subgroups of G for which C_{G}(H) is not p-nilpotent (and not the union of the images of res_{G,C_{G}(H)}^{∗} : V_{C_{G}(H)}(k) → V_{G}(k) as stated there; also in the proof of Theorem 10.2 of that paper, V_{C_{G}(H)} should be replaced by V_{H}, and no other changes are necessary).

The representation theoretic nucleus Θ_{G} is the subset of V_{G}(k) given as the union of the varieties V_{G}(M) as M runs over the finitely generated modules in the principal block B_{0}(kG) with H^{n}(G, M) = 0 for all n. By Theorem 6.4 of [3], it suffices to consider periodic modules in this definition.

We say that a finitely generated kG-module M is a trivial homology module or a TH module if there exists a finite complex (C_{i}, δ_{i} : C_{i} → C_{i−1}) of finitely generated kG-modules and homomorphisms such that the following conditions hold:

(i) Each C_{i} is a projective kG-module, and C_{i} = 0 for i < 0 and for i sufficiently large. (ii) For i > 0, H_{i}(C_{∗}) is a direct sum of copies of the trivial kG-module k. (iii) H_{0}(C_{∗}) ≅ M.

We say that a finitely generated kG-module M is a nuclear homology module or an NH module if it satisfies the same conditions, but with (ii) replaced by:

(ii′) For i > 0, H_{i}(C_{∗}) is a direct sum of copies of the trivial kG-module k and finitely generated modules M′ in B_{0}(kG) with V_{G}(M′) ⊆ Y_{G}.

We write T H and N H for the thick subcategories of stmod(kG) consisting of the direct summands of trivial homology modules and of nuclear homology modules respectively.

Next, we recall from Section 5 of Rickard [5] that given any thick subcategory C of stmod(kG), there are functors E_{C} and F_{C} on StMod(kG) satisfying the following properties:

(a) For any X in StMod(kG), E_{C}(X) is a filtered colimit of objects in C.

(b) For any X in StMod(kG), F_{C}(X) is C-local, in the sense that for any object M in C, \underline{Hom}_{kG}(M, F_{C}(X)) = 0.

(c) There is a triangle in StMod(kG)

E_{C}(X) → X → F_{C}(X) → Ω^{−1}E_{C}(X).

In fact (see the remark after Proposition 5.7 of [5]) the functors E_{C} and F_{C} are characterized by these properties.

Our goal will be to show that the functor F_{N H} is the zero functor on the principal block, which will enable us to prove that every finitely generated module in the principal block is a direct summand of a nuclear homology module.

If V is a closed homogeneous subvariety of V_{G}(k), we write C_{V} for the subcategory of stmod(kG) consisting of the finitely generated modules M with V_{G}(M) ⊆ V. Then the corresponding functors E_{V} = E_{C_{V}} and F_{V} = F_{C_{V}} are given by tensoring with certain (usually infinitely generated) modules e_{V} and f_{V}. These are orthogonal idempotents in StMod(kG), in the sense that e_{V} ⊗ e_{V} ≅ e_{V} ⊕ (projective), f_{V} ⊗ f_{V} ≅ f_{V} ⊕ (projective), and e_{V} ⊗ f_{V} is projective. The triangle for a module X in this situation is given by tensoring the triangle

e_{V} ⟶^{λ_{V}} k ⟶^{μ_{V}} f_{V} → Ω^{−1}e_{V}

with X.

3. Inducing idempotents

Let L be a line through the origin in V_{G}(k). Then by the Quillen stratification theorem, there is an elementary abelian p-subgroup E, uniquely determined up to conjugacy, with the property that L is in the image of res_{G,E}^{∗} : V_{E}(k) → V_{G}(k), but L is not in the image of res_{G,E′}^{∗} : V_{E′}(k) → V_{G}(k) for any proper subgroup E′ of E. In this situation, we say that L originates in E. We write C = C_{G}(E) for the centralizer, and N = N_{G}(E) for the normalizer in G of E. Let ℓ_{0} be a line through the origin in V_{E}(k) with L = res_{G,E}^{∗}(ℓ_{0}), and let D be the subgroup of N_{G}(E) consisting of the elements which stabilize ℓ_{0} setwise. Then ℓ_{0} and D are uniquely determined up to conjugacy in N. Since ℓ_{0} originates in E, the centralizer C is equal to the pointwise stabilizer of ℓ_{0}. Any finite group of automorphisms of the line ℓ_{0} is cyclic of order prime to p, so we have C ⊴ D ≤ N with D/C a cyclic group of order prime to p. Finally, we set ℓ = res_{D,E}^{∗}(ℓ_{0}) ⊆ V_{D}(k), so that L = res_{G,D}^{∗}(ℓ).

