Control of Fusion and Cohomology of Trivial Source Modules

Akihiko Hida ^{∗}

April 2006

Abstract Let G be a finite group and H a subgroup. We give an algebraic proof of Mislin’s theorem which states that the restriction map from G to H on mod-p cohomology is an isomorphism if and only if H controls p-fusion in G. We follow the approach of P. Symonds (Bull. London Math. Soc. 36 (2004) 623-632) and consider the cohomology of trivial source modules.

1 Introduction

Let G be a finite group and k an algebraically closed field of characteristic p > 0. In [13], P. Symonds proved the following.

Theorem 1.1 ([13, Theorem 4.1]). As an inflation functor, the cohomology H^{∗}(−, k) contains every simple cohomological inflation functor as a composition factor.

Using this result, he proved the following thorem of G. Mislin [7] for finite groups.

Theorem 1.2 ([7, Theorem], [11, Theorem 1.1], [13, Theorem 1.1]). Let H be a subgroup of G containing a Sylow p-subgroup of G. Then the following are equivalent. (1)The restriction

res_{G,H} : H^{∗}(G, k) ⟶ H^{∗}(H, k)

∗E-mail address: ahida@math.edu.saitama-u.ac.jp

is an isomorphism. (2)If ^{x}Q ⊆ H then x ∈ HC_{G}(Q), for any x ∈ G and p-subgroup Q ≤ H.

The original result of Mislin is proved for compact Lie groups. The proof in [11] and Symonds’ proof of Theorem 1.1 use also results from topology. The purpose of this paper is to give an algebraic proof of Theorem 1.2 using modular representation theory of finite groups.

Let G be a finite group. Then H^{n}(−, k) is a cohomological Mackey functor for G. The simple cohomological Mackey functors are classified in [15], [16]. They are parametrized by pairs (P, V), where P is a p-subgroup of G and V is a simple k(N_{G}(P)/P)-module (up to conjugation and isomorpism). The following result is obtained from Theorem 1.1 immediately.

Theorem 1.3 ([13, Corollary 4.2]). Let P be a p-subgroup of G and V a simple k(N_{G}(P)/P C_{G}(P))-module. Then the simple Mackey functor S_{P,V}^{G} appears in H^{n}(−, k) as a composition factor for some n ≥ 0.

Actually we can prove Theorem 1.2 using only Theorem 1.3, see section 2. On the other hand, Theorem 1.3 is equivalent to a condition on the cohomology of a trivial source module. Let M be an indecomposable trivial source kG-module with vertex P. Then the Green correspondent of M in N_{G}(P) is a projective cover of some simple kN_{G}(P)-module V as a k(N_{G}(P)/P)- module. We denote the trivial source module M by M_{P,V}^{G}. Then Theorem 1.3 is equivalent to the following.

Theorem 1.4 ([13, Theorem 5.3]). Let P be a p-subgroup of G and V a simple k(N_{G}(P)/P)-module. Then H^{∗}(G, M_{P,V}^{G}) ≠ 0 if and only if C_{G}(P) acts on trivially on V.

We prove Theorem 1.4 in section 4. To reduce the problem to some local subgroup of G, we need a result of Benson [3], which says that every periodic module is induced from some subgroup. In [8], T. Okuyama gives an algebraic proof of Theorem 1.4 independently. Some related results are obtained in [1], [10].

Let H be a subgroup of G and M a kG-module. We denote by M ↓_{H} the restriction of M to H. If N is a kH-module, we denote by N ↑^{G} the induced kG-module. Let M and N be kG-modules. Let

Êxt _{kG}^{n}(M, N) = Ext_{kG}^{n}(M, N)

for n > 0 and

Êxt_{kG}^{0}(M, N) = Hom_{kG}(M, N)/PHom_{kG}(M, N)

where PHom_{kG}(M, N) is the set of all kG-homomorphisms which factor through projective modules. Let

Êxt_{kG}^{∗}(M, N) = ⨁_{n≥0} Êxt_{kG}^{n}(M, N).

It is possible to consider Êxt_{kG}^{n}(M, N) for n < 0, but we need only the non-negative part.

