On the vanishing of group cohomology

D. J. Benson Mathematical Institute

24–29 St. Giles Oxford OX1 3LB Great Britain

J. F. Carlson^{∗} Dept. of Mathematics University of Georgia Athens, GA30602 U.S.A.

G. R. Robinson Dept. of Mathematics U.M.I.S.T. P.O. Box 88 Manchester M60 1QD Great Britain

To Walter Feit on his sixtieth birthday.

1 Introduction

One of the fundamental questions in the cohomology of finite groups is the following. Given a module M for a finite group G, for what values of n do we have H^{n}(G, M) = 0? This paper consists of two related contributions to the study of this question.

Our first theorem is best stated in terms of Tate cohomology.

Theorem 1.1 If R is a commutative ring of coefficients, G a finite group and M an RGmodule, and if Ĥ^{n}(G, M) ≠ 0 for some n, then Ĥ^{n}(G, M) ≠ 0 for infinitely many values of n, positive and negative.

Secondly, we address the question of which groups have the property that every nonprojective module in the principal block has cohomology. We provide some evidence for the following conjecture.

Conjecture 1.2 Suppose G is a finite group and k is a field of characteristic p. Then the following are equivalent.

(i) The centraliser of every element of order p in G is p-nilpotent (i.e., has a normal p-complement).

(ii) For every non-projective module M in the principal block B_{0}(kG), H^{n}(G, M) ≠ 0 for some (and hence for infinitely many) n.

We are able to prove this conjecture as long as p is odd, and more generally we can prove that condition (ii) of the above conjecture implies condition (i) (Proposition 5.3 and Theorem 9.5):

Theorem 1.3 If p is odd then Conjecture 1.2 is true.

∗Partially supported by a grant from the NSF.

We are also able to prove that condition (i) is equivalent to condition (ii) for nonprojective trivial source modules M in the principal block, and that it is equivalent to condition (ii) with M replaced by its Green correspondent. More precisely, we shall prove the following.

Theorem 1.4 Suppose G is a finite group and k is a field of characteristic p. We have the implications (A) ⇔ (A′) ⇔ (A′′) ⇒ (B) ⇔ (B′) ⇒ (C) ⇔ (D) ⇔ (D′) ⇔ (E) ⇒ (F) among the following statements.

(A) Every finitely generated module in B_{0}(kG) is a trivial homology module (or TH module; for the definition, see Section 2).

(A′) Every simple module in B_{0}(kG) is a direct summand of a TH module.

(A′′) Every trivial source module in B_{0}(kG) is a direct summand of a TH module. (B) For every finitely generated non-projective module M in B_{0}(kG), we have

H^{n}(G, M) ≠ 0 for some n.

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

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

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

(D′) The centraliser of every non-trivial p-subgroup of G is p-nilpotent.

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

(F) The Cartan matrix of B_{0}(kG) has only one non-unit principal divisor. In other words, the Brauer characters of the projective modules and the trivial module span all Brauer characters in the principal block.

If p is odd then (D) ⇒ (B).

We also give some evidence for the implication (F) ⇒ (A′). Namely, we show that for groups with dihedral Sylow subgroups in characteristic 2, as well as for A_{6} in characteristic 3 and P SL_{2}(8) in characteristic 2, every simple module in the principal block is a TH module. A new idea seems to be necessary if one is to prove this more generally.

In Section 10 we give an interpretation of the above conjecture in terms of varieties. Namely, we define two subsets of the cohomology variety V_{G} which we call the nucleus Y_{G} and the representation theoretic nucleus Θ_{G}, and we conjecture that these are equal. We prove the containment Θ_{G} ⊇ Y_{G}.

We shall prove that for any module M in the principal block, the variety of M is contained in the union of Θ_{G} and the variety defined by the annihilator of H^{∗}(G, M).

Finally, we formulate a conjecture for arbitrary finite groups which generalises the conjectured implication (F) ⇒ (A′) discussed above. This is given in terms of nuclear homology modules, which are a generalisation of trivial homology modules. For the sake of a non-trivial example, we prove this conjecture for the Mathieu group M_{11} in characteristic 2.

This work grew out of an attempt to study a well known question in group cohomology. If S is a simple module in the principal block B_{0}(kG), is it true that for some value of n > 0 we have H^{n}(G, S) ≠ 0? At present no example is known where this fails. For other contributions to the study of this question, see Linnell [16], and Linnell and Stammbach [17, 18].

This paper was written while the first two authors were visiting UMIST, for a representation theory symposium funded by the SERC. We would like to take this opportunity to thank the SERC and all those involved in making this meeting possible.

2 Trivial homology modules

We begin with the definition. Let R be a commutative ring of coefficients and G a finite group.

Definition 2.1 An RG-module M is said to be a trivial homology module or TH module if there exists a finite complex

C_{r} →^{θ_{r}} C_{r−1} → ⋯ →^{θ_{2}} C_{1} →^{θ_{1}} C_{0}

of RG-modules and homomorphisms such that each C_{i} is a projective RG-module, H_{i}(C_{∗}) is a direct sum of copies of the trivial RG-module R for i > 0, and H_{0}(C_{∗}) ≅ M.

The following theorem was proved in [1, Lemma 6.2].

Theorem 2.2 The trivial RG-module R is a TH module. Furthermore, given any positive integer B, there is a complex (C_{∗}, θ) as in the definition, with H_{0}(C_{∗}) ≅ H_{r}(C_{∗}) ≅ R and H_{i}(C_{∗}) = 0 for 1 ≤ i ≤ B. □

Actually, in [1] it is only shown that such a complex exists for R = Z the rational integers. However if C_{∗} is such a complex of ZG-modules then R ⊗_{Z} C_{∗} is a complex of RG-modules with the desired properties.

Theorem 2.3 Let M be an R-projective RG-module. Let (C_{∗}, θ) be a finite complex of projective RG-modules. Then there exists a cohomology spectral sequence whose E_{2} term is the Tate cohomology

E_{2}^{st} = Êxt_{RG}^{s}(H_{t}(C_{∗}), M)

and which converges to zero.

Proof Let (F_{∗}, ∂) be a doubly infinite projective resolution

(F_{∗}, ∂) : ⋯ → F_{1} →^{∂_{1}} F_{0} →^{∂_{0}} F_{−1} → ⋯

of the trivial RG-module R. That is, the sequence is exact, each F_{i} is projective and ∂_{0}(F_{0}) ≅ R ≅ Ker(∂_{−1}). We wish to investigate the cohomology of the total complex of the double complex Hom_{RG}(F_{∗} ⊗_{R} C_{∗}, M). We have two coboundary maps on this complex, namely (1 ⊗ θ)^{∗} and (∂ ⊗ 1)^{∗}, arising from the boundaries θ on C and ∂ on F, respectively. Consequently we get two spectral sequences converging to the cohomology of the total complex, according to which of the two coboundary maps we use first. Now notice that the complex (F_{∗} ⊗ C_{t}, ∂ ⊗ 1) is a doubly infinite projective resolution of the projective module C_{t}. Hence it is totally split and its cohomology with any coefficients is zero. Therefore the cohomology of the total complex is also zero.

On the other hand, (F_{s}⊗C_{∗}, 1⊗θ) is also a totally split complex since F_{s} is a projective module and C_{t} is R-projective. However, this one has cohomology:

H_{t}(F_{s} ⊗ C_{∗}, 1 ⊗ θ) = F_{s} ⊗ H_{t}(C_{∗}, θ).

The splitting shows that the E_{1} term of the spectral sequence is

E_{1}^{st} ≅ Hom_{RG}(F_{s} ⊗ H_{t}(C_{∗}), M).

To get the E_{2} term we take homology with respect to the map induced by ∂ on F_{∗}. But (F_{∗} ⊗ H_{t}(C_{∗}), ∂ ⊗ 1) is a projective resolution of H_{t}(C_{∗}). So E_{2}^{st} is as given. □

Remark The construction given above is in some sense an algebraic analogue of the equivariant cohomology spectral sequence

H^{s}(BG; H^{t}(X; R)) ⇒ H_{G}^{s+t}(X; R)

for a finite G-CW-complex X, in the case where the action is free, so that H_{G}^{∗}(X; R) ≅ H^{∗}(X/G; R). Use of Tate cohomology gets rid of the finite E_{∞} term.

We are now ready to prove Theorem 1.1. We state it again here in a slightly stronger form.

Theorem 2.4 Given a finite group G, there exists a positive integer r such that for any commutative ring R of coefficients and any RG-module M, if Ĥ^{n}(G, M) = 0 for r + 1 consecutive values of n then Ĥ^{n}(G, M) = 0 for all n positive and negative.

Proof We first prove this in the case where M is R-projective. Let

(C_{∗}, θ) : C_{r} → ⋯ → C_{1} → C_{0}

be a complex of projective RG-modules as in Theorem 2.2 displaying R as a TH module, and consider the spectral sequence described in Theorem 2.3. Suppose that Ĥ^{n}(G, M) is not always zero, but is zero for r + 1 consecutive values of n, say m, m + 1, …, m + r, with either Ĥ^{m−1}(G, M) or Ĥ^{m+r+1}(G, M) non-zero. Now the top and bottom rows of this spectral sequence are

E_{2}^{sr} ≅ Êxt_{RG}^{s}(H_{r}(C_{∗}), M) ≅ Ĥ^{s}(G, M)
E_{2}^{s0} ≅ Êxt_{RG}^{s}(H_{0}(C_{∗}), M) ≅ Ĥ^{s}(G, M)

and so either E_{2}^{m−1,r} ≠ 0 or E_{2}^{m+r+1,0} ≠ 0. However, E_{2}^{st} = 0 for m ≤ s ≤ m + r. The differential d_{k} on the E_{k} term takes E_{k}^{st} to E_{k}^{s+k,t−k+1}, and d_{k} = 0 for k > r + 1. So whichever of E_{2}^{m−1,r} or E_{2}^{m+r+1,0} is non-zero, this group can never get killed at any stage in the spectral sequence. This contradicts the fact that the spectral sequence converges to zero.

We now deal with the general case where M is not necessarily R-projective. In this case, we first remark that since Ĥ^{n}(G, M) does not change when the coefficient ring R is replaced by Z, we may as well assume R is Z. Note at this stage that we have not assumed that M is finitely generated. Now if

0 → M′ → F → M → 0

is a short exact sequence with F a free ZG-module, then M′ is Z-free [12, Theorem I.5.1]. Since Ĥ^{n}(G, F) = 0 (see for example [3, VI.5.3]), Ĥ^{n}(G, M) ≅ Ĥ^{n+1}(G, M′) and so the theorem for M follows from the theorem for M′. □

Remark The proof of the above theorem gives explicit values of r in terms of the generators of the cohomology ring (via the proof of Theorem 2.1 in [1]). Better constants r may be found by refining the argument given.

