# Dual universal coefficient theorem for group cohomology

## Contents

## Statement

### For coefficients in an abelian group

Suppose is a group and is an abelian group. The **dual universal coefficients theorem** relates the homology groups for trivial group action of on and the cohomology groups for trivial group action of on as follows:

First, for any , there is a natural short exact sequence of abelian groups:

Second, the sequence splits (not necessarily naturally), and we get:

### For coefficients in the integers

This is the special case where . In this case, we case:

### Typical case of finitely generated abelian groups

Suppose for some finite group and for some finite group . Suppose further that:

and

Then we have:

where we further have:

where , i.e., the -torsion of .

Also:

Thus, we get overall that:

Finally, suppose:

In this case, the expressions simplify further:

and:

### Typical case of finitely generated abelian groups and coefficients in the integers

Suppose for some finite group and for some finite group . Then:

In other words, we pick the torsion-free part of and the torsion part of (roughly speaking).

## Related facts

### Similar facts for group cohomology

- Universal coefficients theorem for group homology
- Universal coefficients theorem for group cohomology
- Kunneth formula for group homology
- Kunneth formula for group cohomology

### Similar facts for other homology and cohomology theories

- Dual universal coefficients theorem for cohomology (for topological spaces)
- Universal coefficients theorem for homology (for topological spaces)
- Universal coefficients theorem for cohomology (for topological spaces)

### Applications

- This can be used to show the equivalence of the facts: first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms and first homology group for trivial group action equals tensor product with abelianization
- Formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization