DUALITY AND LOCAL GROUP COHOMOLOGY

P R Hewitt

Univ of Toledo

23 March 97

Abstract. Recently, Meierfrankenfeld has published three theorems on the cohomology of a finitary module. They cover the local determination of complete reducibility; the local splitting of group extensions; and the representation of locally split extensions in the double dual. In this note we derive all three by combining a certain duality between homology and cohomology with the continuity of homology.

Local Cohomology

We describe a general framework for three recent results of Meierfrankenfeld’s [M], on the cohomology of finitary modules. It turns out that Theorem A below — which began life as an attempt to understand the third of this trio — includes as corollaries all three. Moreover, it implies some new corollaries, and covers all cohomological degrees.

Throughout, G is a group, k is a commutative field, and L is a local system — an upwardly directed collection of subgroups whose union is G. Modules are k-spaces. If V is a module, then V^{∨} denotes its dual, and ⟨ | ⟩ denotes their pairing.

We will explore the relationship between the cohomology of G, H^{∗}(G, V), with the local cohomology, H^{∗}(L, V). This latter is defined by taking the limit, with respect to restriction maps, of the cohomology groups H^{∗}(L, V), for L in L:

H^{∗}(L, V) := \underleftarrow{lim} H^{∗}(L, V).

We can define analogous limits, C^{∗}(L, V), Z^{∗}(L, V), and B^{∗}(L, V), for cochains, cocycles, and coboundaries, respectively. What we find is that there are canonical isomorphisms C^{∗}(L, V) ≅ C^{∗}(G, V) and Z^{∗}(L, V) ≅ Z^{∗}(G, V). Indeed, the localization C^{∗}(G, V) → C^{∗}(L, V) is defined by restriction to the local subgroups, ϕ ↦ {ϕ|_{L}}. In the reverse direction, the map C^{∗}(L, V) → C^{∗}(G, V) splices an L-sequence of n-cochains {ϕ_{L}} into a cochain on G: g_{1}, …, g_{n} ↦ ϕ_{L}(g_{1}, …, g_{n}), where L is any member of L that contains all the g_{i}. This is well-defined, since in the inverse limit C^{∗}(L, V), ϕ_{L}|_{L′} = ϕ_{L′}, whenever L ⊃ L′.

These maps respect coboundary, and so they induce isomorphisms for the cocycle groups. However, splicing an L-sequence of coboundaries need not yield a coboundary in G. In general we obtain only an embedding B^{∗}(G, V) ↪ B^{∗}(L, V).

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20J05.

Now consider the localization H^{∗}(G, V) → H^{∗}(L, V). An element of H^{i}(L, V) is an inverse system of affine flats {ϕ_{L} + B^{i}(L, V)}, where ϕ_{L}|_{L′} ≡ ϕ_{L′} mod B^{i}(L′, V) whenever L ⊃ L′. The image of H^{i}(G, V) consists of those whose inverse limit is nonempty. Similarly, an element in the kernel of the map H^{i+1}(G, V) → H^{i+1}(L, V) is an inverse system of affine flats {ψ_{L} + Z^{i}(L, V)}, where δψ_{L}|_{L′} = δψ_{L′} whenever L ⊃ L′ — viewed modulo the collection of those inverse systems that have a nonempty limit.

The simple ‘compactness’ proof of [M, Thm 2] applies in arbitrary cohomological degree, to give the following generalization.

Proposition. Let i be a positive integer. a. If dim_{k} B^{i}(L, V) < ∞ for all L in L, then localization H^{i}(G, V) → H^{i}(L, V) is

surjective in degree i. b. If moreover dim_{k} Z^{i}(L, V) < ∞ for all L in L, then additionally localization

H^{i+1}(G, V) → H^{i+1}(L, V) is injective in degree i + 1. □

We omit the proof because we will not be using this result. The problem with this generalization is that it concerns finitary modules only for H^{2}(G, V).

We will write Ext_{G}^{∗}(X, Y) instead of H^{∗}(G, Hom_{k}(X, Y)), and Tor_{∗}^{G}(X, Y) instead of H_{∗}(G, X ⊗_{k} Y). In addition to local cohomology, we could also define local homology: Tor_{∗}^{L}(X, Y) = \underrightarrow{lim} Tor_{∗}^{L}(X, Y). However, this yields nothing new.