Corollary 2.5 H^{∗}(G, M) ≠ 0 if and only if Ĥ^{∗}(G, M) ≠ 0. □

Corollary 2.6 Suppose R is a Noetherian ring and M is a finitely generated RG-module with the property that some power of every element of positive degree in H^{∗}(G, R) annihilates H^{∗}(G, M) (by cup product). Then H^{∗}(G, M) = 0.

Proof By Evens [8, Theorem 6.1, Corollary 6.2] H^{∗}(G, M) is finitely generated as a module over H^{∗}(G, R), and the latter is a finitely generated ring over R. So if some power of every element of positive degree in H^{∗}(G, R) annihilates H^{∗}(G, M), then H^{∗}(G, M) is non-zero only in finitely many degrees. Thus by Theorem 2.4, H^{∗}(G, M) = 0. □

Proposition 2.7 Suppose M_{1} is a TH RG-module, and M_{2} is any RG-module. Then either Êxt_{RG}^{n}(M_{1}, M_{2}) = 0 for all n, or Ĥ^{n}(G, M_{2}) ≠ 0 for infinitely many n, positive and negative.

Proof As in the proof of Theorem 2.4, we may reduce to the case where M_{2} is R-projective. In this case, let

(C_{∗}, θ) : C_{r} → ⋯ → C_{1} → C_{0}

be a complex of projective RG-modules displaying M_{1} as a TH module, and consider the spectral sequence

Êxt_{RG}^{s}(H_{t}(C_{∗}), M_{2}) ⇒ 0

described in Theorem 2.3. If Ĥ^{n}(G, M_{2}) = 0 for all n, then the only possible non-zero row of this spectral sequence is

E_{2}^{s0} = Êxt_{RG}^{s}(M_{1}, M_{2}).

Since the spectral sequence converges to zero, this row too must be zero. Thus if Êxt_{RG}^{n}(M_{1}, M_{2}) ≠ 0 for some n, then Ĥ^{n}(G, M_{2}) ≠ 0 for some n, and hence by Theorem 2.4, for infinitely many n, positive and negative. □

Corollary 2.8 Suppose M is a non-projective TH module. Then Ĥ^{n}(G, M) ≠ 0 for infinitely many n, positive and negative.

Proof Apply the above proposition with M_{1} = M_{2} = M. By [5], for M non-projective, Êxt_{RG}^{∗}(M, M) ≠ 0. We can also see this by noticing that there is a non-trivial element in Êxt_{RG}^{0}(M, M), and using Theorem 2.4. □

This proves the implication (A) ⇒ (B) in Theorem 1.4.

3 Properties of trivial homology modules

The basic constructions here depend on the fact that if

(C_{∗}, θ) : ⋯ → C_{r} → ⋯ → C_{0}

is a complex of projective modules and

(D_{∗}, ϕ) : ⋯ → D_{r} → ⋯ → D_{0}

is exact then any map H_{0}(C) → H_{0}(D) extends to a chain map (C, θ) → (D, ϕ). For a proof of this well known fact, see for example [12, Theorem IV.4.1].

Lemma 3.1 Suppose that M is a TH module and B is a positive integer. Then there exists a complex (C_{∗}, θ) as in the definition, with H_{i}(C_{∗}) = 0 for 1 ≤ i ≤ B.

Proof We prove this by induction on B. Suppose

(C_{∗}, θ) : 0 → C_{r} → ⋯ → C_{0} → 0

is such a complex with H_{i}(C_{∗}) = 0 for 1 ≤ i ≤ B − 1. Let (D_{∗}, ϕ) be a complex as in Theorem 2.2 with H_{i}(D_{∗}) = 0 for 1 ≤ i ≤ r − B. Then we may form a homomorphism of complexes

0 →  C_{r}  → ⋯ → C_{B+1} →  C_{B}  → ⋯ → C_{0} → 0
↓  ↓  ↓
0 → · · · · · · →D_{r−B} ⊗ H_{B}(C_{∗})→ ⋯ →D_{1} ⊗ H_{B}(C_{∗})→D_{0} ⊗ H_{B}(C_{∗})→ 0

By adding a suitable exact complex of projective modules to (C_{∗}, θ), we may convert this into a surjective homomorphism of complexes. By the long exact sequence of homology, the kernel of this is a complex with the desired properties. □

Proposition 3.2 Suppose that 0 → M_{1} → M_{2} →^{σ} M_{3} → 0 is a short exact sequence of RG-modules, and two of the terms are TH modules. Then so is the third.

Proof We first treat the case where M_{2} and M_{3} are TH modules. In this case, let

(C_{∗}, θ) : 0 → C_{r} → ⋯ → C_{0} → 0

be a complex displaying M_{2} as a TH module. Using Lemma 3.1, we choose a complex

(D_{∗}, ϕ) : 0 → D_{s} → ⋯ → D_{0} → 0

displaying M_{3} as a TH module, and with H_{i}(D_{∗}) = 0 for 1 ≤ i ≤ r. The given homomorphism M_{2} → M_{3} now extends to a map of complexes

0 → C_{r} → ⋯ → C_{0} → 0
↓  ↓
0 → D_{s} → ⋯ → D_{r} → ⋯ → D_{0} → 0

Just as in the proof of Lemma 3.1, we now add on an exact sequence of projective modules to (C_{∗}, θ) to make the map surjective, and take the kernel. The long exact sequence of homology shows that this kernel displays M_{1} as a TH module.

To treat the remaining cases, we make the observation that if 0 → M′ → P → M → 0 is a short exact sequence with P projective then M′ is a TH module if and only if M is.

If M_{1} and M_{2} are TH modules, we choose a projective module P and a surjection ν : P → M_{3} with kernel M′_{3}. Applying the snake lemma to the diagram

0 →  0  →  P  → P → 0
↓  (\begin{matrix}1 \\ 0 \end{matrix}) ↓  ↓ν
0 → P ⊕ M_{1} →^{(ν,σ)} P ⊕ M_{2} → M_{3} → 0

we see that there is a short exact sequence 0 → M′_{3} → P ⊕ M_{1} → M_{2} → 0. Since P ⊕ M_{1} and M_{2} are TH modules, so is M′_{3}, and hence so is M_{3}.

If on the other hand M_{1} and M_{3} are TH modules, we choose a projective module P mapping onto M_{2} with kernel M′_{2}. Composing with the surjection M_{2} → M_{3}, we see that P also maps onto M_{3}, and we write M′_{3} for the kernel. Applying the snake lemma to the diagram

0 → 0 → P → P → 0
↓ ↓ ↓
0 → M_{1} → M_{2} → M_{3} → 0

we see that there is a short exact sequence 0 → M′_{2} → M′_{3} → M_{1} → 0. Since M_{1} and M′_{3} are TH modules, so is M′_{2}, and hence so is M_{2}. □

Proposition 3.3 If M_{1} and M_{2} are TH kG-modules where k is a field then so is M_{1}⊗_{k}M_{2}.

Proof Let

(C_{∗}, θ) : 0 → C_{r} → ⋯ → C_{0} → 0

and

(D_{∗}, ϕ) : 0 → D_{s} → ⋯ → D_{0} → 0

be complexes displaying M_{1} and M_{2} as TH modules. We augment these complexes to

C′_{∗} : 0 → C_{r} → ⋯ → C_{0} → M_{1} → 0

and

D′_{∗} : 0 → D_{s} → ⋯ → D_{0} → M_{2} → 0

and form the tensor product

C′_{∗} ⊗ D′_{∗} : 0 → C_{r} ⊗ D_{s} → ⋯ → C_{0} ⊗ M_{2} ⊕ M_{1} ⊗ D_{0} → M_{1} ⊗ M_{2} → 0.

By the Künneth theorem this complex has all its homology a sum of trivial modules, and so its truncation at M_{1} ⊗ M_{2} is a complex displaying M_{1} ⊗ M_{2} as a TH module. □

Remark This works equally well when k is replaced by a hereditary commutative coefficient ring R.

Proposition 3.4 Suppose M is a TH kG-module, where k is a field. Then so is M^{∗} = Hom_{k}(M, k).

Proof Let

(C_{∗}, θ) : 0 → C_{r} → ⋯ → C_{0} → 0

be a complex of projective modules displaying M as a TH module. First we form the dual complex

(C′_{∗}, θ′) : 0 → C′_{0} → ⋯ → C′_{−r} → 0

with C′_{−r} = (C_{r})^{∗}. Now we move copies of k in the homology from negative degree to positive degree one at a time as follows. Choose a complex

(D_{∗}, ϕ) : 0 → D_{s} → ⋯ → D_{0} → 0

giving k as a TH module, with H_{i}(D_{∗}) = 0 for 1 ≤ i ≤ r. Then a non-zero map H_{−r}(C′_{∗}) → H_{0}(D_{∗}) extends to a map of complexes

\begin{aligned}0 → & C′_{0} & → ⋯ → & C′_{−r} & → 0 \\ & ↓ & & ↓ \\ 0 → D_{s} → ⋯ → & D_{r} & → ⋯ → & D_{0} & → 0 \end{aligned} .

We add on an exact sequence of projective modules to (C′_{∗}, θ′) to make the map surjective, and take the kernel. The long exact sequence of homology shows that there is one less copy of k in negative degree in the resulting kernel. Continuing this way we delete copies of k one at a time from negative degree and add copies in positive degree. Eventually we arrive at a complex displaying M^{∗} as a TH module. □

Theorem 3.5 Let k be a field and G a finite group. Suppose every simple module in B_{0}(kG) is a TH module. Then every finitely generated module in B_{0}(kG) is a TH module. Thus every finitely generated non-projective module M in B_{0}(kG) satisfies Ĥ^{n}(G, M) ≠ 0 for infinitely many n, positive and negative.

