Group cohomology
Gruppenkohomologie is a technical tool of mathematics, which was originally the study of groups, but later found applications also in particular in the topology and number theory.
- 2.1 Inhomogeneous cochains
Definition as a derived functor
Definition
It is a finite group. The functor from the category of -modules to the category of abelian groups, which assigns a module invariant under the subgroup elements is left exact. Its nth right derivative is the n-th with cohomology group of coefficients in a module.
Relationship with Ext
The Gruppenkohomologie can also be defined using the functor Ext:
While the group ring of and with the trivial operation is provided.
Definition on cochains
From the description using the functor Ext is seen that the Gruppenkohomologie can be calculated using a once chosen projective resolution of the trivial module. It can be specified explicitly as:
Is
That is, Index is omitted.
The Gruppenkohomologie is then the cohomology of the complex with
And
The elements of this complex are called homogeneous cochains.
Inhomogeneous cochains
The condition of invariance of the cochains allows to reduce the number of copies of one thing: the Gruppenkohomologie can also be defined on the complex of inhomogeneous cochains:
And
For example, is
The non-homogeneous 1- cocycle
Hot crossed homomorphisms.