Theorem 3.1. With the above notation, let e_{ℓ} be the idempotent kD-module corresponding to ℓ and e_{L} be the idempotent kG-module corresponding to L. Then

e_{ℓ} ↑^{G}≅ e_{L} ⊕ (projective).

Proof. Consider the composite map

e_{ℓ} ↑^{G}⟶^{λ_{ℓ}↑^{G}} k_{D} ↑^{G}⟶^{ν} k,

where ν : k_{D} ↑^{G}→ k is the augmentation map. On restriction to E, this becomes (modulo projectives) the composite map

⨁_{g∈N/D} g ⊗ e_{ℓ} ↓_{E}→ ( ⨁_{g∈N/D} g ⊗ k) ⊕ (induced modules) ⟶^{ν↓_{E}} k.

Here, g ∈ N/D means that g runs over a set of left coset representatives of D in N. The first map is the sum of all the maps

g ⊗ e_{ℓ} ↓_{E} ≅ e_{g(ℓ)} ↓_{E}
↓  ↓
g ⊗ k ≅  k

and the second map ν ↓_{E} sends ∑_{i} g_{i}⊗λ_{i} to ∑_{i} λ_{i}. On restriction to a cyclic shifted subgroup corresponding to a point in ℓ_{0}, the summands g ⊗ e_{ℓ} ↓_{E} for g ∉ D give projective modules, while 1 ⊗ e_{ℓ} restricts to give k ⊕ (projective), because ℓ_{0} isn’t fixed by any g ∈ N \ D. Moreover, this copy of k maps isomorphically to 1 ⊗ k in the second module, and then isomorphically to k in the third module. So if we complete to a triangle

e_{ℓ} ↑^{G}→ k → f → Ω^{−1}e_{ℓ} ↑^{G},

then f restricted to this cyclic shifted subgroup is projective.

The module e_{ℓ} ↑^{G} is a filtered colimit of modules in C_{L}, since e_{ℓ} is a filtered colimit of modules in C_{ℓ}. For M in C_{L}, Hom_{k}(M, f) is projective, by a combination of Dade’s lemma (the infinite dimensional version given in Section 3 of [2]) and Chouinard’s theorem [4], so f is C_{L}-local.

By Rickard’s characterization (see the remark after Proposition 5.7 of [5]), the triangle

e_{ℓ} ↑^{G}→ k → f → Ω^{−1}e_{ℓ} ↑^{G}

is isomorphic to

e_{L} → k → f_{L} → Ω^{−1}e_{L}.

Corollary 3.2. If M is a module whose variety V_{G}(M) = {L} with L as above, then M ⊕ (projective) is induced from D.

Proof. If V_{G}(M) ⊆ {L} then using the theorem, we have

M ⊕ (projective) ≅ M ⊗ e_{L} ≅ M ⊗ e_{ℓ} ↑^{G}≅ (M ↓_{D} ⊗e_{ℓ})↑^{G},

and so M ⊕ (projective) is induced from D.

This completes the proof of Theorem 1.5.

4. An equivalence of categories

We can combine the results of the last section with the Mackey decomposition theorem to obtain an equivalence of categories as follows. Let C be the full subcategory of StMod(kG) consisting of kG-modules M with V_{G}(M) ⊆ {L} (or equivalently M ≅ e_{L} ⊗ M), and let C′ be the full subcategory of StMod(kD) consisting of modules M′ with V_{D}(M′) ⊆ {ℓ} (or equivalently M′ ≅ e_{ℓ} ⊗ M′). Using the Mackey decomposition theorem, we see that if M′ is in C′ then M′ ↑^{G}↓_{D} is isomorphic to a direct sum of M′ with a module M′′ satisfying V_{D}(M′′) ∩ {ℓ} = ∅. So we have

e_{ℓ} ⊗ (M′ ↑^{G}↓_{D}) ≅ M′.