Proof This follows easily from Corollary 2.8 and Proposition 3.2. □

Lemma 3.6 Suppose M_{1} and M_{2} ⊕ M_{3} are TH modules and θ : M_{1} → M_{2} is surjective. Then M_{2} ⊕ Ker(θ) is a TH module. Proof This applies by applying Proposition 3.2 to the short exact sequence

0 → M_{2} ⊕ Ker(θ) → M_{1} ⊕ M_{2} ⊕ M_{3} \underrightarrow{( \begin{matrix}θ & 0 & 0 \\ 0 & 0 & 1 \end{matrix} )} M_{2} ⊕ M_{3} → 0

in which the last two terms are TH modules. □ Proposition 3.7 A module M is a direct summand of a TH module if and only if M ⊕ Ω(M) is a TH module, where Ω(M) is the kernel of a map P → M with P projective. Proof Take M_{2} = M and M_{1} = P in the above lemma. □ Corollary 3.8 Suppose M is a non-projective direct summand of a TH module. Then Ĥ^{n}(G, M) ≠ 0 for infinitely many n, positive and negative. Proof By the proposition, M ⊕ Ω(M) is a TH module. But

Ĥ^{n}(G, M ⊕ Ω(M)) ≅ Ĥ^{n}(G, M) ⊕ Ĥ^{n−1}(G, M). □

This proves the implication (A′) ⇒ (B) of Theorem 1.4.

4 Examples in which all modules are TH modules

We start off by looking at p-groups. Proposition 4.1 Let k be a field of characteristic p and G a finite p-group. Then every kG-module is a TH module. Proof This follows from Theorems 3.5 and 2.2. □ Corollary 4.2 Let k be a field of characteristic p and G a finite p-nilpotent group. Then every module in B_{0}(kG) is a TH module. □ Theorem 4.3 Suppose R is the ring of integers in an algebraic number field or its localisation or completion at a prime ideal lying above p, and G is a finite p-group. Then every finitely generated RG-module is a summand of a TH module.

Proof First we deal with modules of finite length. For this purpose it is sufficient to deal with a simple module of finite length. Such a module is of the form R/℘ for some prime ideal ℘ lying above p. Since ℘, regarded as a RG-module, is a summand of a sum of copies of R, it is a summand of a TH module, and hence by Proposition 3.2 so is R/℘.

We now deal with the general finitely generated module. Let F be the field of fractions of R, and first suppose F is a splitting field. Then the left regular representation is F G ≅ ⨁_{i} n_{i}S_{i} where n_{i} = dim(S_{i}). Thus the left regular representation of RG contains a submodule L of finite index with L ≅ ⨁_{i} n_{i}L_{i} and F L_{i} ≅ S_{i}. Since RG/L is a torsion module of finite length, and hence a summand of a TH module, Proposition 3.2 shows that L is a summand of a TH module, and hence each L_{i} is a summand of a TH module. But now any finitely generated RG-module has a submodule isomorphic to a sum of modules of the form L_{i}, with finite quotient, and so another application of Proposition 3.2 completes this case.

In case F is not a splitting field, we extend to a splitting field, form the required complex there, and then restrict back. The restriction of the trivial module R is a sum of trivial modules, so this proves the theorem in this case. □

In the case of a field of coefficients, in order that every simple module in the principal block be a TH module, it is certainly necessary that the characters of the trivial module and the projectives in the principal block should additively generate all characters in the principal block. This is equivalent to the condition that the Cartan matrix of the principal block have only one principal divisor which is not a unit. We now prove a partial converse to this.

Proposition 4.4 Suppose that the Cartan matrix of B_{0}(kG) has only one non-unit principal divisor. If every simple module in B_{0}(kG) is a summand of a TH module, then every simple module in B_{0}(kG) is a TH module.

Proof Suppose P is a projective module in B_{0}(kG), and Q is a maximal submodule of P. Applying Lemma 3.6 with P in place of M_{1} and P/Q in place of M_{2}, we see that Q ⊕ P/Q is a TH module. If Q′ is a maximal submodule of Q, then the same procedure shows that Q′ ⊕ Q/Q′ ⊕ P/Q is a TH module. Continuing this way down a composition series for P, we see that a semisimple module with the same composition factors as P is a TH module.

Now by the condition on the Cartan matrix, given any simple module S, there are modules M_{1} and M_{2}, each a sum of projectives in the principal block and copies of the trivial module, such that the Brauer character of M_{1} is the same as that of M_{2} ⊕ S. By the above argument, semisimple modules M′_{1} and M′_{2} with the same composition factors as M_{1} and M_{2} are TH modules. The proposition now follows by applying Proposition 3.2 to the short exact sequence 0 → M′_{2} → M′_{1} → S → 0. □

The following proposition, which also proves the implication (D) ⇒ (F) in Theorem 1.4, facilitates checking the conditions of Proposition 4.4.

Proposition 4.5 Suppose that the centraliser of every element of order p in G is pnilpotent. Then the Cartan matrix of B_{0}(kG) has only one non-unit principal divisor.

Proof This is well known, but for the convenience of the reader we include a proof.

We shall show that if ϕ is a Brauer character in the principal block, then ϕ−ϕ(1).1 is the character of a virtual projective (i.e., an integral combination of characters of projective modules), where 1 denotes the trivial character. For then the Z-module generated by the Brauer characters in B_{0}(kG) has a basis of the form 1 = ϕ_{1}, ϕ_{2} − ϕ_{2}(1).1, …, ϕ_{l} − ϕ_{l}(1).1, where the ϕ_{i} are the irreducible Brauer characters in the principal block, and this proves the proposition.

To show that ϕ − ϕ(1).1 is the character of a virtual projective, it suffices to show that the class function θ given by ϕ − ϕ(1).1 on p-regular elements and zero on p-singular elements, is a generalised character. For then since it vanishes on p-singular elements it must be a virtual projective.

If y is a p′-element of G, we now show that ϕ(y) ≡ ϕ(1) (mod |C_{G}(y)|_{p}). If x is a non-trivial p-element centralising y, then for any irreducible character ψ in B_{0}(C_{G}(x)) we have ψ(x) = ψ(xy) since C_{G}(x) is p-nilpotent. Thus by Brauer’s second and third main theorems, it follows that if we let ϕ̃ be an ordinary generalised character in B_{0} lifting ϕ (the decomposition map is surjective) then ϕ̃(xy) = ϕ̃(x). If Q is a Sylow p-subgroup of C_{G}(y), and we define ϕ′ on Q by ϕ′(x) = ϕ̃(xy), then ϕ′ is an algebraic integer combination of characters of Q. Thus (ϕ(y) − ϕ(1))/|Q| = (ϕ′ − ϕ̃|_{Q}, 1)_{Q} is an algebraic integer.

It now follows for example from [21, Lemma 2(iii)] that θ is a generalised character of G, so that ϕ − ϕ(1).1 is the character of a virtual projective and we are done. □

Theorem 4.6 Suppose k is a field of characteristic two and G is a finite group with dihedral Sylow two-subgroups. Then every simple module in B_{0}(kG) is a TH module. Thus every non-projective module M in B_{0}(kG) satisfies Ĥ^{n}(G, M) ≠ 0 for infinitely many n, positive and negative.

Proof We use the structure of the projective indecomposable modules given in Erdmann [7]. According to the classification by Gorenstein and Walter, we have three possibilities for G/O_{2′}(G).

(i) P SL_{2}(q) ≤ G/O_{2′}(G) ≤ P ΓL_{2}(q), q odd. We first treat the case where P SL_{2}(q) ≤ G/O_{2′}(G) but P GL_{2}(q) ≰ G/O_{2′}(G). In this case, if q ≡ 1 (mod 4) then there is a short exact sequence

0 → L ⊕ M → Ω^{2}k → k → 0

where L and M are the non-trivial simple modules in the principal block. Thus by Propositions 3.2 and 4.4 every simple module in the principal block is a TH module. If, on the other hand, q ≡ 3 (mod 4), then there is a short exact sequence

0 → L ⊕ M → Ω^{−1}k → k → 0

so that again every simple module in the principal block is a TH module.

In case P GL_{2}(q) ≤ G/O_{2′}(G), if q ≡ 1 (mod 4) then there is a short exact sequence

0 → M ⊕ M′ → Ω^{2}k → k → 0

where M is the non-trivial simple module in the principal block and M′ is a uniserial module. If q ≡ 3 (mod 4) then there is a short exact sequence

0 → M_{1} ⊕ M_{2} → Ω^{−1}k → k → 0

where M_{1} is an extension of M by k, M_{2} is an extension of k by M, and M is the nontrivial simple module in the principal block. Thus by Proposition 3.2 M ⊕ M is a TH module, and hence by Proposition 4.4 every simple module in the principal block is a TH module.

(ii) G/O_{2′}(G) ≅ A_{7} In this case, the two non-trivial simple modules in the principal block are L of dimension 14 and M of dimension 20. There is a short exact sequence

0 → M ⊕ L′ → Ω^{2}k → k → 0

where L′ is an extension of L by itself. Thus M is a summand of a TH module. Now Ω(L′) has only one composition factor not isomorphic to k or M, and this is isomorphic to L. Thus every simple module in the principal block is a TH module.

(iii) G/O_{2′}(G) is isomorphic to a Sylow two subgroup of G. This case is trivial as there is only one simple module in the principal block, namely k. □

Our next example is the group A_{6} in characteristic three. The modules Ω^{n}(k) in this case were computed in [2]. From the information given there, we can see that there is a short exact sequence

0 → k → Ω^{3}(k) → X ⊕ Y → 0

where X and Y are modules with three composition factors. The unique top composition factor of each is the trivial module, and we denote by X̃ and Ỹ the submodules with two composition factors. Thus X ⊕ Y and X̃ ⊕ Ỹ are TH modules, by Proposition 3.2. There is then an exact sequence

0 → L ⊕ M → Ω^{−1}(X ⊕ Y) → X̃^{∗} ⊕ Ỹ^{∗} → 0

where L and M are the two three dimensional simple modules. Thus by another application of Proposition 3.2, L ⊕ M is a TH module. Finally, the short exact sequence

0 → L ⊕ M → X̃ ⊕ Ỹ → N ⊕ N → 0

shows that N ⊕N is a TH module. An application of Proposition 4.4 now shows that every simple module in the principal block is a TH module, so that every non-projective module in the principal block has cohomology. Note that this is an example of wild representation type, so that there is no chance of proving this by classifying the indecomposables. This example also shows that the same holds for A_{7} in characteristic three, since e_{0}.k_{A_{6}} ↑^{A_{7}} ≅ k_{A_{7}}, and every module is a direct summand of a module induced from A_{6}.

Our final example is one of rank three. We write 2^{3} : 7 for the split extension of an elementary abelian group of order eight by an automorphism of order seven. This group has seven simple modules over a large enough field of characteristic two, all in the principal block, which we denote by S_{0} = k, S_{1}, …, S_{6} with the notation chosen so that the second Loewy layer of the projective cover of S_{i} is S_{i+1} ⊕ S_{i+2} ⊕ S_{i+4}, with the indices taken modulo seven. In this case it turns out (and this is quite a hard calculation, which we shall omit for the sake of brevity) that there are short exact sequences

0 → X → Ω^{5}(k) → k ⊕ k ⊕ k ⊕ Ω^{−1}(k) ⊕ Ω^{−1}(k) ⊕ Ω^{−1}(k) → 0
0 → Ω^{−2}(k) → X → Y → 0

in such a way that Y has S_{1}⊕S_{2}⊕S_{4} as a direct summand. We now dualise and use Proposition 4.4 to deduce that every simple module is a TH module. Since e_{0}.k_{2^{3}:7} ↑^{P SL_{2}(8)}≅ k_{P SL_{2}(8)}, it follows that the same is true for P SL_{2}(8) in characteristic two.

