Inflation-restriction exact sequence

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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/NH 2(G/N, AN) →H 2(G, A)

In this sequence, there are maps

  • inflation H 1(G/N, AN) → H 1(G, A)
  • restriction H 1(G, A) → H 1(N, A)G/N
  • transgression H 1(N, A)G/NH 2(G/N, AN)
  • inflation H 2(G/N, AN) →H 2(G, A)

The inflation and restriction are defined for general n:

  • inflation Hn(G/N, AN) → Hn(G, A)
  • restriction Hn(G, A) → Hn(N, A)G/N

The transgression is defined for general n

  • transgression Hn(N, A)G/NHn+1(G/N, AN)

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
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