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
- 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/N → H 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/N → Hn+1(G/N, AN)
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
- 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.
- Hazewinkel, Michiel (1995). Handbook of Algebra, Volume 1. Elsevier. p. 282. ISBN 0444822127. https://archive.org/details/handbookofalgebr0003unse/page/282.
- Koch, Helmut (1997). Algebraic Number Theory. Encycl. Math. Sci.. 62 (2nd printing of 1st ed.). Springer-Verlag. ISBN 3-540-63003-1.
- 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.
- 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. Springer-Verlag. pp. 117–118. ISBN 0-387-90424-7.
 |  |