5 Generating modules with no cohomology

We now give a general method for producing non-projective indecomposable modules M in B_{0}(kG) for which Ĥ^{n}(G, M) = 0 for all values of n.

Lemma 5.1 Let H be any subgroup of G and M_{0} be any indecomposable kH-module which is not in B_{0}(kH). Then for any summand M of M_{0} ↑^{G} and all values of n we have Ĥ^{n}(G, M) = 0.

Proof Let e_{0} be the principal block idempotent for kG. By Shapiro’s lemma we have

Ĥ^{n}(G, e_{0}.M_{0} ↑^{G}) = Ĥ^{n}(G, M_{0} ↑^{G}) ≅ Ĥ^{n}(H, M_{0}) = 0.

Thus e_{0}.M_{0} ↑^{G} is a module with no cohomology, and so M also has no cohomology. □

Here are some examples of this phenomenon with M non-projective. We write D^{λ} for the simple kΣ_{n}-module corresponding to the p-regular partition λ, where p is the characteristic of k, as in James [15].

Proposition 5.2 Let M_{0} denote the module D^{((r−1)p,p−1)} for Σ_{rp−1}, and let M = e_{0}.M_{0} ↑^{Σ_{rp}}. Then M is a non-projective uniserial module of length 3 with composition factors D^{((r−1)p,p)}, D^{(rp−1,1)}, D^{((r−1)p,p)}, and with no cohomology.

Proof The character of M is easily seen to be as given, by referring to James [15, Theorem 24.15]. Since D^{(rp−1,1)} ↓_{Σ_{rp−1}} does not involve D^{((r−1)p,p−1)}, the Nakayama relations show that the module M is not a direct sum. Since it is self dual it must be uniserial. It is clearly not projective. □

Another example may be obtained by inducing a 10 dimensional simple module from P SL_{2}(11) to M_{11} in characteristic two. This simple module is in a block of defect one and is periodic of period one. Since 10 ⊗ 5 ≅ 10 ⊕ (projective) for P SL_{2}(11), we have

10↑^{M_{11}} ⊕ (projective) = (10 ⊗ 5)↑^{M_{11}}

= ((10_{M_{11}})↓_{P SL_{2}(11)} ⊗5)↑^{M_{11}}
= 10 ⊗ 5↑^{M_{11}}
= 10 ⊗ (16 ⊕ 44)
= 10 ⊗ 44 ⊕ (projective)

and so this “explains” why Ext_{M_{11}}^{∗}(10, 44) = 0. The diagram for this module e_{0}.10_{P SL_{2}(11)} ↑^{M_{11}} ≅ e_{0}.(10 ⊗ 44) is as follows.

10  44
1  10  1
44  10

Can such an induced module ever have a simple summand in the principal block? Such a simple module would be a counterexample to the well known conjecture that every simple module in the principal block must have cohomology, but it seems hard to construct a counterexample this way.

The following proposition is a more systematic way of using the above technique.

Proposition 5.3 Suppose G has a non-trivial p-subgroup P such that C_{G}(P) is not pnilpotent. Then there is a non-projective trivial source kG-module M in the principal block with vertex P, such that Ĥ^{n}(G, M) = 0 for all values of n.

Proof Since C_{G}(P) is not p-nilpotent, we can find a p′-element y ∈ C_{G}(P) with y ∉ O_{p′}C_{G}(P) = O_{p′}N_{G}(P). The intersection of the kernels of the simple modules in B_{0}(kN_{G}(P)) is O_{p′,p}N_{G}(P), which does not contain y, and so we may choose a simple module S in B_{0}(kN_{G}(P)) for which y is not in the kernel of the action on S.

Thus S ↓_{⟨y⟩×P} contains some one dimensional submodule M_{0} on which y acts nontrivially, and so M_{0} is not in B_{0}(⟨y⟩ × P). By the Nakayama relations we have

Hom_{N_{G}(P)}(M_{0} ↑^{N_{G}(P)}, S) ≅ Hom_{⟨y⟩×P} (M_{0}, S ↓_{⟨y⟩×P}) ≠ 0

and so M_{0} ↑^{N_{G}(P)} has an indecomposable summand M_{1} in the principal block.

Since P is in the kernel of M_{0}, it is also in the kernel of M_{1}, which therefore has vertex exactly P. The Green correspondent M of M_{1} is then a non-projective summand of M_{0} ↑^{G} in the principal block. Since M_{0} is not in the principal block, we have Ĥ^{n}(G, M) = 0 for all values of n by Lemma 5.1. □

This proposition proves the implications (C) ⇒ (D) and (E) ⇒ (D) in Theorem 1.4.

6 Periodic modules

In this section, we show that if all periodic modules in the principal block have cohomology, then so do all non-projective modules. We use this to analyse the case of a normal vertex.

We shall need to use the language of varieties for modules, and our terminology will be the same as in [1]. In particular, we denote by V_{G} the maximal ideal spectrum of the cohomology ring H^{·}(G, k) = H^{∗}(G, k) in case p = 2 and of the even cohomology ring H^{·}(G, k) = H^{ev}(G, k) if p is odd. For a module M, V_{G}(M) is the closed homogeneous subvariety of V_{G} defined by the annihilator of Ext_{kG}^{∗}(M, M). If ζ ≠ 0 is an element of H^{2n}(G, k) represented by a homomorphism ζ̂ : Ω^{2n}(k) → k, we denote its kernel by L_{ζ}. The following proposition, which is the same as [1, Proposition 3.1], summarises some of the properties of these varieties. The proofs are a culmination of the work of many people.

Proposition 6.1 Let M and N be kG-modules.

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

(ii) dim V_{G}(M) is equal to the complexity of M.

(iii) V_{G}(M ⊕ N) = V_{G}(M) ∪ V_{G}(N).

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

(v) V_{G}(Ω^{n}(M)) = V_{G}(M) for all n.

(vi) For ζ ∈ H^{2n}(G, k), V_{G}(L_{ζ}) = V_{G}(⟨ζ⟩), the hypersurface of V_{G} determined by ζ. (vii) The dimension of V_{G} = V_{G}(k) is equal to the p-rank of G (p is the characteristic

of k).

Lemma 6.2 If M_{1} is a TH kG-module and Ĥ^{n}(G, M_{2}) = 0 for all n, then Ĥ^{n}(G, M_{1} ⊗ M_{2}) = 0 for all n.

Proof This follows from Propositions 2.7 and 3.4 since

Ĥ^{n}(G, M_{1} ⊗ M_{2}) ≅ Êxt_{kG}^{n}(M_{1}^{∗}, M_{2}). □

Lemma 6.3 Suppose M_{1} is an indecomposable kG-module and M_{2} is a TH module. Then every non-projective summand of M_{1} ⊗ M_{2} is in the same block as M_{1}.

Proof Tensor M_{1} with a complex displaying M_{2} as a TH module. The part in any block other than that of M_{1} is projective and exact except in degree zero, so the homology in degree zero is projective. □

Theorem 6.4 Suppose that M is a module in B_{0}(kG) with variety V_{G}(M) ≠ {0}, and Ĥ^{n}(G, M) = 0 for all n. If V is a closed homogeneous subset of V_{G}(M), then there is a module M′ in B_{0}(kG) with Ĥ^{n}(G, M′) = 0 for all n, and V_{G}(M′) = V.

Proof Choose elements of cohomology ζ_{1}, …, ζ_{s} with

V = V_{G}(M) ∩ V_{G}(⟨ζ_{1}⟩) ∩ ⋯ ∩ V_{G}(⟨ζ_{s}⟩),

and let M′ be the non-projective part of M ⊗ L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{s}}. By Proposition 3.2, each L_{ζ_{i}} is a TH module, and hence by Proposition 3.3 so is L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{s}}, so that by Lemma 6.2 Ĥ^{n}(G, M′) = 0 for all n. We have

V_{G}(M′) = V_{G}(M ⊗ L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{s}})
= V_{G}(M) ∩ V_{G}(L_{ζ_{1}}) ∩ ⋯ ∩ V_{G}(L_{ζ_{s}})
= V_{G}(M) ∩ V_{G}(⟨ζ_{1}⟩) ∩ ⋯ ∩ V_{G}(⟨ζ_{s}⟩)
= V.

By Lemma 6.3, M′ is in the principal block. □

Corollary 6.5 If there is a non-projective module in the principal block with no cohomology, then there is a non-projective periodic module in the principal block with no cohomology.

Proof This is the case dim V = 1 of the above theorem. □

This completes the proof of the implication (B) ⇔ (B′) in Theorem 1.4.

Lemma 6.6 Let G be a finite group with O_{p}(G) ≠ 1 and O_{p′}(G) = 1. Suppose the centraliser of every element of order p in O_{p}(G) is p-nilpotent. Then every non-trivial p′-element in G acts fixed point freely (i.e., fixing only the identity element) on O_{p}(G).

Proof If a non-trivial p′-element y centralises an element x of order p in O_{p}(G), then ⟨y⟩ × ⟨x⟩ acts by conjugation on O_{p}(G). Since C_{G}(x) is p-nilpotent, y acts trivially on C_{O_{p}(G)}(x). So by Thompson’s A × B lemma [10, Theorem 5.3.4], y ∈ C_{G}(O_{p}(G)). Since C_{G}(O_{p}(G)) is p-nilpotent, it follows that y ∈ O_{p′}(G) = 1. □

Lemma 6.7 Suppose a cyclic p′-group T acts on a p-group P. If L is a line through the origin in V_{P} which is invariant under the action of T, and on which T acts faithfully, then there is an element ζ ∈ H^{2n}(P, k) for some n with the following properties.

(i) L ∩ V_{P} (⟨ζ⟩) = {0}.

(ii) The one dimensional subspace ⟨ζ⟩ of H^{2n}(P, k) is T -invariant, and affords a faithful one dimensional representation of T.

Proof The line L corresponds to a T -invariant ideal I ⊆ H^{·}(P, k). The map

H^{·}(P, k) → H^{·}(P, k)/I = k[L]

splits as a map of graded kT -modules, since the order of T is coprime to p. Let ρ be such a splitting.

