# nLab Hochschild-Serre spectral sequence

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

cohomology

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 ${H}^{n}\left(G,A\right)$ form the derived functors of the invariants functor $A↦{A}^{G}=\left\{a\in A\mid ga=a,g\in G\right\}$.

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

${A}^{G}=\left({A}^{K}{\right)}^{G/K}\phantom{\rule{thinmathspace}{0ex}}.$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

${E}_{2}^{p,q}={H}^{p}\left(G/H,{H}^{q}\left(H,A\right)\right)$E^{p,q}_2 = H^p(G/H,H^q(H,A))

and it is converging to the group cohomology ${E}_{\infty }^{n}={H}^{n}\left(G,A\right)$.

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

Revised on October 29, 2012 20:02:18 by Urs Schreiber (131.174.188.167)