Hochschild-Serre exact sequence
This sequence is termed the Hochschild-Serre exact sequence or the inflation-restriction exact sequence.
where stands for the trivial group.
The inflation-restriction exact sequence or Hochschild-Serre exact sequence is a long exact sequence of cohomology groups given as follows:
Here, denotes the subgroup of comprising those elements fixed pointwise by every element of . Further:
- The homomorphisms are inflation homomorphisms corresponding to the quotient map .
- The homomorphisms are restriction homomorphisms to coupled with the observation that the image is invariant under the natural -action induced from the action by conjugation on . PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE])
- The homomorphisms are termed transgression homomorphisms.
Case of central subgroup
A special case of interest is where is a central subgroup of . In this case, the -action on is trivial, and so simply becomes .
First five terms
The first five terms are important in that they form a five-term exact sequence of significance.