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:
- H p(G/N, H q(N, A)) ⇒ H p+q(G, A)
and there is a spectral sequence of homological type
- H p(G/N, H q(N, A)) ⇒ H p+q(G, A).
The same statement holds if G is a profinite group, N is a closed normal subgroup and H* denotes the continuous cohomology.
Example: Cohomology 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 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
Example: 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.
- 0 → H 1(G/N, AN) → H 1(G, A) → H 1(N, A)G/N → H 2(G/N, AN) →H 2(G, A).
The spectral sequence is an instance of the more general Grothendieck spectral sequence of the composition of two derived functors. Indeed, H∗(G, -) is the derived functor of (−)G (i.e. taking G-invariants) and the composition of the functors (−)N and (−)G/N is exactly (−)G.
A similar spectral sequence exists for group homology, as opposed to group cohomology, as well.
- Kevin Knudson. Homology of Linear Groups. Birkhäuser. Example A.2.4
- Minoru Nakaoka (1960), "Decomposition Theorem for Homology Groups of Symmetric Groups", Annals of Mathematics, Second Series, 71 (1): 16–42, 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 Math., 87 (2): 145–151
- McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, 58 (2nd ed.), Cambridge University Press, doi:10.2277/0521567599, 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
- Hochschild, G.; Serre, Jean-Pierre (1953), "Cohomology of group extensions", Transactions of the American Mathematical Society, 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, 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, Zbl 0948.11001, MR1737196