Hochschild-Serre spectral sequence
Special and general types
Cohomology and Extensions
The Hochschild-Serre spectral sequence is a spectral sequence that expresses group cohomology by a special case of the Grothendieck spectral sequence.
Let be a group, a normal subgroup and a left -module. The group cohomology groups form the derived functors of the invariants functor .
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 -page is
and it is converging to the group cohomology .
There is a similar spectral sequence for group homology obtained as a Grothendieck spectral sequence for the two-stage computation of coinvariants.