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 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 Example A.2.4

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., for a brief summary see section 2 of

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:§ 6.8.3

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., Theorem 8bis.12

A similar spectral sequence exists for Lie algebras. , Chapter 2 section 5.1

References

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