Lyndon–Hochschild–Serre spectral sequence
In mathematics, especially in the fields of group cohomology, homological algebra and number theory, the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup N and the quotient group G/''N to the cohomology of the total group G''. The spectral sequence is named after Roger Lyndon, Gerhard Hochschild, and Jean-Pierre Serre.
Statement
Let be a group and be a normal subgroup. The latter ensures that the quotient is a group, as well. Finally, let be a -module. Then there is a spectral sequence of cohomological typeand there is a spectral sequence of homological type
where the arrow means convergence of spectral sequences.
The same statement holds if is a profinite group, is a closed'' normal subgroup and denotes the continuous cohomology.
Examples
Homology of the Heisenberg group
The spectral sequence can be used to compute the homology of the Heisenberg group G with integral entries, i.e., matrices of the formThis group is a central extension
with center corresponding to the subgroup with. The spectral sequence for the group homology, together with the analysis of a differential in this spectral sequence, shows that
Cohomology of wreath products
For a group G, the wreath product is an extensionThe resulting spectral sequence of group cohomology with coefficients in a field k,
is known to degenerate at the -page.
Properties
The associated five-term exact sequence to the group cohomology is the usual inflation-restriction exact sequence:The associated five-term exact sequence to the group homology is the coinflation-corestriction exact sequence:
Generalizations
The spectral sequence is an instance of the more general Grothendieck spectral sequence of the composition of two derived functors. Indeed, is the derived functor of and the composition of the functors and is exactly.A similar spectral sequence exists for group homology, as opposed to group cohomology, as well.