Lyndon–Hochschild–Serre spectral sequence

Statement

The precise statement is as follows:

Let G be a group and 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 of cohomological type

H ^p(G/N, H ^q(N, A)) ⇒ H ^p+q(G, A)

and there is a spectral sequence of homological type

H _p(G/N, H _q(N, A)) ⇒ H _p+q(G, A).

The same statement holds if G is a profinite group, N is a closed normal subgroup and H* denotes the continuous cohomology.

Example: Cohomology of the Heisenberg group

The spectral sequence can be used to compute the homology of the Heisenberg group G with integral entries, i.e., matrices of the form

( 1 a b 0 1 c 0 0 1 ) ,   a , b , c ∈ Z .

This group is an extension

0 → Z → H → Z ⊕ Z → 0

corresponding to the subgroup with a=c=0. The spectral sequence for the group homology, together with the analysis of a differential in this spectral sequence, shows that

H i ( G , Z ) = { Z i = 0 , 3 Z ⊕ Z i = 1 , 2 0 i > 3.

Example: Cohomology of wreath products

For a group G, the wreath product is an extension

1 → G p → G ≀ Z / p → Z / p → 1.

The resulting spectral sequence of group cohomology with coefficients in a field k,

H r ( Z / p , H s ( G p , k ) ) ⇒ H r + s ( G ≀ Z / p , k ) ,

is known to degenerate at the E 2 -page.

Properties

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).

Generalizations

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.