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 ∈ 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

• 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)

The inflation and restriction are defined for general n:

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

The transgression is defined for general n

• transgression Hⁿ(N, A)^G/N → Hⁿ⁺¹(G/N, A^N)

only if Hⁱ(N, A)^G/N = 0 for i ≤ n − 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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