Hochschild-Serre exact sequence

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

General case
Suppose $$G$$ is a group, $$H$$ is a normal subgroup of $$G$$, and $$A$$ is an abelian group with a $$G$$-action on it, i.e., a homomorphism $$\varphi:G \to \operatorname{Aut}(A)$$. We thus have a short exact sequence:

$$1 \to H \to G \to G/H \to 1$$

where $$1$$ 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:

$$1 \to H^1(G/N;A^N) \stackrel{\operatorname{inf}}{\to} H^1(G;A) \stackrel{\operatorname{res}}{\to} H^1(N;A)^{G/N} \stackrel{d_i}{\to} H^2(G/N;A^N) \stackrel{\operatorname{inf}}{\to} H^2(G;A) \to \dots $$

Here, $$A^N$$ denotes the subgroup of $$A$$ comprising those elements fixed pointwise by every element of $$N$$. Further:


 * The homomorphisms $$H^i(G/N;A^N) \stackrel{\operatorname{inf}}{\to} H^i(G;A)$$ are inflation homomorphisms corresponding to the quotient map $$G \to G/N$$.
 * The homomorphisms $$H^i(G;A) \to H^i(N;A)^{G/N}$$ are restriction homomorphisms to $$H^i(N;A)$$ coupled with the observation that the image is invariant under the natural $$G/N$$-action induced from the action by conjugation on $$N$$. )
 * The homomorphisms $$H^i(N;A)^{G/N} \to H^{i+1}(G/N;A^N)$$ are termed transgression homomorphisms.

Case of central subgroup
A special case of interest is where $$H$$ is a central subgroup of $$G$$. In this case, the $$G/N$$-action on $$H^i(N;A)$$ is trivial, and so $$H^i(N;A)^{G/N}$$ simply becomes $$H^i(N;A)$$.

First five terms
The first five terms are important in that they form a five-term exact sequence of significance.

$$1 \to H^1(G/N;A^N) \stackrel{\operatorname{inf}}{\to} H^1(G;A) \stackrel{\operatorname{res}}{\to} H^1(N;A)^{G/N} \stackrel{d_i}{\to} H^2(G/N;A^N) \stackrel{inf}{\to} H^2(G;A)$$