2 Mackey functor and fusion

In this section, we review the results of Symonds [13, section 3 and section 5]. Here we consider only Mackey functors for a finite group G though Symonds considered inflation functors and global Mackey fanctors. Fix a p-subgroup Q of G. Let T_{G}(Q, H) = {x ∈ G | ^{x}Q ⊆ H} for H ≤ G. Then H(resp. C_{G}(Q)) acts on T_{G}(Q, H) on the left (resp. right). Let

H\T_{G}(Q, H)/C_{G}(Q) = {HxC_{G}(Q) | x ∈ T_{G}(Q, H)}

be the set of orbits. We constract a Mackey functor M_{Q} as follows. Let M_{Q}(H) = k(H\T_{G}(Q, H)/C_{G}(Q)), the k-space with basis H\T_{G}(Q, H)/C_{G}(Q), for H ≤ G. Let

c_{g} : M_{Q}(H) ⟶ M_{Q}(^{g}H)
HxC_{G}(Q) ⟼ ^{g}H(gx)C_{G}(Q)

for g ∈ G. Furthermore we define two maps

t_{K}^{H} : M_{Q}(K) ⟶ M_{Q}(H)
KxC_{G}(Q) ⟼ HxC_{G}(Q)
r_{K}^{H} : M_{Q}(H) ⟶ M_{Q}(K)

HxC_{G}(Q) ⟼ ∑_{h∈[K\H], hx∈T_{G}(Q,K)} K(hx)C_{G}(Q)

for K ≤ H ≤ G, where [K\H] is a representatives of the cosets of K in H.

Theorem 2.1. (1) Let Q be a p-subgroup of G. Then M_{Q} constructed above is a cohomological Mackey functor for G. (2)For any subgroup H of G, C_{G}(H) acts trivially on the kN_{G}(H)-module M_{Q}(H). (3)Let P be a p-subgroup of G. Then the Mackey functor M_{P} contains the simple Macky functor S_{P,V}^{G} for any simple k(N_{G}(P)/P C_{G}(P))-module V.

Proof. (1)It is easy to check that M_{Q} is a Mackey functor. By the same proof as in [13, Lemma 3.3], it is cohomological. (2)Let x ∈ T_{G}(Q, H). Then ^{x^{−1}yx}z = z for any y ∈ C_{G}(H) and z ∈ Q. Hence C_{G}(H)x ⊆ xC_{G}(Q). (3)By the definition of M_{P}, M_{P} (P) = k(N_{G}(P)/P C_{G}(P)). This contains every simple k(N_{G}(P)/P C_{G}(P))-module as a composition factor. On the other hand, if R is a proper subgroup of P, then M_{P} (R) = 0 by the definition of M_{P}. Hence M_{P} contains S_{P,V}^{G} for any simple k(N_{G}(P)/P C_{G}(P))-module V.

The following result is clear from the defiition of M_{Q}.

Proposition 2.2. Let Q be a p-subgroup of G and Q ≤ H ≤ G. Then

dim M_{Q}(H) = 1

if and only if

T_{G}(Q, H) = HC_{G}(Q).

Now let explain how we can obtain Theorem 1.2 from Theorem 1.3 following [13]. Assume that H ≤ G and H contains a Sylow p-subgroup of G. Then

res_{G,H} : H^{∗}(G, k) ⟶ H^{∗}(H, k)

is an isomorphism if and only if

r_{H}^{G} : S_{P,V}^{G} (G) ⟶ S_{P,V}^{G} (H)

is an isomorphism for any p-subgroup P and simple k(N_{G}(P)/P C_{G}(P))- module V by Theorem 1.3. On the other hand, these maps are isomorphisms if and only if r_{H}^{G} : M_{Q}(G) ⟶ M_{Q}(H) is an isomorphism for any p-subgroup Q by Theorem 2.1. By Proposition 2.2, these maps are isomorphisms if and only if T_{G}(Q, H) = HC_{G}(Q) for any p-subgroup Q of H.

Next we consider the cohomology of some trivial source modules. Let P be a p-subgroup of G and V a simple k(N_{G}(P)/P)-module. Let P_{V} be the projective cover of V as a k(N_{G}(P)/P)-module. Let M_{P,V}^{G} be the Green correspondent of P_{V} with respect to (G, N_{G}(P), P). Then by [13, section 5], the simple Mackey functor S_{P,V}^{G} is a composition factor of H^{n}(−, k) if and only if H^{n}(G, M_{P,V^{∗}}^{G}) ≠ 0 where V^{∗} = Hom_{k}(V, k). Hence Theorem 1.3 is equivalent to Theorem 1.4.

Note that if S_{P,V}^{G} is a composition factor of H^{n}(−, k) then C_{G}(P) acts trivially on S_{P,V}^{G} (P) = V. So the only if part of Theorem 1.4 is clear.

3 Periodic modules

In this section we state some results on the cohomology and the variety of kG-modules. If ζ ∈ H^{n}(G, k), then ζ is considered as a kG-homomorphism Ω^{n}(k) ⟶ k. Let L_{ζ} be the kernel of this map if ζ ≠ 0. Set L_{ζ} = Ω^{n}(k)⊕Ω(k) if ζ = 0. Then there exists a short exact sequence

0 ⟶ L_{ζ} ⟶ Ω^{n}(k) ⊕ (projective) ⟶ k ⟶ 0.