Since every object in C is induced from an object in C′ by Corollary 3.2, it follows that the functors (e_{ℓ} ⊗ −) ∘ res_{G,D} : C → C′ and ind_{D,G} : C′ → C are mutually inverse equivalences of categories.

Lemma 4.1. If M is a kG-module in C, which lies in the principal block B_{0}(kG), then e_{ℓ} ⊗M ↓_{D} is a direct sum of a projective module and a module in the principal block B_{0}(kD).

Conversely, if M is a kG-module in C with no summand in the principal block B_{0}(kG), then e_{ℓ} ⊗ M ↓_{D} is a direct sum of a projective module and a module with no summand in the principal block B_{0}(kD).

Proof. Let e be a block idempotent of kG, and let Br_{E} : Z(kG) → Z(kD) be the Brauer map with respect to E. If b is any block of kD, say with defect group R, then E ≤ R ≤ C, and so RC_{G}(R) ≤ C. So the Brauer correspondent b^{G} is defined, and by Brauer’s third main theorem, b^{G} is equal to B_{0}(kG) if and only if b = B_{0}(kD). It follows that if e_{0} is the principal block idempotent of kG and e_{1} is the principal block idempotent of kD, then Br_{E}(e_{0}) = e_{1}, and Br_{E}(e)e_{1} ≠ 0 if and only if e = e_{0}.

If M is a finitely generated kG-module with e.M = M, then Nagao’s lemma says that

M ↓_{D}≅ Br_{E}(e).M ↓_{D} ⊕M_{1}

where M_{1} is a direct sum of modules which are projective relative to subgroups Q ≤ C with E ≰ Q. Since the variety of e_{ℓ} ⊗ M ↓_{D} has trivial intersection with the image of V_{E′} → V_{D} for any proper subgroup E′ of E, it follows that

e_{ℓ} ⊗ M ↓_{D}≅ Br_{E}(e).(e_{ℓ} ⊗ M ↓_{D}) ⊕ M_{2}

where M_{2} is projective.

If M = e.M is not finitely generated, express it as a filtered colimit of finitely generated modules M_{α} in C. Each e_{ℓ} ⊗ M_{α} ↓_{D} may be written as a direct sum of Br_{E}(e).(e_{ℓ} ⊗ M_{α} ↓_{D}) and a projective module killed by Br_{E}(e). There are no maps between these two types of summands, so when we pass to the colimit, we obtain a decomposition of e_{ℓ} ⊗ M ↓_{D} of the desired form.

Theorem 4.2. The functors (e_{ℓ} ⊗ −) ∘ res_{G,D} : C → C′ and ind_{D,G} : C′ → C are mutually inverse equivalences of categories, and induce mutually inverse equivalences between the full subcategories B_{0}(kG) ∩ C and B_{0}(kD) ∩ C′.

Proof. This follows immediately from the lemma and the discussion preceding it.

5. The main theorems

We continue with the same notation. Namely, L is a line through the origin in V_{G}(k) originating in an elementary abelian p-subgroup E of G. We set C = C_{G}(E), N = N_{G}(E) and L = res_{G,E}^{∗}(ℓ_{0}), with ℓ_{0} a line through the origin in V_{E}(k). We set D equal to the stabilizer in N of the line ℓ_{0}. We set ℓ_{1} = res_{C,E}^{∗}(ℓ_{0}) ⊆ V_{C}(k) and ℓ = res_{D,E}^{∗}(ℓ_{0}) ⊆ V_{D}(k).

Lemma 5.1. Suppose that C is p-nilpotent. Then for any module M′ in B_{0}(kD) satisfying V_{D}(M′) = {ℓ}, we have Ĥ^{n}(D, M′) ≠ 0 for some n.

