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Hochschild-Serre spectral sequence

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Homological algebra

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

Revised on November 22, 2013 04:40:58 by Urs Schreiber (82.169.114.243)