nLab Hochschild-Serre spectral sequence

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

group theory

Contents

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.

References

Revised on September 8, 2016 08:15:05 by Anonymous Coward (134.58.253.57)