Alternating bihomomorphism of finitely generated abelian groups arises as skew of 2-cocycle

Statement

In terms of 2-cocycles

Suppose $G$ and $A$ are finitely generated abelian groups. Suppose $\lambda :G\times G\to A$ is an alternating bihomomorphism of groups from $G$ to $A$. Then, there exists a 2-cocycle for trivial group action $c:G\times G\to A$ such that $\operatorname {Skew} c=\lambda$, i.e.,:

$\!c(g,h)-c(h,g)=\lambda (g,h)\ \forall \ g,h\in G$

In terms of cohomology groups

Suppose $G$ and $A$ are finitely generated abelian groups. Consider the homomorphism:

$H^{2}(G,A){\stackrel {\operatorname {Skew} }{\to }}\bigwedge ^{2}(G,A)$

which sends a cohomology class to the skew of any 2-cocycle representing it (that this is a homomorphism arises from the fact that skew of 2-cocycle for trivial group action of abelian group is alternating bihomomorphism). Then, this homomorphism is surjective, i.e., every alternating bihomomorphism arises from some cohomology class.

Facts used

1. Structure theorem for finitely generated abelian groups
2. Symplectic decomposition of an alternating bilinear form taking values in a local principal ideal ring
3. Orthogonal direct sum of cocycles is cocycle
4. Symplectic decomposition of an alternating bilinear form taking values in integers

Proof

First part:reduction to the case where $A$ is either infinite cyclic or cyclic of prime power order

We first show that the problem can be reduced to the case that $A$ is a cyclic group.

Given: Finitely generated abelian groups $A$ and $G$, an alternating bihomomorphism $\lambda :G\times G\to A$.

To prove: Assuming that we can solve the problem if $A$ were replaced by an infinite cyclic group or a cyclic group of prime power order, we can solve the problem in the general case, i.e., we can find a 2-cocycle $c$ such that $\operatorname {Skew} (c)=\lambda$.

Proof:

Step no.	Assertion/construction	Facts used	Given data/assumptions used	Previous steps used	Explanation
CR 1	We can find cyclic groups $A_{1},A_{2},\dots ,A_{n}$ such that $A=A_{1}\oplus A_{2}\oplus \dots \oplus A_{n}$, such that each $A_{i}$ is either infinite cyclic or cyclic of prime power order	Fact (1)	$A$ is finitely generated and abelian		Given+Fact direct
CR 2	Let $\pi _{i}$ be the projection of $A$ on to the factor $A_{i}$. Then, define $\lambda _{i}=\pi _{i}\circ \lambda$. Each $\lambda _{i}$ is an alternating bihomomorphism from $G$ to $A_{i}$.			Step (CR1)
CR3	We can find cocycles $c_{i}:G\times G\to A_{i}$such that $\operatorname {Skew} (c_{i})=\lambda _{i}$.		Assumption that we can solve the reduced problem.	Step (CR1): Each $A_{i}$ is cyclic; Step (CR2):$\lambda _{i}$ is an alternating bihomomorphism
CR4	Consider the mapping $c=c_{1}\oplus c_{2}\oplus \dots \oplus c_{n}$. $c$ is a 2-cocycle $G\times G\to A$ and $\operatorname {Skew} (c)=\lambda$.			Step (CR3)

Solution in the case of cyclic of prime power order

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