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

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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_isuch 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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