Inflation-restriction exact sequence

In mathematics, the inflation-restriction exact sequence is an exact sequence occurring in group cohomology and is a special case of the five-term exact sequence arising from the study of spectral sequences.

Specifically, let G be a group, N a normal subgroup, and A an abelian group which is equipped with an action of G, i.e., a homomorphism from G to the automorphism group of A. The quotient group G/N acts on
 * AN = { a ∈ A : na = a for all n ∈ N}.

Then the inflation-restriction exact sequence is:


 * 0 → H1(G/N, AN) → H1(G, A) → H1(N, A)G/N → H2(G/N, AN) →H2(G, A)

In this sequence, there are maps
 * inflation H1(G/N, AN) → H1(G, A)
 * restriction H1(G, A) → H1(N, A)G/N
 * transgression H1(N, A)G/N → H2(G/N, AN)
 * inflation H2(G/N, AN) →H2(G, A)

The inflation and restriction are defined for general n:
 * inflation Hn(G/N, AN) → Hn(G, A)
 * restriction Hn(G, A) → Hn(N, A)G/N

The transgression is defined for general n only if Hi(N, A)G/N = 0 for i ≤ n &minus; 1.
 * transgression Hn(N, A)G/N → Hn+1(G/N, AN)

The sequence for general n may be deduced from the case n = 1 by dimension-shifting or from the Lyndon–Hochschild–Serre spectral sequence.