 Inflation-restriction exact sequence

For a normal subgroup $N$ of $G$ and a $G$-module $M$, the low-degree exact sequence begins $0\to H^1(G/N,M^N)\to H^1(G,M)\to H^1(N,M{)}^{G/N}\to H^2(G/N,M^N)$. The first map inflates cocycles along $G\to G/N$; the second restricts them. If the restriction is trivial, subtract a coboundary to make the cocycle vanish on $N$ and descend to the quotient. Thus for a finite Galois extension, the kernel of restriction on Galois cohomology with finite coefficients is finite.

The inflation-restriction exact sequence is an important concept in homological algebra and algebraic topology, particularly in the study of groups and cohomology theories. It relates the cohomology groups of different spaces or algebraic structures through the use of restriction and inflation maps.