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 → H 1(G/N, AN) → H 1(G, A) → H 1(N, A)G/N → H 2(G/N, AN) →H 2(G, A)
In this sequence, there are maps
inflation H 1(G/N, AN) → H 1(G, A) restriction H 1(G, A) → H 1(N, A)G/N transgression H 1(N, A)G/N → H 2(G/N, AN) inflation H 2(G/N, AN) →H 2(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
transgression Hn(N, A)G/N → Hn+1(G/N, AN) only if Hi(N, A)G/N = 0 for i ≤ n − 1. The sequence for general n may be deduced from the case n = 1 by dimension-shifting or from the Lyndon–Hochschild–Serre spectral sequence.
Notes
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