nLab Hochschild-Serre spectral sequence

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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 GG be a group, KGK\subset G a normal subgroup and AA a left GG-module. The group cohomology groups H n(G,A)H^n(G,A) form the derived functors of the invariants functor AA G={aA|ga=a,gG}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. A^G = (A^K)^{G/K} \,.

The Hochschild–Serre spectral sequence is the Grothendieck spectral sequence for the composition of these functors. Its E 2E_2-page is

E 2 p,q=H p(G/K,H q(K,A)) E^{p,q}_2 = H^p(G/K,H^q(K,A))

and it is converging to the group cohomology E n=H n(G,A)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

Last revised on February 26, 2018 at 18:00:32. See the history of this page for a list of all contributions to it.