Since T acts faithfully on L, T also acts faithfully on k[L], which is a polynomial ring in one generator in some degree. A generator of this polynomial ring therefore spans a one dimensional space in H^{2n}(P, k)/I for some n, on which T acts faithfully, and therefore its image under ρ has the desired properties. □

Proposition 6.8 Suppose the centraliser of every element of order p in O_{p}(G) is pnilpotent. If M is an indecomposable periodic module in B_{0}(kG), whose vertex is contained in O_{p}(G), then Ĥ^{n}(G, M) ≠ 0 for some n.

Proof First, we remark that since O_{p′}(G) is the kernel of B_{0}(kG), we may assume O_{p′}(G) = 1. If O_{p}(G) = 1, the proposition is trivial, so we may assume O_{p}(G) ≠ 1, so that the previous lemma applies.

We now apply Clifford theory with respect to the subgroup O_{p}(G). Let M_{0} be an indecomposable kO_{p}(G)-module such that M is a summand of M_{0} ↑^{G}. Letting G_{0} be the inertia group of M_{0}, namely G_{0} = {g ∈ G | g ⊗ M_{0} ≅ M_{0}}, so that O_{p}(G) ⊆ G_{0} ⊆ G, Clifford theory tells us that M is induced from G_{0} [13, Theorem VII.9.6]. Thus by Shapiro’s lemma we may assume G = G_{0}, so that M_{0} is inertial.

Now the variety V_{O_{p}(G)}(M_{0}) is (projectively) connected [6], and hence is a single line through the origin in V_{O_{p}}(G). Since M_{0} is inertial, this line must be invariant under the action of G/O_{p}(G). Let E be a minimal elementary abelian subgroup of O_{p}(G) such that M_{0} ↓_{E} is not projective. Then E is well defined up to conjugacy in O_{p}(G), and since M_{0} is inertial, every G-conjugate of E is O_{p}(G)-conjugate to E. Thus G = O_{p}(G).N_{G}(E) by the Frattini argument. Moreover, V_{E}(M_{0}) is a union of lines, and N_{O_{p}(G)}(E) acts transitively on these lines, as does N_{G}(E). Thus if L is one such line, G = O_{p}(G).Stab_{G}(L), where Stab_{G}(L) is the subgroup of N_{G}(E) stabilising L (setwise). It follows that Stab_{G}(L)/C_{G}(E) has a one dimensional faithful representation (on L) in characteristic p, so that it is a cyclic p′-group. Now C_{G}(E) ⊆ O_{p}(G) by Lemma 6.6, so that since G = O_{p}(G).Stab_{G}(L), it follows that G/O_{p}(G) is a cyclic p′-group which acts faithfully on the line L.

We now apply Lemma 6.7 to see that there is an element ζ ∈ H^{2n}(O_{p}(G), k) for some n, such that T acts faithfully on the one dimensional subspace generated by ζ, and L ∩ V_{O_{p}(G)}(⟨ζ⟩) = {0}. Thus ⟨ζ⟩ is represented by a homomorphism Ω^{2n}(k) → ε of kGmodules, where ε is a one dimensional representation whose kernel is O_{p}(G). Denote by L_{ζ} the kernel, so that there is a short exact sequence

0 → L_{ζ} → Ω^{2n}(k) → ε → 0.

Since

V_{O_{p}(G)}(L_{ζ} ⊗ M) = V_{O_{p}(G)}(L_{ζ} ⊗ M_{0})
= V_{O_{p}(G)}(L_{ζ}) ∩ V_{O_{p}(G)}(M_{0})
= V_{O_{p}(G)}(⟨ζ⟩) ∩ V_{O_{p}(G)}(M_{0})
= 0,

it follows that L_{ζ} ⊗ M is a projective kG-module, and so the short exact sequence

0 → L_{ζ} ⊗ M → Ω^{2n}(k) ⊗ M → ε ⊗ M → 0

shows that Ω^{2n}(M) ≅ ε ⊗ M. Since there is some non-zero element of Ĥ^{0}(G, ε^{r} ⊗ M) for some r (M must have some top composition factor!), we deduce that

Ĥ^{−2nr}(G, M) ≅ Ĥ^{0}(G, Ω^{2nr}(M))
≅ Ĥ^{0}(G, ε^{r} ⊗ M)
≠ 0. □

Theorem 6.9 Suppose the centraliser of every element of order p in G is p-nilpotent. Then for every non-projective indecomposable module M in B_{0}(kG) with vertex D and Green correspondent f(M), we have Ĥ^{n}(N_{G}(D), f(M)) ≠ 0 for some n.

Proof Since the Green correspondent of a module in the principal block is again in the principal block, we may replace G by N_{G}(D). By Corollary 6.5, we may assume M is periodic. We now apply Proposition 6.8. □

This completes the proof of the implication (D) ⇒ (E) in Theorem 1.4.

7 Trivial source modules

The main obstacle to proving the implication (D) ⇒ (B) of Theorem 1.4, and hence Conjecture 1.2, using Theorem 6.9 is that all the cohomology of the Green correspondent of a periodic module may actually be in the image of the trace map (transfer) from proper subgroups of the vertex. When the module has trivial source, this cannot happen, as we shall see in the next theorem. This theorem gives the implication (D) ⇒ (C) of Theorem 1.4.

Theorem 7.1 Let G be a finite group in which the centraliser of every element of order p is p-nilpotent. Suppose M is a trivial source module in B_{0}(kG) with vertex D and Green correspondent f(M). If H^{∗}(N_{G}(D), f(M)) ≠ 0 then H^{∗}(G, M) ≠ 0.

Proof We must show that if H^{∗}(N_{G}(D), f(M)) ≠ 0 then there are elements which are not in the image of the trace map from proper subgroups of D, since then the theorem follows from the Green correspondence. Now D is in the kernel of the action of N_{G}(D) on f(M), and f(M) is projective as a module for N_{G}(D)/D, since M is a trivial source module. The theorem now follows from the next lemma.

Lemma 7.2 Suppose M is an indecomposable trivial source module in B_{0}(kG) whose vertex D ≠ 1 is normal in G. Let Z be the subgroup of Z(D) generated by elements of order p, and suppose that C_{G}(Z) is p-nilpotent. Then

H^{∗}(G, M)/ ∑_{H<D} tr_{H,G}H^{∗}(H, M) ≠ 0.

Proof We regard k(G/D) = k_{D} ↑^{G} as a G/D-G/D-bimodule. Then H^{∗}(G, k_{D} ↑^{G}) is a right k(G/D)-module, and the Shapiro isomorphism

H^{∗}(G, k_{D} ↑^{G}) →^{Sh} H^{∗}(D, k)

is a right k(G/D)-module isomorphism. Let e be a primitive idempotent in k(G/D) with M ≅ k(G/D).e. Then

H^{∗}(G, M) = H^{∗}(G, k_{D} ↑^{G} .e)
= H^{∗}(G, k_{D} ↑^{G}).e
= H^{∗}(D, k).e ⊆ H^{∗}(D, k).

Letting π : k_{D} ↑^{G}↓_{D}→ k_{D} be the natural kD-module homomorphism, for any proper subgroup H of D we have the following diagram, which the reader will easily check is commutative.

H^{∗}(H, M) ↪ H^{∗}(H, k_{D} ↑^{G}↓_{H})
↓ tr  ↓ tr  ↘π_{∗}
H^{∗}(D, M) ↪ H^{∗}(D, k_{D} ↑^{G}↓_{D})  H^{∗}(H, k)
↓ tr  ↓ tr  ↘π_{∗}  ↓ tr  ↘0
H^{∗}(G, M) ↪ H^{∗}(G, k_{D} ↑^{G}) →^{Sh} H^{∗}(D, k) →^{res} H^{∗}(Z, k)

The composite of transfer from H^{∗}(H, k) to H^{∗}(D, k) and restriction from H^{∗}(D, k) to H^{∗}(Z, k) is zero because Z is central, by the Mackey formula. It thus suffices to produce an element ζ ∈ H^{∗}(D, k) with res_{D,Z}(ζ.e) ≠ 0, so that ζ.e ∈ H^{∗}(G, M) cannot be a sum of transfers from proper subgroups of D.

To construct such an element ζ, we argue as follows. We know that C_{G}(Z) is pnilpotent, and since O_{p′}C_{G}(Z) ≤ O_{p′}(G) is in the kernel of B_{0}(kG), we may assume without loss of generality that O_{p′}C_{G}(Z) = 1, so that C_{G}(Z) is a p-group. So every simple k(G/D)-module appears as a composition factor of H^{n}(Z, k) for some n. The point is that G/C_{G}(Z) acts faithfully on Z and hence also on a polynomial subring E^{∗}(Z) of H^{∗}(Z, k).

Now for any n, there is a k(G/C_{G}(Z))-module homomorphism H^{n}(Z, k) → H^{p^{a}n}(Z, k) where p^{a} = |D : Z|, given by ξ → ξ^{p^{a}} = res_{D,Z}norm_{Z,D}(ξ), where norm_{Z,D} is the Evens norm map. This map is injective on E^{∗}(Z), and so every simple k(G/C_{G}(Z))-module is a composition factor of res_{D,Z}(H^{n}(D, k)) for some n. Since res_{D,Z} is a k(G/C_{G}(Z))- module homomorphism, it follows that there exists an element ζ ∈ H^{∗}(D, k) such that res_{D,Z}(ζ.e) = res_{D,Z}(ζ).e ≠ 0. □

We now prove the implication (A′′) ⇒ (A′) of Theorem 1.4.

Theorem 7.3 Suppose that every permutation module in B_{0}(kG) is a direct summand of a TH module. Then every finitely generated module in B_{0}(kG) is a direct summand of a TH module.

Proof Every module induced from a p-subgroup has a filtration by permutation modules, since every module for the p-subgroup has a filtration by one dimensional modules. Thus by Proposition 3.2, every module induced from a p-subgroup is a summand of a TH module. But every module is a direct summand of a module induced from its vertex. □

Proof of Theorem 1.4 The implications (A) ⇒ (A′), (A) ⇒ (A′′), (B) ⇒ (B′) and (B) ⇒ (C) are trivial. The implication (A) ⇒ (B) is Corollary 2.8, (A′) ⇒ (B) is Corollary 3.8, (A′′) ⇒ (A′) is Theorem 7.3, (B′) ⇒ (B) is Corollary 6.5, (C) ⇒ (D) and (E) ⇒ (D) follow from Proposition 5.3, (D) ⇒ (E) is Theorem 6.9, (D) + (E) ⇒ (C) is Theorem 7.1, and (D) ⇒ (F) is Proposition 4.5. The implication (D) ⇔ (D′) follows easily from the fact that any subgroup of a p-nilpotent group is p-nilpotent.

