Lyndon–Hochschild–Serre spectral sequence
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In mathematics, especially in the fields of group cohomology, homological algebra and number theory the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup N and the quotient group G/N to the cohomology of the total group G.
The precise statement is as follows:
Let G be a finite group, N be a normal subgroup. The latter ensures that the quotient G/N is a group, as well. Finally, let A be a G-module. Then there is a spectral sequence:
- H p(G/N, H q(N, A)) ⇒ H p+q(G, A).
The same statement holds if G is a profinite group and N is a closed normal subgroup.
The associated five-term exact sequence is the usual inflation-restriction exact sequence:
- 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)
The spectral sequence is an instance of the more general Grothendieck spectral sequence of the composition of two derived functors. Indeed, H∗(G, -) is the derived functor of (−)G (i.e. taking G-invariants) and the composition of the functors (−)N and (−)G/N is exactly (−)G.
[edit] References
- Lyndon, Roger B. (1948), "The cohomology theory of group extensions", Duke Mathematical Journal 15 (1): 271–292, doi:, ISSN 0012-7094
- Hochschild, G.; Serre, Jean-Pierre (1953), "Cohomology of group extensions", Transactions of the American Mathematical Society 74: 110–134, doi:, MR0052438, ISSN 0002-9947
- Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Berlin: Springer-Verlag, MR1737196, ISBN 978-3-540-66671-4
