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.

Statement

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

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

$\left ( \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array} \right ), \ a, b, c \in \mathbb Z.$

This group is an extension

$0 \to \mathbb Z \to H \to \mathbb Z \oplus \mathbb Z \to 0$

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^[1]

$H_i (G, \mathbb Z) = \left \{ \begin{array}{cc} \mathbb Z & i=0, 3 \\ \mathbb Z \oplus \mathbb Z & i=1,2 \\ 0 & i>3. \end{array} \right.$

Example: Cohomology of wreath products

For a group G, the wreath product is an extension

$1 \to G^p \to G \wr \mathbb Z / p \to \mathbb Z / p \to 1.$

The resulting spectral sequence of group cohomology with coefficients in a field k,

$H^r(\mathbb Z/p, H^s(G^p, k)) \Rightarrow H^{r+s}(G \wr \mathbb Z/p, k),$

is known to degenerate at the $E_2$-page.^[2]

Properties

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

0 → H ¹(G/N, A^N) → H ¹(G, A) → H ¹(N, A)^G/N → H ²(G/N, A^N) →H ²(G, A).

Generalizations

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.^[3]

References

1. Kevin Knudson. Homology of Linear Groups. Birkhäuser. Example A.2.4
2. Minoru Nakaoka (1960), "Decomposition Theorem for Homology Groups of Symmetric Groups", Annals of Mathematics, Second Series 71 (1): 16–42, 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
3. 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 8^bis.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, MR 1737196