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.
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.
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
The spectral sequence for the group homology, together with the analysis of a differential in this spectral sequence, shows that[1] 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
[2] The associated five-term exact sequence is the usual inflation-restriction exact sequence: The spectral sequence is an instance of the more general Grothendieck spectral sequence of the composition of two derived functors.
(i.e., taking G-invariants) and the composition of the functors
A similar spectral sequence exists for group homology, as opposed to group cohomology, as well.