Let V_{G}(k) be the maxmal ideal spectrum of H^{∗}(G, k). Let M be a finitely generated kG-module. Let V_{G}(M) be the homogeneous closed subset of V_{G}(k) defined by the annihilator of Ext_{kG}^{∗}(M, M) in H^{∗}(G, k). This closed subset is called the variety of M. Let ζ_{1}, …, ζ_{m} be homogeneous elements in H^{∗}(G, k). We denote by V_{G}(ζ_{1}, …, ζ_{m}) the closed subset defined by the ideal (ζ_{1}, …, ζ_{m}). It is known that V_{G}(⊗_{i=1}^{m}L_{ζ_{i}}) = V_{G}(ζ_{1}, …, ζ_{m}). For detailes, see [2, Chapter 5]. We need the following result in the next section. Note that the proof of this Proposition in [3] uses Rickard’s idempotent module [9] which is not finitely generated.

Proposition 3.1 ([3, Theorem 3.1]). Let E be a subgroup of oreder p in G and H = N_{G}(E). Let l̃ = res_{G,E}^{∗}(V_{E}(k)) and l = res_{H,E}^{∗}(V_{E}(k)). Let M be a kG-module with V_{G}(M) = l̃. If M ↓_{H}= X ⊕ Y where V_{H}(X) = l and V_{H}(Y) ∩ l = {0}, then

X ↑^{G}≅ M ⊕ Z

for some projective module Z.

4 Cohomology of trivial source modules

In this section we prove Theorem 1.4.

Lemma 4.1. Let P be a p-subgroup of G and V a simple k(N_{G}(P)/P C_{G}(P))- module. If G ≥ H ≥ P, then there exists a simple k(N_{H}(P)/P C_{H}(P))- module V′ such that M_{P,V′}^{H} is a direct summand of M_{P,V}^{G} as a kH-module.

Proof. Let P_{V} be a projective cover of V as a k(N_{G}(P)/P)-module. There exists a composition factor V′ of V ↓_{N_{H}(P)} such that Hom_{N_{H}(P)}(P_{V} , V′) ≠ 0. Then V′ is a k(N_{H}(P)/P C_{H}(P))-module and P_{V′} is a direct summand of P_{V} ↓_{N_{H}(P)}, where P_{V′} is a projective cover of V′ as a k(N_{H}(P)/P)-module. So, P_{V′} is a direct summand of U ↓_{N_{H}(P)} for some indecomposable direct summand U of M_{P,V}^{G} ↓_{H}. Then U ≅ M_{P,V′}^{H} by Burry-Carlson-Puig theorem [14, Exercise (20.5)].

Next we prove some results on the module L_{ζ}.

Lemma 4.2. Let X and Y be kG-modules. Let ζ ∈ H^{n}(G, k). (1)If Êxt_{kG}^{∗}(X, Y) = 0, then Êxt_{kG}^{∗}(X ⊗ L_{ζ}, Y) = 0. (2)Let I be a prime ideal of H^{∗}(G, k) and ζ ∈ I. If ann Êxt_{kG}^{∗}(X, Y), the annihilator of Êxt_{kG}^{∗}(X, Y) in H^{∗}(G, k), is contained in I, then

ann Êxt_{kG}^{∗}(X ⊗ L_{ζ}, Y) ⊆ I.

Proof. (1)By the long exact sequence of cohomology, we have the following exact sequence,

Êxt_{kG}^{i}(Ω^{n}(X), Y) ⟶ Êxt_{kG}^{i}(X ⊗ L_{ζ}, Y) ⟶ Êxt_{kG}^{i+1}(X, Y)

for any i ≥ 0. Then the result follows since Êxt_{kG}^{i}(Ω^{n}(X), Y) ≅ Êxt_{kG}^{i+n}(X, Y). (2)Let A = H^{∗}(G, k) and B = Êxt_{kG}^{∗}(X, Y). Since there exists an exact sequence,

Êxt_{kG}^{i}(X, Y) ⟶ Êxt_{kG}^{i+n}(X, Y) ⟶ Êxt_{kG}^{i}(X ⊗ L_{ζ}, Y)

we have ann Êxt_{kG}^{∗}(X ⊗ L_{ζ}, Y) ⊆ \sqrt{ann_{A}B/ζB}. If ann_{A}B/ζB is not contained in I, then B_{I} = (ζB)_{I} and B_{I} = IB_{I} since ζ ∈ I, where B_{I} is the localization of B with respect to I. Hence B_{I} = 0 but this contradicts the assumption that ann_{A}B ⊆ I.