Proof. The argument for this is given in the proof of Proposition 6.8 of [3]; we repeat it here for convenience. Let C̄ = C/O_{p′}(C) and D̄ = D/O_{p′}(C). Then C̄ is a p-group, and D̄/C̄ is a cyclic p′-group. By Lemma 6.7 of [3], we may choose a homogeneous element ζ ∈ H^{m}(C̄, k) = H^{m}(C, k) for some m, so that ℓ_{1} ∩ V_{C}(⟨ζ⟩) = {0}, and so that the one dimensional subspace ⟨ζ⟩ ⊆ H^{∗}(C, k) is D̄/C̄-invariant and affords a faithful one dimensional representation of D̄/C̄. For a suitable one dimensional representation ε of D with kernel C, ζ may be regarded as an element of Ext_{kD}^{m}(k, ε). Thus ζ is represented by a homomorphism ζ̂ : Ω^{m}(k) → ε, and we write L_{ζ} for the kernel of ζ̂. So there is a short exact sequence of kD-modules

0 → L_{ζ} → Ω^{m}(k) → ε → 0.

Tensoring with M′, we obtain a short exact sequence

0 → L_{ζ} ⊗ M′ → Ω^{m}(k) ⊗ M′ → ε ⊗ M′ → 0.

The tensor product theorem for varieties (Theorem 10.8 of [2]) implies that L_{ζ} ⊗ M′ is projective, and so we obtain a stable isomorphism Ω^{m}(M′) ≅ ε ⊗ M′.

Since M′ is non-projective, for some value of r we have

Êxt_{kD}^{0}(ε^{r}, M′) = \underline{Hom}_{kG}(ε^{r}, M′) ≠ 0,

where ε^{r} denotes the rth tensor power of ε. This is because every simple module in B_{0}(kD) is isomorphic to some such ε^{r}. Thus

Ĥ^{mr}(D, M′) ≅ Ĥ^{0}(D, Ω^{−mr}(M′)) ≅ Ĥ^{0}(D, ε^{−r} ⊗ M′) ≅ Êxt_{kD}^{0}(ε^{r}, M′) ≠ 0.

Here, ε^{−r} denotes the rth tensor power of the dual module ε^{∗}.

Theorem 5.2. Suppose that M is a module in B_{0}(kG) with V_{G}(M) = {L}, and that C is p-nilpotent. Then Ĥ^{n}(G, M) ≠ 0 for some n.

Proof. By Theorem 4.2, there is a module M′ in B_{0}(kD) with M′ ↑^{G}≅ M ⊕ (projective). By Shapiro’s lemma we have Ĥ^{n}(G, M) ≅ Ĥ^{n}(D, M′). By Lemma 5.1, this is nonzero for some n.

Corollary 5.3. Suppose that M is a non-projective kG-module in B_{0}(kG) with the property that V_{G}(M) contains no closed homogeneous subset of the nucleus Y_{G}. Then Ĥ^{n}(G, M) ≠ 0 for some n.

Proof. We use the argument given in Theorem 6.4 of [3] to reduce to the complexity one case. Let K be an algebraically closed extension of k of large transcendence degree. Since M is non-projective, V_{G}(M) contains a closed homogeneous irreducible subset V which is not contained in Y_{G}. So V_{G}(K ⊗_{k} M) contains a generic line L for V. Choose elements ζ_{1}, …, ζ_{s} ∈ H^{∗}(G, K) so that

V_{G}(K ⊗_{k} M) ∩ V_{G}⟨ζ_{1}⟩ ∩ ⋯ ∩ V_{G}⟨ζ_{s}⟩ = {L}.

Here, V_{G}⟨ζ_{i}⟩ is the collection of closed homogeneous subsets of the hypersurface V_{G}⟨ζ_{i}⟩ defined by ζ_{i}. Then we have

V_{G}((K ⊗_{k} M) ⊗_{K} L_{ζ_{1}} ⊗_{K} ⋯ ⊗_{K} L_{ζ_{s}}) = {L}.