Lemma 1. If X and Y are G-modules, then Tor_{∗}^{L}(X, Y) = Tor_{∗}^{G}(X, Y).

Proof. First of all, note that if P_{∗} = C_{∗}(G, ℤ), and V is any G-module, then \underrightarrow{lim} P_{∗} ⊗_{L} V = P_{∗} ⊗_{G} V. To see this, observe that P_{∗} ⊗_{G} V = (P_{∗} ⊗ V)/[P_{∗} ⊗ V, G] and [P_{∗} ⊗ V, G] = ∑_{L}[P_{∗} ⊗ V, L].

Since the maps P_{∗} ⊗_{L} V → P_{∗} ⊗_{G} V are surjective, we obtain that the boundaries of the limit are limits of boundaries. Now if an element maps to 0 in a direct limit then it maps to 0 at some finite stage. This tells us that the cycles of the limit are limits of cycles. □

Although local cohomology can differ from global cohomology, we will see that they are the same for the dual of a module.

Main Theorem

In this section we first record a duality between homology and cohomology. We then exploit this duality, together with the continuity of homology, to obtain a theorem on the local detection of cohomology, valid in all degrees.

Lemma 2. If f: V → V is a linear endomorphism, then

ann_{V} (im(f^{∨})) = ker(f) ann_{V^{∨}}(im(f)) = ker(f^{∨})
ann_{V} (ker(f^{∨})) = im(f) ann_{V^{∨}}(ker(f)) = im(f^{∨})

Proof. The first two equalities follow from the identity ⟨f(x) | λ⟩ = ⟨x | f^{∨}(λ)⟩. The third follows from the first, since any subspace of V is the annihilator of its annihilator. The last follows from composition of the canonical isomorphisms ann_{V^{∨}}(ker(f)) = im(f)^{∨} = im(f^{∨}). □

The following is a special case of [B, Prop 2.8.5], but we include a direct proof.

DUALITY AND LOCAL GROUP COHOMOLOGY 3

Lemma 3. If X and Y are L-modules, then Ext_{L}^{∗}(X, Y^{∨}) = Tor_{∗}^{L}(X, Y)^{∨}. Proof. Let P_{∗} = C_{∗}(G, ℤ), and note the canonical isomorphism

Hom(P_{∗}, Hom_{k}(X, Y^{∨})) = (P_{∗} ⊗ X ⊗_{k} Y)^{∨}.

This isomorphism can be described as follows: a functional λ in (P_{∗} ⊗ X ⊗_{k} Y)^{∨} is paired to an additive map f: P_{∗} → Hom_{k}(X, Y^{∨}) when they satisfy the relation

⟨y | f(c)(x)⟩ = ⟨c ⊗ x ⊗ y | λ⟩.

Now if we take fixed points for L in this isomorphism, we obtain that

Hom_{L}(P_{∗}, Hom_{k}(X, Y^{∨})) = (P_{∗} ⊗_{L} (X ⊗_{k} Y))^{∨}.

To finish, apply Lemma 2 to the boundary of P_{∗} ⊗_{L} (X ⊗_{k} Y). □ Theorem A. If X and Y are G-modules, then localization

Ext_{G}^{∗}(X, Y^{∨}) → Ext_{L}^{∗}(X, Y^{∨})

is an isomorphism. Proof. Apply Lemmas 1 and 3, and the fact that the dual of a direct limit of k-spaces is the inverse limit of the duals of those spaces. □ Corollary 1. Locally trivial coclasses are trivial in the double dual. Proof. Extension of scalars from Y to Y^{∨∨} factors through localization:

Ext_{G}^{∗}(X, Y) → Ext_{G}^{∗}(X, Y^{∨∨})
↓  ∥
Ext_{L}^{∗}(X, Y) → Ext_{L}^{∗}(X, Y^{∨∨}). □

Corollary 2. If X is an arbitrary G-module and Y is a finite-dimensional Gmodule, then Ext_{G}^{∗}(X, Y) is determined locally. □

In the next section we derive several more corollaries of this theorem, including all three theorems from [M].

