(also nonabelian homological algebra)
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group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Hochschild-Serre spectral sequence is a spectral sequence that expresses group cohomology by a special case of the Grothendieck spectral sequence.
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
The Hochschild–Serre spectral sequence is the Grothendieck spectral sequence for the composition of these functors. Its $E_2$-page is
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.
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
Last revised on February 26, 2018 at 18:00:32. See the history of this page for a list of all contributions to it.