Last update: 01 February 2012
Let be a group and let be an abelian group with an action of by automorphisms, i.e. let be a module. Define abelian groups
where the operation in
is given by
- The cochains are the elements of
- The coboundary map is
- The cocycles are the elements of
- The coboundaries are the elements of
- The cohomology group of with coefficients in is the abelian group
Let be a field and let be a finite subgroup of .
be a cocycle so that
By Dedekind's lemma we may choose such that
is nonzero. If then
and is a coboundary.
(Dedekind's Lemma.) Let be a subfield of .
- Distinct embeddings of into are linearly independent.
- Distinct characters are linearly independent.
- Distinct elements of are linearly independent in .
Let be a finite extension of . Let
be algebra homomorphisms. Let
be such that
So, by induction,
we may choose such that
and conclude that
Notes and References
Where are these from?