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 
"A""N" = { "a" ∈ "A" : "na" = "a" for all "n" ∈ "N"}.
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")
In this sequence, there are maps
The inflation and restriction are defined for general "n":
The transgression is defined for general "n" 
only if "H""i"("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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