PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 2, February 1998, Pages 353–356 S 0002-9939(98)04128-8

A NOTE ON THE COHOMOLOGY OF FINITARY MODULES

U. MEIERFRANKENFELD

(Communicated by Ronald M. Solomon)

Abstract. Let G be a group, D a division ring and V a DG-module. V is called finitary provided that V/C_{V} (g) is finite dimensional for all g ∈ G. We investigate the first and second degree cohomology of finitary modules in terms of a local system for G.

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 conditions (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 KG-submodules 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 the supremum of all positive integers t such that I^{t} is isomorphic to a KH-submodule of V. Pick h ∈ H with

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors August 20, 1996.

1991 Mathematics Subject Classification. Primary 20J05.

[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 the 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 KG-submodule 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 an 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 an 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 an 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} be 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 it 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 an 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 Dmodules 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 preceding 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

\begin{aligned}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). \end{aligned}

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 University Press, Cambridge (1986). MR 89b:20001; MR 95b:20002

[KW] O. Kegel and B. A. F. Wehrfritz, Locally Finite Groups, North Holland (1973). MR 57:9848

Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

   E-mail address: meier@math.msu.edu