Alternating bihomomorphism of finitely generated abelian groups arises as skew of 2-cocycle
In terms of 2-cocycles
In terms of cohomology groups
Suppose and are finitely generated abelian groups. Consider the homomorphism:
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.
- Structure theorem for finitely generated abelian groups
- Symplectic decomposition of an alternating bilinear form taking values in a local principal ideal ring
- Orthogonal direct sum of cocycles is cocycle
- Symplectic decomposition of an alternating bilinear form taking values in integers
First part:reduction to the case where is either infinite cyclic or cyclic of prime power order
We first show that the problem can be reduced to the case that is a cyclic group.
Given: Finitely generated abelian groups and , an alternating bihomomorphism .
To prove: Assuming that we can solve the problem if 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 such that .
|Step no.||Assertion/construction||Facts used||Given data/assumptions used||Previous steps used||Explanation|
|CR 1||We can find cyclic groups such that , such that each is either infinite cyclic or cyclic of prime power order||Fact (1)||is finitely generated and abelian||Given+Fact direct|
|CR 2||Let be the projection of on to the factor . Then, define . Each is an alternating bihomomorphism from to .||Step (CR1)|
|CR3||We can find cocycles such that .||Assumption that we can solve the reduced problem.||Step (CR1): Each is cyclic; Step (CR2): is an alternating bihomomorphism|
|CR4||Consider the mapping . is a 2-cocycle and .||Step (CR3)|