Finitary Cohomology

We now turn to the cohomology of finitary modules. The following characterization of finitary groups — due to Meierfrankenfeld — is the key to understanding their cohomology. Lemma 4. The finitary group on V is exactly the centralizer of V^{∨∨}/V. Proof. Let x be a transformation of V. Apply Lemma 2 to x − 1 twice: between V and V^{∨}, and also between V^{∨} and V^{∨∨}. Next, use the fact that a space equals its double dual if and only if it is finite dimensional. □

In light of this characterization, consider the long exact sequence associated to the extension V ↪ V^{∨∨}:

(†) ⋯ → Ext_{G}^{i−1}(U, V^{∨∨}/V) → Ext_{G}^{i}(U, V) → Ext_{G}^{i}(U, V^{∨∨}) → · · · .

Theorem B. If U is an arbitrary G-module and V is a finitary G-module, then the locally trivial portion of Ext_{G}^{i}(U, V) lies in the image of Ext_{G}^{i−1}(U, V^{∨∨}/V).

Proof. Apply Theorem A and Lemma 4 to (†). □

Corollary 3. Let U be an arbitrary module, and V a finitary module. a. If U = [U, G], then localization Ext_{G}^{1}(U, V) → Ext_{L}^{1}(U, V) is injective. b. If Hom(G, k) = 0, then localization H^{2}(G, V) → H^{2}(L, V) is injective.

Proof. If U = [U, G] then Ext_{G}^{0}(U, V^{∨∨}/V) = Hom_{kG}(U, V^{∨∨}/V) = 0, whence we obtain part a. If Hom(G, k) = 0 then Ext_{G}^{1}(k, V^{∨∨}/V) = Hom(G, V^{∨∨}/V) = 0, which yields part b. □

Corollary 4. [M, Thm 1] If V is locally completely reducible, then [V, G] is completely reducible.

Proof. Let W < [V, G]. We show that W is a direct summand. Set U = [V, G]/W. Since [V, G] = [V, G, G], Hom_{G}(U, W^{∨∨}/W) = 0. Now apply Corollary 3a. □

Corollary 5. [M, Thm 2] If H^{1}(L, V) and C_{V} (L) are finite-dimensional for every L in L, then localization H^{2}(G, V) → H^{2}(L, V) is injective.

Proof. If V is infinite dimensional, apply Corollary 2. If not, the exact sequence

⋯ → V/ C_{V} (L) → H^{1}(L, C_{V} (L)) → H^{1}(L, V) → · · · .

tells us that H^{1}(L, k) = 0, since otherwise H^{1}(L, V) would be infinite dimensional. Hence H^{1}(G, k) = 0. Now apply Corollary 3b. □

Corollary 6. [M, Thm 3] If V ↪ W is a locally split extension of V by a trivial G-module, then there is a canonical injection W/ C_{W} (G) ↪ [V^{∨}, G]^{∨}.

Proof. Note that [V^{∨}, G]^{∨} = V^{∨∨}/ C_{V^{∨∨}} (G), and apply Theorem A. □

Two Examples

For the first example, let Ω be a set of some infinite cardinal ℵ, let G be the finitary symmetric group on Ω, let k be of characteristic 2, and let V be the natural permutation module kΩ. The Eckmann-Shapiro Lemma (see [B, Cor 2.8.4]) tells us that H^{1}(G, V^{∨}) = H^{2}(G, V^{∨}) = k. Now V embeds in V^{∨}, with G-trivial quotient (it is a quotient of V^{∨∨}/V). The associated long exact sequence contains the fragment

⋯ → H^{1}(G, V^{∨}) → H^{1}(G, V^{∨}/V) → H^{2}(G, V) → H^{2}(G, V^{∨}) → · · · .

Thus dim_{k} H^{2}(G, V) = dim_{k} H^{1}(G, V^{∨}/V) = 2^{ℵ}. On the other hand, if we express G as a union of finite symmetric groups, then we have that H^{2}(L, V) = k. To see this, let Γ and Γ′ be cofinite subsets, with Γ ⊃ Γ′. Set F = C_{G} (Ω), F′ = C_{G} (Ω′), and Δ = Ω − Ω′. We have that H^{2}(F, V) = k ⊕ H^{2}(F, kΔ) ⊕ H^{2}(F, kΩ′), while H^{2}(F′, V) = k ⊕ H^{2}(F′, kΩ′). Thus, taking successively smaller Γ′, we see that none of the classes in H^{2}(F, V) that take values in kΓ survive in the limit.

