Lyndon–Hochschild–Serre spectral sequence

Short description: Topic in mathematics

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 $G$ 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

$H_p(G/N,H_q(N,A))⟹H_{p+q}(G,A)$,

where the arrow '$⟹$' means convergence of spectral sequences.

The same statement holds if $G$ is a profinite group, $N$ is a closed normal subgroup and $H^{*}$ 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

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

This group is a central extension

$0\to \mathbb{\mathbb{Z}}\to G\to \mathbb{\mathbb{Z}}\oplus \mathbb{\mathbb{Z}}\to 0$

with center $\mathbb{\mathbb{Z}}$ corresponding to the subgroup with $a=b=0$. The spectral sequence for the group homology, together with the analysis of a differential in this spectral sequence, shows that

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

Cohomology of wreath products

For a group G, the wreath product is an extension

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

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

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

is known to degenerate at the $E_2$-page.

Properties

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

$0\to H^1(G/N,A^N)\stackrel{\text{inf}}{\to }{H}^1(G,A)\stackrel{\text{res}}{\to }{H}^1(N,A{)}^{G/N}\stackrel{d}{\to }{H}^2(G/N,A^N)\stackrel{\text{inf}}{\to }{H}^2(G,A).$

The associated five-term exact sequence to the group homology is the coinflation-corestriction exact sequence:

$H_2(G,A)\stackrel{\text{coinf}}{\to }{H}_2(G/N,A_N)\stackrel{d}{\to }{H}_1(N,A{)}_{G/N}\stackrel{\text{cor}}{\to }{H}_1(G,A)\stackrel{\text{coinf}}{\to }{H}_1(G/N,A_N)\to 0.$

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.

A similar spectral sequence exists for Lie algebras.

References

• 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 
• 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