A note on the cohomology of finitary modules

U. Meierfrankenfeld

May 17, 2002

In this note we prove the following three theorems on the cohomology of finitary modules in terms of the cohomology of a local system of subgroups:

Theorem 1 Let G be a group, K a field, V a finitary KG-module and L a local system of subgroups of G. Suppose that, for all H ∈ L, V is completely reducible as a KH-module. Then [V, G] is completely reducible as a KG-module.

Theorem 2 Let G be a group, D a division ring, V a finitary DG-module, L a local system of subgroups of G and H an extension of V by G, (i.e. H/V ≅ G). Suppose that the following holds for all L in L:

(i) The extension of V by L in H splits.

(ii) V/C_{V} (L) is finite dimensional.

(iii) H^{1}(L, V) is finite dimensional.

Then H splits over V.

Theorem 3 Let G be a group, D a division ring, L a local system of subgroups of G, W a DG-module and V a DG-submodule of W such that W = V + C_{W} (H) for all H ∈ L. Then there exists a canonical DG-monomorphism from W/C_{W} (G) to [V^{∗}, G]^{∗}, where Y^{∗} denotes the dual of a module Y.

We remark that conditons (ii) and (iii) in Theorem 2 are automatically fulfilled if all members of L are finite groups generated by elements whose order is coprime to the characteristic of D.

Proof of Theorem 1: Let H ∈ L. Then [V, H] = [V, H, H] and so [V, G] = [V, G, G]. Hence we may assume that V = [V, G]. Let W be the sum of all the irreducible KGsubmodules in V, where W = 0 if G has no irreducible submodules in V. We need to show that W = V.

So suppose that V ≠ W. Then [V, G] ≰ W and we may assume that [V, H] ≰ W for all H ∈ L. Let H ∈ L and let I_{H} be the set of irreducible KH-submodules I in [V, H] with I ≰ W. For I ∈ I_{H} let m(I) be supremum of all positive integers t such that I^{t} is isomorphic to a KH-submodule of V. Pick h ∈ H with [I, h] ≠ 0. Then m(I) · deg_{I}(h) ≤ deg_{V} (h). In particular, m(I) is finite. Note that there exists a unique KH-submodule Î in V isomorphic to I^{m(I)}, namely Î is the submodule generated by all the H submodules in V isomorphic to I. Let K(I) = Hom_{KH}(I, I) and d(I) = dim_{K} K(I). Since dim_{K}[I, h] = dim_{K(I)}[I, h] · dim_{K} K(I), d(I) ≤ deg_{V} (h) and so d(I) is finite. Let m be the minimum of all m(I), I ∈ I_{H}, H ∈ L and d the minimum of all d(I), I ∈ I_{H}, H ∈ L, m(I) = m.

Pick H ∈ L and I ∈ I_{H} with m(I) = m and d(I) = d. Without loss H ≤ F for all F ∈ L. Let F ∈ L. Since V is completely reducible as a KF -module, there exists J ∈ I_{F} such that I is isomorphic to a KH submodule of J. Let a be a positive integer such that I^{a} is isomorphic to a KH-submodule of J. Then I^{a·m(J)} is isomorphic to a KH-submodule of V and so a · m(J) ≤ m. By minimal choice of m, m ≤ m(J). Thus a = 1 and m(J) = m. In particular, Î ≤ Ĵ and there exists a unique KH-submodule U in J isomorphic to I. Hence K(J) acts on U and restriction K(J) |_{U} of K(J) to U is contained in K(U). Since dim_{K} K(U) = dim_{K} K(I) = d ≤ dim_{K} K(J), we conclude that K(J)|_{U} = K(U). It is now easy to see that every irreducible KH submodule of Î lies in an irreducible KF -submodule of Ĵ. Hence ⟨I^{F} ⟩ is an irreducible KF -module for all F ∈ L and ⟨I^{G}⟩ is an irreducible KGsubmodule in V not contained in W. This contradiction completes the proof of Theorem 1.

The following definition and lemma are used in the proof of Theorem 2.

Definition 4 (a) Let R be a ring, A a set, M an R-module and for a ∈ A let ρ_{a} : A → M be a bijection. Then A is called an affine R-module provided that for all a, b, c in A, ρ_{a}(b) + ρ_{b}(c) = ρ_{a}(c).

(b) Let R be a ring, A and B affine R-modules and π : A → B. Then π is called an affine R-homomorphism if for some a in A and b in B, ρ_{b}πρ_{a}^{−1} is a R-homomorphism of modules.

(c) Let R be a ring and A an affine R-module. A subset B of A is called an affine R-submodule if ρ_{a}(B) is a R-submodule of M for some a in A.

Remark: Let M be an R-module and define ρ_{x} : M → M, y → y − x. Then M is an affine R-module. Moreover, if A is any affine R-module with M as underlying module, then for all a in A, ρ_{a} is an isomorphism of affine R-modules. Finally if a, b are in A and C is a subset of A, then ρ_{a}(C) = ρ_{b}(C) + ρ_{a}(b) and so C is an affine submodule if and only if ρ_{a}(C) is the coset of a R-submodule in M.

Lemma 5 Let G be a group, R a ring and V an RG-module. Let A_{G} be the set of complements to V in V >◁ G. Then (a) A_{G} is an affine R-module.

(b) Let H ≤ G, then the canonical map from A_{G} to A_{H} is affine.

(c) Let I_{G} = {G^{v} | v ∈ V }. Then I_{G} is an affine RG submodule of A_{G}, I_{G} ≅ V/C_{V} (G) and A_{G}/I_{G} ≅ H^{1}(G, V).

