Inflation-restriction exact sequence

Last updated December 19, 2019

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 →H¹(G/N, A^N) →H¹(G, A) →H¹(N, A)^G/N→H²(G/N, A^N) →H²(G, A)

In this sequence, there are maps

inflationH¹(G/N, A^N) → H¹(G, A)
restrictionH¹(G, A) → H¹(N, A)^G/N
transgressionH¹(N, A)^G/N → H²(G/N, A^N)
inflationH²(G/N, A^N) →H²(G, A)

The inflation and restriction are defined for general n:

inflationHⁿ(G/N, A^N) → Hⁿ(G, A)
restrictionHⁿ(G, A) → Hⁿ(N, A)^G/N

The transgression is defined for general n

transgressionHⁿ(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]

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References

↑ Gille & Szamuely (2006) p.67
↑ Gille & Szamuely (2006) p. 68

Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
Hazewinkel, Michiel (1995). Handbook of Algebra, Volume 1. Elsevier. p. 282. ISBN 0444822127.
Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci. 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. 323 (2nd ed.). Springer-Verlag. pp. 112–113. ISBN 3-540-37888-X. Zbl 1136.11001.
Schmid, Peter (2007). The Solution of The K(GV) Problem. Advanced Texts in Mathematics. 4. Imperial College Press. p. 214. ISBN 1860949703.
Serre, Jean-Pierre (1979). Local Fields . Graduate Texts in Mathematics. 67. Translated by Greenberg, Marvin Jay. Springer-Verlag. pp. 117–118. ISBN 0-387-90424-7. Zbl 0423.12016.

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[GS67-1] Gille & Szamuely (2006) p.67

[GS68-2] Gille & Szamuely (2006) p. 68