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 the study of spectral sequences.

Specifically, let G be a group, N a normal subgroup, and A an abelian group with equipped with an action of G, i.e., a homomorphism from G to the automorphism group of A. Then the quotient group G/N acts on A^N = { a $\in$ A : ga = a for all g $\in$ G}. Then the inflation-restriction exact sequence is:

0 → H ¹(G/N, A^N) → H ¹(G, A) → H ¹(N, A)^G/N → H ²(G/N, A^N) →H ²(G, A)