# Hochschild-Serre exact sequence

## Definition

This sequence is termed the **Hochschild-Serre exact sequence** or the **inflation-restriction exact sequence**.

### General case

Suppose is a group, is a normal subgroup of , and is an abelian group with a -action on it, i.e., a homomorphism . We thus have a short 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.