The implications so far give (A′′) ⇒ (A′) ⇒ (F). But by Proposition 4.4, (A′) and (F) together imply (A).

The final statement about odd primes will be proved in the next two sections. □

8 A stable equivalence of module categories

In this and the next section, we shall prove Conjecture 1.2 for p odd. The basic idea is that if the centralisers of p-elements are p-nilpotent then we can find a suitable subgroup H of G with the properties that the module categories of the principal blocks of H and G are equivalent modulo projectives, and with H of such a tight shape that we can prove the theorem there.

In this section, we provide a proof of an unpublished theorem of M. Broué, which enables us to descend to a suitable subgroup, and in the next section we go through the group theory required to find a subgroup of the right shape. The proof of Broué’s theorem presented here is perhaps closer in spirit to a proof by Alperin of the same result (also unpublished).

Definition 8.1 A subgroup H of a finite group G is said to be weakly p-embedded if H ≠ G, p | |H|, and N_{G}(Q) = N_{H}(Q)O_{p′}N_{G}(Q) (note that O_{p′}N_{G}(Q) = O_{p′}C_{G}(Q)) whenever Q ≠ 1 is a p-subgroup of H.

Note in particular that if H is weakly p-embedded in G then H contains a Sylow p-subgroup of G. Also by Alperin’s fusion theorem [10, Section 7.2] H controls strong p-fusion in G and so by the stable element method [14, Section X.12] we have H^{∗}(G, k) ≅ H^{∗}(H, k). We shall see that the stronger hypothesis is sufficient to guarantee that H^{∗}(G, M) ≅ H^{∗}(H, M) for any module M in the principal block.

The following is a variant of a result of Nagao [19].

Lemma 8.2 Let M be an indecomposable module in a block B of kG, and let Q ≠ 1 be a p-subgroup of G. Let X be an indecomposable summand of M ↓_{N_{G}(Q)} such that X ↓_{Q} has a summand with vertex Q. Then X lies in a block b of kN_{G}(Q) for which b^{G} = B.

Proof Let e be the central idempotent in kG corresponding to B. We may write e = e_{1} + e_{2}, where e_{1} = Br_{Q}(e) is an idempotent in Z(kN_{G}(Q)) and e_{2} is an idempotent in ∑_{P <Q} Tr_{P,Q}(kG)^{P}.

We have M ↓_{N_{G}(Q)}= e_{1}.M ↓_{N_{G}(Q)} ⊕e_{2}.M ↓_{N_{G}(Q)}. Let E = End_{k}(M), and σ : kG → E be the homomorphism given by the action of G on M. Then σ(e_{2}) ∈ ∑_{P <Q} Tr_{P,Q}(E^{P}), so that by Rosenberg’s lemma any primitive idempotent of E^{Q} occurring in the decomposition of σ(e_{2}) is in Tr_{P,Q}(E^{P}) for some P < Q. Thus every summand of e_{2}.M ↓_{Q} has vertex properly contained in Q, and so X is a summand of e_{1}.M ↓_{N_{G}(Q)}= Br_{Q}(e).M ↓_{N_{G}(Q)} and is hence in a block b with b^{G} = B. □

Theorem 8.3 (M. Broué) Suppose H is a weakly p-embedded subgroup of G, and let M be an indecomposable module in B_{0}(kG) with non-trivial vertex P ⊆ H. Then we have the following.

(i) There is an indecomposable module f(M) in B_{0}(kH), also with vertex P, such that M ↓_{H}= f(M)⊕ (projective).

(ii) f(M)↑^{G}≅ M⊕ (projective) ⊕ (modules outside the principal block).

Proof (i) Suppose Q ≠ 1 is a vertex of an indecomposable summand of M ↓_{H}. By [4]; [20, Cor. 1.4], there is a bijection between indecomposable summands of M ↓_{H} with vertex P and indecomposable summands of M ↓_{N_{H}(Q)} with vertex Q.

Let T be an indecomposable summand of M ↓_{N_{H}(Q)} with vertex Q. Let X be an indecomposable summand of M ↓_{N_{G}(Q)} such that T is a summand of X ↓_{N_{H}(Q)}. Then X ↓_{Q} has a summand with vertex Q (namely a source of T) and so by the lemma and Brauer’s third main theorem, X is in B_{0}(kN_{G}(Q)). Since H is weakly p-embedded in G, N_{G}(Q) = N_{H}(Q)O_{p′}N_{G}(Q). Since O_{p′}N_{G}(Q) is the kernel of B_{0}(N_{G}(Q)), we thus have T = X ↓_{N_{H}(Q)}, and X has vertex Q. Again applying [4]; [20, Cor. 1.4], we see that Q = P, and X is the Green correspondent of M. Taking f(M) to be the Green correspondent of T yields the result.

(ii) Let Q be a non-trivial p-subgroup of H, and consider the indecomposable summands of f(M)↑^{G}↓_{N_{G}(Q)} in the principal block of kN_{G}(Q).

Since H is weakly p-embedded in G, H controls strong p-fusion in G, and so Q ≤ H^{g} if and only if g ∈ HN_{G}(Q). It follows using Mackey decomposition that the indecomposable summands of f(M) ↑^{G}↓_{N_{G}(Q)} with vertex Q all occur in f(M) ↓_{N_{H}(Q)}↑^{N_{G}(Q)}. We claim that there can be such a summand with vertex Q in the principal block of kN_{G}(Q) if and only if Q is H-conjugate to P, in which case there is just one such summand, namely X (the Green correspondent of M).

Now for any indecomposable kN_{H}(Q)-module U, O_{p′}N_{H}(Q) acts either without fixed points or trivially on U. If O_{p′}N_{H}(Q) acts without fixed points on U, then O_{p′}N_{G}(Q) acts without fixed points on U ↑^{N_{G}(Q)}, so U ↑^{N_{G}(Q)} has no summands in B_{0}(kN_{G}(Q)). If O_{p′}N_{H}(Q) acts trivially on U, then the fixed points of O_{p′}N_{G}(Q) on U ↑^{N_{G}(Q)} have the same dimension as U. In this case, since N_{G}(Q) = N_{H}(Q)O_{p′}N_{G}(Q), we may extend U to a kN_{G}(Q)-module Ũ which is a summand of U ↑^{N_{G}(Q)}. Hence Ũ is the unique indecomposable summand of U ↑^{N_{G}(Q)} on which O_{p′}N_{G}(Q) acts trivially. This summand has the same vertex as U, and lies in B_{0}(kN_{G}(Q)) if and only if U lies in B_{0}(N_{H}(Q)).

It follows that f(M) ↓_{N_{H}(Q)}↑^{N_{G}(Q)} has no summand in B_{0}(kN_{G}(Q)) with vertex Q unless f(M) ↓_{N_{H}(Q)} has a summand with vertex Q in B_{0}(kN_{H}(Q)). If there is such a summand, then Q is a vertex for f(M), and the Green correspondent of f(M) is the unique such summand. Thus we may suppose Q = P, in which case this summand is isomorphic to T, and X is the unique indecomposable summand of T ↑^{N_{G}(Q)} in B_{0}(kN_{G}(Q)) with vertex Q. The result now follows by another application of [4]; [20, Cor. 1.4]. □

Corollary 8.4 Suppose H is a weakly p-embedded subgroup of G and M is a kG-module. Then the restriction map is an isomorphism

res_{G,H} : H^{∗}(G, M) → H^{∗}(H, M) = H^{∗}(H, f(M)).

Proof To deduce this from the theorem and Shapiro’s lemma, we must show that if M is an indecomposable module not in B_{0}(kG) then M ↓_{H} has no non-projective summand in B_{0}(kH). If M ↓_{H} has a summand in B_{0}(kH) with vertex Q ≠ 1 then M ↓_{N_{H}(Q)} has a summand T in B_{0}(N_{H}(Q)) with vertex Q. Lemma 8.2 and Brauer’s third main theorem show that this cannot happen. □

9 Some group theory

For p odd, and under the assumption that the centraliser of every element of order p is pnilpotent, we shall show that there is a suitable weakly p-embedded subgroup H, for which we can prove that every non-projective module in the principal block has cohomology. We begin with some lemmas.

Lemma 9.1 Suppose C_{G}(x) is p-nilpotent whenever x is an element of order p in G. Then the same is true for every section of G.

Proof This is clear for subgroups, so it suffices to consider quotients \overline{G} = G/N for N a normal subgroup of G. We proceed by induction on |G|. We may assume that O_{p′}(G) = 1.

Suppose that q is a prime divisor of |N| and that a Sylow q-subgroup Q of N is not normal. Then by the Frattini argument G = N.N_{G}(Q) and G/N ≅ N_{G}(Q)/N_{N}(Q). Since N_{G}(Q) is a proper subgroup of G we are done by induction. This shows that N is nilpotent. Since O_{p′}(N) = 1 this implies that N is a p-group.

Now C_{G}(N) is p-nilpotent by hypothesis, and so it is a p-group, as O_{p′}(G) = 1. So C_{G}O_{p}(G) ≤ C_{G}(N) ≤ O_{p}(G), and hence G is p-constrained. Thus by a well known result of Bender (see for example [14, Lemma X.1.6]), O_{p′}C_{G}(x) ≤ O_{p′}(G) = 1 and hence C_{G}(x) is a p-group, whenever x is an element of order p in G. Since N is a p-group, for y a non-trivial p′-element of G we have C\overline{G}(\overline{y}) = \overline{C_{G}(y)} (where the bar denotes passage from G to \overline{G} = G/N), and so C\overline{G}(\overline{y}) is a p′-group. Thus whenever \overline{x} is a non-trivial element of order p in \overline{G}, C\overline{G}(\overline{x}) is a p-group. This completes the proof of the lemma. □

Proposition 9.2 Suppose p is odd, and that the centraliser of every element of order p in G is p-nilpotent. Let H = N_{G}(ZJ(S)), with S a Sylow p-subgroup of G (here, J(S) is the Thompson subgroup; see for example [10, Section 8.2]). Then H is weakly p-embedded in G.

Proof We first claim that H controls strong p-fusion in G. If not, then by a theorem of Glauberman [9, Theorem B] Qd(p) is involved in G, where Qd(p) denotes the semi-direct product (Z/p × Z/p) : SL_{2}(p) with the natural action. But this contradicts Lemma 9.1 since the centraliser of an element of order p in Qd(p) but not in O_{p}Qd(p) has order 2p^{2}, and is not p-nilpotent.