Let A be a k-algebra and B an A-module. We say an element a ∈ A is B-regular if B ≠ 0 and ab ≠ 0 for any nonzero element b ∈ B.

Lemma 4.3. Let E be a subgroup of order p in the center of G and ζ ∈ H^{m}(G, k). Let M be a k(G/E)-module. Suppose that H^{∗}(G, M) ≠ 0. If res_{G,E}(ζ) is not nilpotent, then ζ is H^{∗}(G, M)-regular.

Proof. This lemma is proved using the argument in [6, Proof of Theorem 10.3.1]. It is also proved in [5, Proposition 4] for M = k. We include the proof using the method in [5, Proposition 4].

Let v ∈ H^{n}(G, M). Assume that v ≠ 0 and ζv = 0. Let

μ_{1}^{∗} : H^{∗}(G, k) ⟶ H^{∗}(E, k) ⊗ H^{∗}(G, k)
μ_{2}^{∗} : H^{∗}(G, M) ⟶ H^{∗}(E, k) ⊗ H^{∗}(G, M)

be the homomorphisms induced by the multiplication μ : E × G ⟶ G in G. Then

μ_{1}^{∗}(ζ) = res_{G,E}(ζ) ⊗ 1 + ζ̃
μ_{2}^{∗}(v) = ∑ v_{j}

where ζ̃ ∈ ∑_{i=1}^{m} H^{m−i}(E, k) ⊗ H^{i}(G, k) and v_{j} ∈ H^{n−j}(E, k) ⊗ H^{j}(G, M). Take a minimal k such that v_{k} ≠ 0. Then

0 = μ_{2}^{∗}(ζv) = (res_{G,E}(ζ) ⊗ 1)v_{k} + w

with w ∈ H^{∗}(E, k) ⊗ (⊕_{j>k}H^{j}(G, M)). Hence res_{G,E}(ζ) is a divisor of 0 in H^{∗}(E, k). This is a contradiction since E is a cyclic group.

Corollary 4.4. Let P be a p-subgroup of G and E a subgroup of order p in the cenetr of P. Let M be a P -projective kG-module. Suppose that E acts on M trivially and H^{∗}(G, M) ≠ 0. Let ζ ∈ H^{∗}(G, k). If res_{G,E}(ζ) is not nilpotent, then ζ is H^{∗}(G, M)-regular. In particular, ann H^{∗}(G, M) ⊆ \sqrt{Ker res_{G,E}}.

Proof. Since M is P -projective, the restriction

res_{G,E} : H^{∗}(G, M) ⟶ H^{∗}(P, M)

is injective. Hence the result follows from Lemma 4.3.

Now we prove Theorem 1.4 by induction on the order of G. Let P be a p-subgroup of G and V a simple k(N_{G}(P)/P C_{G}(P))-module. Note that if P = 1, then M_{P,V}^{G} is the trivial kG-module and the rsult is clear. In the rest of this section, we assume that P ≠ 1. Let E be a subgroup of order p in the center of P. First, we consider the case G = C_{G}(E).

Proposition 4.5. If G = C_{G}(E), then H^{∗}(G, M_{P,V}^{G}) ≠ 0.

Proof. The following argument is based on the idea of [12]. Let M = M_{P,V}^{G}. Then M ≅ M_{P/E,V}^{G/E} as k(G/E)-modules since E is central in G. By the induction, we may assume that H^{∗}(G/E, M) ≠ 0. Take minimal n ≥ 0 such that H^{n}(G/E, M) ≠ 0. Consider the spectral sequence,

E_{2}^{pq} = H^{p}(G/E, H^{q}(E, M)) ≅ H^{p}(G/E, M) ⊗ H^{q}(E, k) ⇒ H^{p+q}(G, M).

Since E_{2}^{pq} = 0 for any p < n and any q ≥ 0, E_{∞}^{n0} ≅ E_{2}^{n0} = H^{n}(G/E, M) ≠ 0.

Let H = N_{G}(E) and C = C_{G}(E). Then H/C is a cyclic group of order prime to p and H/C acts on H^{∗}(C, k). We have the following result which is the analogy to [4, Lemma 6.7].

Lemma 4.6. There exists ζ ∈ H^{∗}(C, k) such that res_{C,E}(ζ) is not nilpotent and the one dimensional subspace kζ affords a faithful representation of H/C.

