In mathematics,–Hochschild– 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 type
and 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 form
This 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 extension
The 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 is the usual inflation-restriction 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 (i.e., taking G-invariants) and the composition of the functors and is exactly .
A similar spectral sequence exists for group homology, as opposed to group cohomology, as well.
References
(paywalled)