nLab Hochschild-Serre spectral sequence

Idea

The Hochschild-Serre spectral sequence is a spectral sequence that expresses group cohomology by a special case of the Grothendieck spectral sequence.

Statement

Let $G$ be a group, $K\subset G$ a normal subgroup and $A$ a left $G$-module. The group cohomology groups $H^n(G,A)$ form the derived functors of the invariants functor $A\mapsto A^G = \{ a\in A | g a = a, g\in G\}$.

The invariants can be computed in two stages, hence as the composite of two functors as

$$A^G = (A^K)^{G/K} \,.$$

The Hochschild–Serre spectral sequence is the Grothendieck spectral sequence for the composition of these functors. Its $E_2$-page is

$$E^{p,q}_2 = H^p(G/K,H^q(K,A))$$

and it is converging to the group cohomology $E^n_\infty = H^n(G,A)$.

There is a similar spectral sequence for group homology obtained as a Grothendieck spectral sequence for the two-stage computation of coinvariants.

Related concepts

• Lie algebra cohomology

References

• Gerhard Hochschild, Jean-Pierre Serre, Cohomology of Lie algebras, Annals of Mathematics, Second Series, Vol. 57, No. 3 (May, 1953), pp. 591-603 (JSTOR)

• James Milne, section 14 of Lectures on Étale Cohomology