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, A^N) → H ¹(G, A) → H ¹(N, A)^G/N → H ²(G/N, A^N) →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.

References

• Lyndon, Roger C. (1948), "The cohomology theory of group extensions", Duke Mathematical Journal 15 (1): 271–292, doi:10.1215/S0012-7094-48-01528-2, ISSN 0012-7094 
• Hochschild, G.; Serre, Jean-Pierre (1953), "Cohomology of group extensions", Transactions of the American Mathematical Society (American Mathematical Society) 74 (1): 110–134, doi:10.2307/1990851, ISSN 0002-9947, JSTOR 1990851, MR0052438 
• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR1737196