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
editLet 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
editHomology of the Heisenberg group
editThe 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
editFor 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
editThe associated five-term exact sequence is the usual inflation-restriction exact sequence:
Generalizations
editThe 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
edit- ^ 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
- ^ 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
- ^ 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
- Lyndon, Roger C. (1948), "The cohomology theory of group extensions", Duke Mathematical Journal, 15 (1): 271–292, doi:10.1215/S0012-7094-48-01528-2, ISSN 0012-7094 (paywalled)
- Hochschild, Gerhard; Serre, Jean-Pierre (1953), "Cohomology of group extensions", Transactions of the American Mathematical Society, 74 (1): 110–134, doi:10.2307/1990851, ISSN 0002-9947, JSTOR 1990851, MR 0052438
- Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001