Next, we note that in Lemma 6.3 of [3], although M_{2} needs to be finitely generated, M_{1} does not. So every non-projective summand of M_{1} ⊗ L_{ζ} is in the same block as M_{1}. So every non-projective summand of (K ⊗_{k} M) ⊗_{K} L_{ζ_{1}} ⊗_{K} ⋯ ⊗_{K} L_{ζ_{s}} is in B_{0}(KG), and by the theorem, we have

Ĥ^{n}(G, (K ⊗_{k} M) ⊗_{K} L_{ζ_{1}} ⊗_{K} ⋯ ⊗_{K} L_{ζ_{s}}) ≠ 0

for infinitely many values of n, positive and negative.

Similarly, in Lemma 6.2 of [3], although M_{1} must be finitely generated, M_{2} need not be. So if ζ is a homogeneous element in cohomology, then Ĥ^{n}(G, M_{2}) = 0 for all n implies Ĥ^{n}(G, L_{ζ} ⊗ M_{2}) = 0 for all n. So we may deduce that Ĥ^{n}(G, K ⊗_{k} M) ≠ 0 for infinitely many values of n, positive and negative. Finally, this implies that the same is true of Ĥ^{n}(G, M).

Proposition 5.4. If M is an N H-local kG-module, then V_{G}(M) contains no closed homogeneous subset of the nucleus Y_{G}.

Proof. If V_{G}(M) contains a closed homogeneous subset V of Y_{G}, then

\underline{Hom}_{kG}(E_{V} (M), M) ≠ 0,

while if M is N H-local, E_{N H}(M) = 0. However, any map from E_{V} (M) to M factors through E_{N H}(M), because the subcategory of stmod(kG) consisting of modules with variety in V is contained in N H.

Theorem 5.5. If M is a module in B_{0}(kG), then E_{N H}(M) ≅ M and F_{N H}(M) = 0.

Proof. Consider the variety of F_{N H}(M). By Proposition 5.4, it contains no closed homogeneous subset of the nucleus Y_{G}. So if F_{N H}(M) is nonzero in StMod(kG) (i.e., nonprojective), its variety must contain some closed homogeneous subset which is not in the nucleus. Then by Corollary 5.3, we have Ĥ^{n}(G, F_{N H}(M)) ≠ 0 for infinitely many values of n. So for some n, we have \underline{Hom}_{kG}(Ω^{n}(k), F_{N H}(M)) ≠ 0. Since Ω^{n}(k) is an NH module, this contradicts the fact that F_{N H}(M) is N H-local. It follows that F_{N H}(M) = 0, and therefore that E_{N H}(M) ≅ M.

Proof of Theorem 1.2. By Theorem 5.5, if M is in B_{0}(kG), then E_{N H}(M) ≅ M. So M is a filtered colimit of NH modules, and since it is finitely generated, it follows that it is a direct summand of an NH module.

Proof of Corollary 1.3. It is shown in Corollary 10.12 of [3] that this follows from Theorem 1.2.

Theorem 5.6. Suppose that the centralizer of every element of order p in G is p-nilpotent. Then every finitely generated module in the principal block is a trivial homology module.

Proof. The condition on G is equivalent to the condition that Y_{G} = {0}. So under these conditions, nuclear homology modules are the same as trivial homology modules. So the theorem follows from Theorem 1.2, using Theorem 3.5 and Propositions 4.4 and 4.5 of [3].

Proof of Theorem 1.4. It is proved in [3] that (A) ⇔ (A′) ⇔ (A′′) ⇒ (B) ⇔ (B′) ⇒ (C) ⇔ (D) ⇔ (D′) ⇔ (E). It is clear that (A′′′) ⇒ (A′), (B′′) ⇒ (B) and (B′′) ⇒ (B′′′) ⇒ (B′). Theorem 5.6 shows that (D) ⇒ (A′′′). Finally, Corollary 5.3 shows that (D) ⇒ (B′′).

Proof of Theorem 1.1. This is just the statement that (B′′) ⇔ (D) in Theorem 1.4, so this is now proved.

References

[1] D. J. Benson, J. F. Carlson and J. Rickard. Complexity and varieties for infinitely generated modules. To appear in Math. Proc. Camb. Phil. Soc.

[2] D. J. Benson, J. F. Carlson and J. Rickard. Complexity and varieties for infinitely generated modules, II. Preprint, 1995.

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

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

[5] J. Rickard. Idempotent modules in the stable category. Preprint, 1994.

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

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