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