Proof. Let ξ(≠ 0) ∈ H^{2}(E, k). We can constract, using Evens norm map, a homogeneous elememt ζ_{0} ∈ H^{m}(C, k) such that res_{C,E}(ζ_{0}) = ξ^{p^{a}} for some a ≥ 0 ([2, Lemma 5.6.2]). Let J_{m} = Ker res_{C,E} ∩ H^{m}(C, k). Then H/C acts on H^{m}(C, k)/J_{m} = k(ζ_{0} + J_{m}) faithfully. The short exact sequence

0 ⟶ J_{m} ⟶ H^{m}(C, k) ⟶ k(ζ_{0} + J_{m}) ⟶ 0

splits as a sequence of k(H/C)-modules. So there exists ζ ∈ H^{m}(C, k) such that ζ ∉ J_{m} and kζ affords a faithful representation of (H/C).

Proposition 4.7. If G = N_{G}(E), then H^{∗}(G, M_{P,V}^{G}) ≠ 0.

Proof. Let M = M_{P,V}^{G} and C = C_{G}(E). By Lemma 4.1, there exists a simple k(N_{C}(P)/P C_{C}(P))-module V′ such that M_{P,V′}^{C} is a direct summand of M ↓_{C}. By Proposition 4.5, we have H^{∗}(C, M) ≠ 0. There exists v(≠ 0) ∈ H^{∗}(C, M) such that the one dimensional subspace kv is G/C invariant. Let ζ ∈ H^{∗}(C, k) as in Lemma 4.6. Then ζ is H^{∗}(C, M)-regular by Lemma 4.3 and ζ^{b}v is G/C invariant for some b ≥ 0. Hence H^{∗}(G, M) ≅ H^{∗}(C, M)^{G/C} ≠ 0.

Finally we consider the general case. Let H = N_{G}(E) and C = C_{G}(E). Let ζ_{1}, …, ζ_{m} be a homogeneous generator of \sqrt{Ker res_{G,E}}. Let l̃= res_{G,E}^{∗}(V_{E}(k)) and l = res_{H,E}^{∗}(V_{E}(k)). Consider the tensor product L = ⊗_{i=1}^{m}L_{ζ_{i}}. Then V_{G}(L) = l̃. We can decompose L ↓_{H} as L ↓_{H} ≅ X ⊕ Y where V_{H}(X) = l and V_{H}(Y) ∩ l = {0}. Then by Proposition 3.1,

L ⊕ (projective) ≅ X ↑^{G} .

Let η_{i} = res_{G,H}(ζ_{i}). There exists η_{0} ∈ H^{∗}(H, k) such that V_{H}(η_{0}, η_{1}, …, η_{m}) = l and V_{H}(Y) ∩ V_{H}(η_{0}) = {0}. Then Y ⊗ L_{η_{0}} is projective since

V_{H}(Y ⊗ L_{η_{0}}) = V_{H}(Y) ∩ V_{H}(L_{η_{0}}) = {0}

(see [2, Theorem 5.1.1]). We may assume that η_{0}Ext_{kH}^{∗}(X, X) = 0 since some power of η_{0} annihilates Ext_{kH}^{∗}(X, X). Then, modulo projectives,

⨂_{i≥0} L_{η_{i}} ≅ (X ⊕ Y) ⊗ L_{η_{0}} ≅ X ⊗ L_{η_{0}} ≅ Ω(X) ⊕ Ω^{n}(X)

for some n ≥ 0 by [2, Proposition 5.9.5].

Proposition 4.8. H^{∗}(G, M_{P,V}^{G}) ≠ 0.

Proof. By Lemma 4.1, there exists a simple k(N_{H}(P)/P C_{H}(P))-module V′ such that M_{P,V′}^{H} is a direct summand of M_{P,V}^{G} ↓_{H}. Let M = M_{P,V}^{G} and M′ = M_{P,V′}^{H}. Then H^{∗}(H, M′) ≠ 0 by Proposition 4.7. Let I = \sqrt{Ker res_{H,E}} = \sqrt{(η_{0}, η_{1}, …, η_{m})}. Then ann_{H^{∗}(H,k)}H^{∗}(H, M′) ⊆ I by Corollary 4.4. So

ann_{H^{∗}(H,k)}Êxt_{kH}^{∗}(X, M′) ⊆ ann_{H^{∗}(H,k)}Êxt_{kH}^{∗}(⨂_{i≥0} L_{η_{i}}, M′) ⊆ I

by Lemma 4.2 (2). In particular,

Êxt_{kG}^{∗}(L, M) ≅ Êxt_{kG}^{∗}(X ↑^{G}, M) ≅ Êxt_{kH}^{∗}(X, M) ≠ 0.

Then H^{∗}(G, M) ≠ 0 by Lemma 4.2 (1). This completes the proof.

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