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.

The sequence for general n may be deduced from the case n = 1 by dimension-shifting or from the Lyndon–Hochschild–Serre spectral sequence.

Notes

References

• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
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• 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 978-3-540-37888-4. Zbl 1136.11001.
• Schmid, Peter (2007). The Solution of The K(GV) Problem. Advanced Texts in Mathematics. Vol. 4. Imperial College Press. p. 214. ISBN 978-1860949708.
• Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics. Vol. 67. Translated by Greenberg, Marvin Jay. Springer-Verlag. pp. 117–118. ISBN 0-387-90424-7. Zbl 0423.12016.

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