Proof of the Lemma: Identify V and G with their images in the semidirect product V >◁ G. So V >◁ G = V G.

(a) Let M_{G} the set of functions f : V G/V → V with f(ab) = f(a)^{b^{−1}} + f(b) for all a, b in V G/V, i.e M_{G} is the set of derivations for G on V. Note that M_{G} is an R-module via (r · f)(a) = r · f(a). For K, L in A_{G} define ρ_{K}(L) ∈ M_{G} by ρ_{K}(L)(V a) = v, whenever a ∈ K and v ∈ V with va ∈ L. Then ρ_{K} is a bijection from A_{G} onto M_{G} (see for example [As, 17.1]).

Let K, L, N be in A_{G} and a in K. Put b = ρ_{K}(L)(V a)a and c = ρ_{L}(N)(V b)b. Then V a = V b = V c, b ∈ L , c ∈ N and c = ρ_{L}(N)(V a)ρ_{K}(L)(V a)a. Thus ρ_{K}(L) + ρ_{L}(N) = ρ_{K}(N). (Here we write the binary operation on V multiplicatively whenever V is regarded as a subgroup of V >◁ G).

(b) For L in A_{G} let π(L) = L ∩ V H. Then its is easy to check that ρ_{H}πρ_{G}^{−1} is just the restriction map M_{G} → M_{H}, ϕ → ϕ_{V H/V}. Thus π is affine.

(c) Define α : V → M by α(v)(a) = v^{a} − v. Then ker α = C_{V} (G) and α(V) = ρ_{G}(I_{G}) is the set of inner derivations. In particular H^{1}(G, V) = M/α(V) ≅ A_{G}/I_{G} and (c) holds.

Proof of Theorem 2: Let L ∈ L. By (i) we may view V >◁ L as a subgroup of H and by part (a) of the Lemma , A_{L} is a affine D-module and by (ii),(iii) and Part (c) of the Lemma, A_{L} is finite dimensional. For L and K in L with L ≤ K let π_{K,L} be the affine map defined in Part (b) of the Lemma . We claim that the inverse limit of (π_{K,L})_{L≤K} is not empty. Note that finite dimensional affine D-modules fulfill the descending chain condition on affine subspaces and so a set of affine subspaces whose intersection is empty has a finite subset whose intersection is empty. Moreover, images and inverse images of affine subspaces under affine maps are affine. Now the proof in [KW, 1K1] that inverse limits of non-empty finite sets are not empty carries over word for word, except that ”subset” has to be replaced by ”affine subspace”. Let (C_{L})_{L∈Λ} be an element in the inverse limit. Then ⋃{C_{L}|L ∈ L} is a complement to V in H and Theorem 2 is proved.

Proof of Theorem 3: For X ≤ V^{∗} let X^{⊥} = {v ∈ V |x(v) = 0 for all x ∈ X}. We will first prove that:

(∗) For all K ≤ G, [V^{∗}, K]^{⊥} = C_{V} (K).

Indeed, let x ∈ V^{∗}, k ∈ K and v ∈ V. Then

[x, k](v) = (x^{k} − x)(v) = x^{k}(v) − x(v) = x(v^{k^{−1}}) − x(v) = x([v, k^{−1}])

It follows that v ∈ [V^{∗}, K]^{⊥} if and only if [v, K] ≤ V^{∗⊥} = 0 and so if and only if v ∈ C_{V} (K).

Let H ∈ L. Define a map a_{H} : W → [V^{∗}, H]^{∗} by a_{H}(w)(x) = x(u) where x ∈ [V^{∗}, H], w ∈ W and u ∈ V with w ∈ u + C_{W} (H). Note that by (*) this definition does not depend on the choice of u. If K ≤ H with K ∈ L then C_{W} (H) ≤ C_{W} (K) and so w ∈ u + C_{W} (K) and a_{H}(w)(x) = a_{K}(w)(x) for all x ∈ [V^{∗}, K]. Define a : W → [V^{∗}, G]^{∗} by a(w)(x) = a_{H}(w)(x) whenever w ∈ W, x ∈ [V^{∗}, G] and H ∈ L with x ∈ [V^{∗}, H]. By the preceeding observation and since L is a local system this definition does not depend on the choice of H. Let w ∈ W with a(w) = 0. Then a_{H}(w) = 0 for all H ∈ L and so u ∈ [V^{∗}, H]^{⊥}, where u is as above. By (*), u ∈ C_{V} (H) and so w ∈ C_{W} (H) for all H ∈ L. Thus ker a = C_{W} (G). It remains to show that a is a DG-homomorphism. Clearly a is D-linear. Let w, x, u and H be as above and g ∈ G. We may assume without loss that g ∈ H. Then w^{g} ∈ u^{g} + C_{W} (H) and so

a(w^{g})(x) = a_{H}(w^{g})(x) = x(u^{g}) = x^{g^{−1}}(u) =
= a_{H}(w)(x^{g^{−1}}) = a(w)(x^{g^{−1}}) = a(w)^{g}(x).

Thus a(w^{g}) = a(w)^{g} and a is a DG-homomorphism, completing the proof of Theorem 3.

References

[As] M. Aschbacher, Finite Group Theory, Cambridge Unversity Press, Cambridge (1986)

[KW] O. Kegel and B.A.F. Wehrfritz, Locally Finite Groups, North Holland (1973).

Department of Mathematics

Michigan State University

East Lansing. Michigan 48824

meier@math.msu.edu