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 A^N = { a $\in$ A : na = a for all n $\in$ 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

• inflation H ¹(G/N, A^N) → 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, A^N)
• inflation H ²(G/N, A^N) →H ²(G, A)

References

• Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. Vol. 62 (2nd printing of 1st ed.). Springer-Verlag. ISBN 3-540-63003-1. Zbl 0819.11044.
• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.). Springer-Verlag. pp. 112–113. ISBN 3-540-37888-X. Zbl 1136.11001.