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)) \implies 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 → H1(G/N, AN) → H1(G, A) → H1(N, A)G/N → H2(G/N, AN) →H2(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.

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