Now since H controls strong p-fusion in G, whenever Q is a non-trivial p-subgroup of H, we have N_{G}(Q) = N_{H}(Q)C_{G}(Q). We claim that N_{H}(Q) contains a Sylow p-subgroup of N_{G}(Q). For if T is a Sylow p-subgroup of G containing a Sylow p-subgroup of N_{G}(Q) then T^{g} = S for some g ∈ G. Both Q and Q^{g} are in S, and since H controls strong p-fusion in G, Q^{g} = Q^{h} for some h ∈ H, and hence N_{G}(Q)^{g} = N_{G}(Q)^{h}. Thus S^{h^{−1}} contains a Sylow p-subgroup of N_{G}(Q), hence so does H.

It now follows that we have N_{G}(Q) = N_{H}(Q)O_{p′}N_{G}(Q) since C_{G}(Q) is p-nilpotent. □

In order to make use of this, we investigate the structure of N_{G}(ZJ(S)) under these hypotheses.

Lemma 9.3 Suppose p is odd, and that the centraliser of every element of order p in G is p-nilpotent. Let H = N_{G}ZJ(S), with S a Sylow p-subgroup of G, and let \overline{H} = H/O_{p′}(H). Then one of the following occurs.

(i) \overline{H} is a Frobenius group whose Frobenius kernel is its Sylow p-subgroup.

(ii) |\overline{H}| is odd, O^{p}(\overline{H}) is a Frobenius group whose kernel is its Sylow p-subgroup and whose Frobenius complement is cyclic. Also \overline{H}/O^{p}(\overline{H}) is cyclic.

(O^{p}(G) denotes the smallest normal subgroup of G for which the quotient is a p-group)

Proof Let \overline{Q} = O_{p}(\overline{H}). Then by Lemma 6.6, \overline{H}/\overline{Q} is faithfully represented as a group of linear transformations on \overline{Q}/Φ(\overline{Q}), in which all non-trivial p′-elements act without non-trivial fixed points. If p |̸ |\overline{H} : \overline{Q}|, then we are in case (i), so we may suppose that p| |\overline{H} : \overline{Q}|.

By the argument used in the proof of Lemma 9.1, the centraliser in \overline{H} of a non-trivial p-element is a p-group, and the same holds in \overline{H}/\overline{Q}. Now if \overline{H} contains an involution \overline{t} then since \overline{t} acts on \overline{Q}/Φ(\overline{Q}) without non-trivial fixed points, it must act as minus the identity and so \overline{t}\overline{Q} is central in \overline{H}/\overline{Q}. Since p| |\overline{H} : \overline{Q}| this is a contradiction, and so \overline{H}/\overline{Q} has odd order, so that by the Feit-Thompson theorem it is solvable.

Let L = \overline{H}/\overline{Q}. Then an element of order p in L cannot have a non-trivial fixed point on O_{p′}(L) since its centraliser is a p-group. Thus by a theorem of Thompson [10, Theorem 10.2.1] O_{p′}(L) is nilpotent. Also, whenever r ≠ p is a prime divisor of |L|, the Sylow r-subgroup of L is cyclic, because otherwise, as r is odd, the Sylow r-subgroup contains Z/r×Z/r and so some non-trivial r-element would centralise a non-trivial element of \overline{Q}. This forces O_{p′}(L) to be cyclic. Since L is solvable and O_{p}(L) = 1, C_{L}O_{p′}(L) ≤ O_{p′}(L) (see for example [10, Theorem 6.3.2]), so that L/O_{p′}(L) is abelian, since it acts faithfully on a cyclic group. Since the centraliser of an element of order p is a p-group, L/O_{p′}(L) is a cyclic p-group. □

Before we prove the main theorem of this section, we need a cohomological lemma.

Lemma 9.4 Let M be a kG-module, and H a normal subgroup of G of index a power of p. If H^{∗}(H, M) ≠ 0 then H^{∗}(G, M) ≠ 0.

Proof Examine the spectral sequence

E_{2}^{st} = H^{s}(G/H, H^{t}(H, M)) ⇒ H^{s+t}(G, M).

Choose n minimal with H^{n}(H, M) ≠ 0. Then E_{2}^{0n} = Hom_{G,H}(k, H^{n}(H, M)) ≠ 0 as G/H is a p-group. Since E_{2}^{st} = 0 for t < n, it follows that E_{∞}^{0n} ≠ 0, and so H^{n}(G, M) ≠ 0. □

Theorem 9.5 Suppose p is odd, and that the centraliser of every element of order p in G is p-nilpotent. If M is a non-projective module in B_{0}(kG) then H^{∗}(G, M) ≠ 0.

Proof Let H = N_{G}ZJ(S), with S a Sylow p-subgroup of G. By Proposition 9.2 H is weakly p-embedded in G. By Theorem 8.3 and its corollary we have

H^{∗}(G, M) ≅ H^{∗}(H, f(M)) ≅ H^{∗}(H/O_{p′}(H), f(M)),

and f(M) is in B_{0}(kH). Thus by Lemma 9.3 it suffices to prove the theorem in the case where G has one of the two structures listed there. In the first case the theorem follows from Corollary 6.5 and Proposition 6.8. In the second case, we have the following situation. G has a normal subgroup H = O^{p}(G) with G/H a non-trivial cyclic p-group. H is a Frobenius group with kernel O_{p}(G), and G/O_{p}(G) is a Frobenius group with kernel a cyclic p′-group. M is an indecomposable non-projective kG-module. Note that kG has only one block, as does kH in this situation. If M ↓_{H} is non-projective, we may apply Lemma 9.4, and then the theorem follows from Corollary 6.5 and Proposition 6.8 as before. So suppose M ↓_{H} is projective. Set N = O_{p}(G), and let S be a Sylow p-subgroup of G. Then S ∩ S^{g} = N for g ∈ G \ S.

We choose M_{0} an indecomposable summand of M ↓_{S} such that M is a summand of M_{0} ↑^{G}, and write M_{0} ↑^{G}= M ⊕ X. Then M_{0} ↓_{N} is a summand of M ↓_{N}, so M_{0} ↓_{N} is projective. Now M_{0} ↑^{G}↓_{S}= M_{0} ⊕ Y, where Y is projective relative to N since S ∩ S^{g} = N for g ∈ G \ S. Hence X ↓_{S} is projective relative to N, so that X is projective relative to N. On the other hand, X ↓_{N} is a summand of

M_{0} ↑^{G}↓_{N}= ⨁_{g∈G/S}(g ⊗ M_{0})↓_{N}

which is projective, as M_{0} ↓_{N} is projective. Hence X ↓_{N} is projective, and since X is projective relative to N it follows that X is projective. Hence for every n > 0,

H^{n}(G, M) = H^{n}(G, M_{0} ↑^{G}) = H^{n}(S, M_{0}) ≠ 0. □

Remark Gorenstein [11] has classified the finite groups in which the centraliser of every involution is 2-nilpotent. It may be that the best way to prove Conjecture 1.2 in case p = 2 is to go through the cases in this theorem individually. We have already initiated this process in Section 4 by treating the case where the Sylow 2-subgroups are dihedral. An example which we have not been able to tackle by the methods presented here is the split extension (Z/2)^{2n} : SL_{2}(2^{n}), where the normal subgroup is formed from the natural module by restricting the field of coefficients to F_{2}.

If G has a split (B, N) pair of Lie rank greater than two of characteristic p, then there is a non-projective module M in B_{0}(kG) with H^{∗}(G, M) = 0. The argument goes as follows. Suppose false. For p = 2, we must have that C_{G}(t) is 2-nilpotent for each involution t, by Proposition 5.3. By standard properties of (B, N) pairs, this forces the centraliser of every involution of G/Z(G) to be a 2-group. The classification of such groups implies that G/Z(G) has Lie rank at most two. In fact, the only example of Lie rank greater than one is SL(3, 2).

For p odd, the arguments of this section show that there is a p-local subgroup N which controls strong fusion. By the Borel-Tits theorem, we may assume that N ≤ P for some parabolic subgroup P ≥ B. Then P also controls strong fusion, and so we may from now on assume N = P. Now whenever P_{J} is a parabolic containing B, we have P_{J} = N_{G}(U_{J}) = C_{G}(U_{J})N_{N}(U_{J}) as N controls fusion. But C_{G}(U_{J}) ≤ Z(G) × Z(U_{J}) by standard properties of (B, N) pairs. Thus C_{G}(U_{J}) ≤ N (as P = N_{G}(U) for some p-subgroup U, and as U_{J} ≤ B ≤ P). Hence P_{J} ≤ P. Now, however, ⟨P_{J} | J ⊂ I⟩ ≤ P, whereas if G has rank greater than one, then G = ⟨P_{J} | J ⊂ I⟩ (as the right hand group contains all generating reflections of W).

10 Varieties and the nucleus

In this section, we investigate the varieties of modules in the principal block of an arbitrary finite group.

Definition 10.1 The nucleus Y_{G} is the subvariety of V_{G} given as the union of the images of the maps res^{∗} : V_{C_{G}(H)} → V_{G} induced by res : H^{·}(G, k) → H^{·}(C_{G}(H), k) where H runs over the set of subgroups of G for which C_{G}(H) is not p-nilpotent.

The representation theoretic nucleus Θ_{G} is the subset of V_{G} given as the union of the varieties V_{G}(M) as M runs over modules in B_{0}(kG) with H^{n}(G, M) = 0 for all n. (Note that by Theorem 6.4, it suffices to consider periodic modules in this definition.)

Theorem 10.2 Θ_{G} ⊇ Y_{G}.

Proof If H is a subgroup of G for which C_{G}(H) is not p-nilpotent, then Proposition 5.3 provides a module M in B_{0}(kG) with H^{n}(G, M) = 0 for all n, and

V_{G}(M) = Im(res^{∗} : V_{C_{G}(H)} → V_{G}). □

Conjecture 10.3 Θ_{G} = Y_{G}.

Remark It does not even seem to be clear that Θ_{G} is a variety.

Definition 10.4 If M is a kG-module, then we define X_{G}(M) to be the closed homogeneous subvariety of V_{G} defined by the annihilator in H^{·}(G, k) of H^{∗}(G, M) = Ext_{kG}^{∗}(k, M). Note that this is the same as the annihilator of Êxt_{kG}^{∗}(k, M) by Theorem 2.4, and hence also equal to the annihilator of Ext_{kG}^{∗}(M, k) = Ext^{∗}(k, M^{∗}) since by Tate duality

Êxt_{kG}^{n}(k, M) ≅ (Êxt_{kG}^{−n}(M, k))^{∗}.

The following two lemmas give the elementary properties of the variety X_{G}(M).

