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 mapsinflation H'' 1 → H 1restriction ''H 1 → H'' 1G/''Ntransgression H'' 1G/''N → H'' 2inflation ''H 2 →H'' 2
The inflation and restriction are defined for general n:inflation ''Hn'' → Hnrestriction ''Hn'' → HnG/''N
The transgression is defined for general n'' transgression ''Hn''G/''N → H''n+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.