Hochschild-Serre exact sequence

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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. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE])
  • 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)