Lemma 10.5 (i) X_{G}(M) ⊆ V_{G}(M).

(ii) V_{G}(M) = ⋃_{S simple} X_{G}(M ⊗ S).

(iii) X_{G}(M) = {0} if and only if H^{∗}(G, M) = 0.

Proof The first two parts of this are well known, and can be deduced, for example, from the discussion in [5]. The third part follows from Corollary 2.6. □

Lemma 10.6 If M is a TH module then we have the following.

(i) X_{G}(M) = V_{G}(M).

(ii) If M′ is any module then X_{G}(M ⊗ M′) ⊆ V_{G}(M) ∩ X_{G}(M′).

Proof Let

(C_{∗}, θ) : C_{r} → ⋯ → C_{1} → C_{0}

be a complex of projective kG-modules displaying M as a TH module.

(i) Consider the spectral sequence

E_{2}^{st} = Êxt_{kG}^{s}(H_{t}(C_{∗}), M) ⇒ 0

of Theorem 2.3. If ζ ∈ H^{∗}(G, k) annihilates Êxt_{kG}^{∗}(k, M) then ζ annihilates E_{2}^{st} whenever t > 0. Since multiplication by ζ commutes with the differentials, this means that some power of ζ must annihilate E_{2}^{∗0} = Êxt_{kG}^{∗}(M, M) in order that the spectral sequence can abut to zero. This forces V_{G}(M) ⊆ X_{G}(M). The reverse inclusion is Lemma 10.5(i).

(ii) Consider the spectral sequence

E_{2}^{st} = Êxt_{kG}^{s}(H_{t}(C_{∗}) ⊗ M′, k) ⇒ 0.

If ζ ∈ H^{∗}(G, k) annihilates Êxt_{kG}^{∗}(M′, k) then ζ annihilates E_{2}^{st} whenever t > 0. So some power of ζ must annihilate E_{2}^{∗0} = Êxt_{kG}^{∗}(M ⊗M′, k). This forces X_{G}(M ⊗M′) ⊆ X_{G}(M′). The other inclusion is given by X_{G}(M ⊗M′) ⊆ V_{G}(M ⊗M′) ⊆ V_{G}(M) using Lemma 10.5(i) and Proposition 6.1(iv). □

Remark It seems likely that we always have equality in part (ii) of the above lemma, but we have been unable to prove this except in the case where M = L_{ζ}.

Theorem 10.7 For any module M in B_{0}(kG),

V_{G}(M) ⊆ X_{G}(M) ∪ Θ_{G}.

Proof Suppose V_{G}(M) ⊈ X_{G}(M) ∪ Θ_{G}. Then there is a non-zero closed homogeneous subvariety (for example a line)

{0} ≠ V = V_{G}(M) ∩ V_{G}(⟨ζ_{1}⟩) ∩ … ∩ V_{G}(⟨ζ_{s}⟩)

with V ∩ (X_{G}(M) ∪ Θ_{G}) = {0}. Now on the one hand we have

V_{G}(M ⊗ L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{s}}) = V_{G}(M) ∩ V_{G}(⟨ζ_{1}⟩) ∩ ⋯ ∩ V_{G}(⟨ζ_{s}⟩)
= V
⊈ Θ_{G}

so that by definition of Θ_{G} we have H^{∗}(G, M ⊗ L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{s}}) ≠ 0. On the other hand we have

X_{G}(M ⊗ L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{s}}) = X_{G}(M) ∩ X_{G}(⟨ζ_{1}⟩) ∩ ⋯ ∩ X_{G}(⟨ζ_{s}⟩)
⊆ X_{G}(M) ∩ V = {0}

so that by Lemma 10.5(iii) we have H^{∗}(G, M ⊗ L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{s}}) = 0. This contradiction completes the proof of the theorem. □

Definition 10.8 We say a kG-module M is a nuclear homology module or NH module if there exists a finite complex

C_{r} →^{θ_{r}} C_{r−1} → ⋯ →^{θ_{2}} C_{1} →^{θ_{1}} C_{0}

of kG-modules and homomorphisms such that the following conditions hold. (i) Each C_{i} is a projective kG-module. (ii) For i > 0, H_{i}(C_{∗}) is a direct sum of copies of the trivial kG-module k and modules

M′ in the principal block with V_{G}(M′) ⊆ Y_{G}.

(iii) H_{0}(C_{∗}) ≅ M.

Proposition 10.9 (i) Suppose M is an NH module and B is a positive integer. Then there exists a complex (C_{∗}, θ) as in the definition, with H_{i}(C_{∗}) = 0 for 1 ≤ i ≤ B.

(ii) Suppose 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence of RG-modules, and two of the terms are NH modules. Then so is the third.

(iii) If M_{1} and M_{2} are NH modules then so is M_{1} ⊗_{R} M_{2}.

(iv) If M is an NH module, then so is M^{∗} = Hom_{k}(M, k).

(v) Suppose every simple module in B_{0}(kG) is an NH module. Then every finitely generated module in B_{0}(kG) is an NH module.

(vi) Suppose M_{1} and M_{2} ⊕ M_{3} are NH modules and θ : M_{1} → M_{2} is surjective. Then M_{2} ⊕ Ker(θ) is a TH module.

(vii) A module M is a direct summand of an NH module if and only if M ⊕ Ω(M) is an NH module.

Proof The proofs of these are the same as the proofs of the corresponding results in Section 3. □

Conjecture 10.10 Every module in B_{0}(kG) is a nuclear homology module.

Proposition 10.11 If M is an NH module and H^{∗}(G, M) = 0 then V_{G}(M) ⊆ Y_{G}.

Proof Let

(C_{∗}, θ) : C_{r} → ⋯ → C_{0}

be a complex displaying M as an NH module. Consider the spectral sequence

E_{2}^{st} = Êxt_{kG}^{s}(H_{t}(C_{∗}), M) ⇒ 0

of Proposition 2.3. Suppose ζ ∈ Ker(res_{G,C_{G}(H)} : H^{∗}(G, k) → H^{∗}(H, k)) for every subgroup H with C_{G}(H) not p-nilpotent (i.e., ζ vanishes on Y_{G}). Then by the definition of an NH module, ζ annihilates E_{2}^{st} whenever t > 0. Thus some power of ζ annihilates E_{2}^{∗0} = Êxt_{kG}^{∗}(M, M). This proves that V_{G}(M) ⊆ Y_{G}. □

Corollary 10.12 Conjecture 10.10 implies Conjecture 10.3 □

Example We shall prove Conjecture 10.10 in case G is the Mathieu group M_{11} and k is a field of characteristic 2. The simple modules in the principal block in this case are k, M of dimension 44 and N of dimension 10. The minimal resolution of k as a kG-module

was described in [2]. By examining Ω^{4}(k), we see using Proposition 10.9 that \begin{matrix}M \\ | \\ M \end{matrix} ⊕ \begin{matrix}M \\ | \\ k \\ | \\ N \end{matrix}

is a TH module, and hence an NH module. Now Y_{G} is the one dimensional subvariety of V_{G} corresponding to the (unique) conjugacy class of involutions, and V_{G}(N) = Y_{G}, so

that N is an NH module. Thus by Proposition 10.9(ii), \begin{matrix}M \\ | \\ M \end{matrix} ⊕ M is an NH module. By

stripping all copies of k and N from P_{k}, we see that M ⊕ M is an NH module, and hence so is M by the above. Thus by Proposition 10.9(v) every finitely generated module in the principal block is an NH module, so by Corollary 10.12 we have Θ_{G} = Y_{G}. In particular, every non-projective, non-periodic module has cohomology, since the variety cannot be in Y_{G} for dimensional reasons.

References

[1] D. J. Benson and J. F. Carlson. Complexity and multiple complexes. Math. Zeit. 195 (1987), 221–238.

[2] D. J. Benson and J. F. Carlson. Diagrammatic methods for modular representations and cohomology. Comm. in Algebra 15 (1987), 53–121.

[3] K. S. Brown. Cohomology of groups. Graduate Texts in Mathematics 87, Springer-Verlag, Berlin/New York 1982.

[4] D. W. Burry and J. F. Carlson. Restrictions of modules to local subgroups. Proc. Amer. Math. Soc. 84 (1982), 181–184.

[5] J. F. Carlson. Complexity and Krull dimension. Representations of Algebras, Puebla, Mexico 1980. Springer Lecture Notes in Mathematics 903, 62–67, Springer-Verlag, Berlin/New York 1981.

[6] J. F. Carlson. The variety of an indecomposable module is connected. Invent. Math. 77 (1984), 291–299.

[7] K. Erdmann. Principal blocks of groups with dihedral Sylow 2-subgroups. Comm. in Algebra 5 (1977), 665–694.

[8] L. Evens. The cohomology ring of a finite group. Trans. Amer. Math. Soc. 101 (1961), 224–239.

[9] G. Glauberman. A characteristic subgroup of a p-stable group. Canad. J. Math. 20 (1968), 1101–1135.

[10] D. Gorenstein. Finite Groups. Chelsea, New York 1968.

[11] D. Gorenstein. Finite groups the centralisers of whose involutions have normal 2complements. Canad. J. Math. 21 (1969), 335–357.

[12] P. J. Hilton and U. Stammbach. A course in homological algebra. Graduate Texts in Mathematics 4, Springer-Verlag, Berlin/New York, 1971.

[13] B. Huppert and N. Blackburn. Finite Groups II. Grundlehren der mathematischen Wissenschaften 242, Springer-Verlag, Berlin/New York, 1982.

[14] B. Huppert and N. Blackburn. Finite Groups III. Grundlehren der mathematischen Wissenschaften 243, Springer-Verlag, Berlin/New York, 1982.

[15] G. D. James. The representation theory of the symmetric groups. Springer Lecture Notes in Mathematics 682, Springer-Verlag, Berlin/New York 1978.

[16] P. A. Linnell. Cohomology of finite soluble groups. J. Algebra 107 (1987), 53–62.

[17] P. A. Linnell and U. Stammbach. The block structure and Ext of p-soluble groups. J. Algebra 108 (1987), 280–282.

[18] P. A. Linnell and U. Stammbach. The cohomology of p-constrained groups. J. Pure Appl. Algebra 49 (1987), 273–279.

[19] H. Nagao. A proof of Brauer’s theorem on generalized decomposition numbers. Nagoya Math. J. 22 (1963), 73–77.

[20] L. Puig. Pointed groups and construction of characters. Math. Z. 176 (1981), 265–292.

[21] G. R. Robinson. Blocks, isometries, and sets of primes. Proc. London Math. Soc. 51 (1985), 432–448.