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 precise statement is as follows:
Let G be a group and N be a normal subgroup. The latter ensures that the quotient G/N is a group, as well. Finally, let A be a G-module. Then there is a spectral sequence of cohomological type
and there is a spectral sequence of homological type
The same statement holds if G is a profinite group, N is a closed normal subgroup and H* 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 an extension
corresponding to the subgroup with a=c=0. The spectral sequence for the group homology, together with the analysis of a differential in this spectral sequence, shows that
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.
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