Lyndon–Hochschild–Serre spectral sequence

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.

A similar spectral sequence exists for group homology, as opposed to group cohomology, as well.[1]

References

  1. ^ McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, 58 (2nd ed.), Cambridge University Press, doi:10.2277/0521567599, ISBN 978-0-521-56759-6, MR1793722 , Theorem 8bis.12