Inflation-restriction exact sequence explained
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 = .
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
- Book: Gille . Philippe . Szamuely . Tamás . Central simple algebras and Galois cohomology . Cambridge Studies in Advanced Mathematics . 101 . Cambridge . . 2006 . 0-521-86103-9 . 1137.12001 .
- Book: Hazewinkel, Michiel . 282 . Handbook of Algebra, Volume 1 . Elsevier . 1995 . 0444822127 .
- Book: Koch, Helmut . Algebraic Number Theory . . 1997 . 3-540-63003-1 . 0819.11044 . Encycl. Math. Sci. . 62 . 2nd printing of 1st .
- Book: 112–113 . Cohomology of Number Fields . 323 . Grundlehren der Mathematischen Wissenschaften . Jürgen . Neukirch . Jürgen Neukirch . Alexander . Schmidt . Kay . Wingberg . 2nd . . 2008 . 978-3-540-37888-4 . 1136.11001 .
- Book: Schmid, Peter . 214 . The Solution of The K(GV) Problem . 4 . Advanced Texts in Mathematics. Imperial College Press . 2007 . 978-1860949708 .
- Book: Serre, Jean-Pierre . Jean-Pierre Serre
. Jean-Pierre Serre . . Marvin Greenberg. Marvin Jay. Greenberg . . 67 . . 1979 . 0-387-90424-7 . 0423.12016 . 117–118 .
Notes and References
- Gille & Szamuely (2006) p.67
- Gille & Szamuely (2006) p. 68
