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

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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

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Homology of the Heisenberg group

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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[1]

 

Cohomology of wreath products

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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.[2]

Properties

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The associated five-term exact sequence is the usual inflation-restriction exact sequence:

 

Generalizations

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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.[3]

References

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  1. ^ Knudson, Kevin (2001). Homology of Linear Groups. Progress in Mathematics. Vol. 193. Basel: Birkhäuser Verlag. doi:10.1007/978-3-0348-8338-2. ISBN 3-7643-6415-7. MR 1807154. Example A.2.4
  2. ^ Nakaoka, Minoru (1960), "Decomposition Theorem for Homology Groups of Symmetric Groups", Annals of Mathematics, Second Series, 71 (1): 16–42, doi:10.2307/1969878, JSTOR 1969878, for a brief summary see section 2 of Carlson, Jon F.; Henn, Hans-Werner (1995), "Depth and the cohomology of wreath products", Manuscripta Mathematica, 87 (2): 145–151, CiteSeerX 10.1.1.540.1310, doi:10.1007/BF02570466, S2CID 27212941
  3. ^ McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, vol. 58 (2nd ed.), Cambridge University Press, ISBN 978-0-521-56759-6, MR 1793722, Theorem 8bis.12