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
Then the inflation-restriction exact sequence is:
In this sequence, there are maps
The inflation and restriction are defined for general n:
- inflation Hn → Hn
- restriction Hn → HnG/N
The transgression is defined for general n
- transgression HnG/N → Hn+1
only if HiG/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.
