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 AN = { a A : na = a for all n N}. Then the inflation-restriction exact sequence is:

0 H 1(G/N, AN) H 1(G, A) H 1(N, A)G/N H 2(G/N, AN) 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 Hi(N, A)G/N = 0 for in  1.[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.[2]

References

  1. Gille & Szamuely (2006) p.67
  2. Gille & Szamuely (2006) p. 68


This article is issued from Wikipedia - version of the 2/20/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.