Although very little of the 2-cohomology of the symmetric group is detected locally, all of the 2-cohomology for the alternating subgroup is detected locally, by Corollary 3. Moreover, all of the 2-cohomology for the symmetric group is detected locally when we replace k by a field of any other characteristic.

For the second example, let k have odd characteristic, let V be an infinitedimensional k-space with a nondegenerate symplectic form, and let G be the finitary symplectic group. The form gives an embedding of V in V^{∨}. Again, the quotient V^{∨}/V is G-trivial. The extension V ↪ V^{∨} is nonsplit since C_{V^{∨}} (G) = 0. However, if we let L be the collection of symplectic groups of the finite-dimensional, nondegenerate subspaces of V, then we find that the extension is locally split. (See [CPS, Tbl 4.5, pp 186–187].) By Corollary 5, any G-trivial extension of V^{∨} is split.

Skew Fields

Of the three theorems in [M], Meierfrankenfeld assumes that k is commutative only in the first. The results above can be extended to the case of a skew field k, by carefully tracking whether k should multiply from the left or right — or both. The guiding principle is the duality of Lemma 3, and so we must ensure that all the homology and cohomology groups are k-spaces. For Lemma 1 we need only take X to be a right k-space and Y a left k-space. However, for Lemma 3 and Theorem A we must further assume that X is a k-kG-bimodule. This is tantamount to requiring that the G-action on X be defined over the center of k. Note that Lemma 3 no longer follows from [B, Prop 2.8.5] when k is not commutative. However, the direct proof given above works in general, if assume that X is a bimodule.

In the applications to finitary modules, we must assume similarly that U is a bimodule. In most of the corollaries this causes no difficulty. In particular, since we are taking U = k in the application to [M, Thm 2] and [M, Thm 3], we find that these are subsumed by Theorem B even when k is not commutative. However, in trying to apply Theorem B to [M, Thm 1], we find that we must assume that the G-action on V is defined over the center of k — which in effect echoes Meierfrankenfeld’s hypothesis for this theorem.

Recently — and independently — Wehrfritz proves Corollary 4 assuming only that k is finite-dimensional over its center [W_{1}]. In fact Theorem A can be extended to this situation, by tensoring with a splitting field. If ζ is the center of k, and μ is a maximal commutative subfield of k, then Hom_{k}(X, Y^{∨}) ⊗_{ζ} μ = Hom_{μ}(X, Y^{∨}). So, Ext_{G}^{∗}(X, Y^{∨}) ⊗_{ζ} μ = Ext_{L}^{∗}(X, Y^{∨}) ⊗_{ζ} μ, whence Ext_{G}^{∗}(X, Y^{∨}) = Ext_{L}^{∗}(X, Y^{∨}), since field extension is faithfully flat.

The example in [W_{2}] shows that, without some hypothesis on the field, Corollary 4 would be false. (See also [ShW, Ex 1.18].)

Acknowledgment

It is a pleasure to record my thanks to Ulrich Meierfrankenfeld, for his many insights, suggestions, and patient explanations; and also to the Math Department at Michigan State University, for their generous hospitality.

References

[M] U Meierfrankenfeld, A note on the cohomology of finitary modules, Proc AMS (to appear).

[B] D Benson, Representations and Cohomology I, Cambridge University Press, 1991.

[CPS] E Cline, B Parshall, L Scott, Cohomology of finite groups of Lie type, I, Publ Math IHES 45 (1975), 169–191.

[W_{1}] B Wehrfritz, The complete reducibility of locally completely reducible finitary linear groups,

Bull LMS (to appear). [W_{2}] B Wehrfritz, Complete reducibility in skew linear groups, Jour LMS 28 (1983), 301–309.

[ShW] M Shirvani, B Wehrfritz, Skew Linear Groups, Cambridge University Press, 1986.