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

Let ${\displaystyle G}$ be a group and ${\displaystyle N}$ be a normal subgroup. The latter ensures that the quotient ${\displaystyle G/N}$ is a group, as well. Finally, let ${\displaystyle A}$ be a ${\displaystyle G}$-module. Then there is a spectral sequence of cohomological type

${\displaystyle H^{p}(G/N,H^{q}(N,A))\Longrightarrow H^{p+q}(G,A)}$

and there is a spectral sequence of homological type

${\displaystyle H_{p}(G/N,H_{q}(N,A))\Longrightarrow H_{p+q}(G,A)}$,

where the arrow '${\displaystyle \Longrightarrow }$' means convergence of spectral sequences.

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

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

This group is a central extension

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

with center ${\displaystyle \mathbb {Z} }$ corresponding to the subgroup with ${\displaystyle a=b=0}$. The spectral sequence for the group homology, together with the analysis of a differential in this spectral sequence, shows that^[1]

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

Cohomology of wreath products

For a group G, the wreath product is an extension

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

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

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

is known to degenerate at the ${\displaystyle E_{2}}$-page.^[2]

Properties

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

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

The associated five-term exact sequence to the group homology is the coinflation-corestriction exact sequence:^[3]

${\displaystyle H_{2}(G,A)\xrightarrow {\text{coinf}} H_{2}(G/N,A_{N})\xrightarrow {d} H_{1}(N,A)_{G/N}\xrightarrow {\text{cor}} H_{1}(G,A)\xrightarrow {\text{coinf}} H_{1}(G/N,A_{N})\to 0.}$

Generalizations

The spectral sequence is an instance of the more general Grothendieck spectral sequence of the composition of two derived functors. Indeed, ${\displaystyle H^{*}(G,-)}$ is the derived functor of ${\displaystyle (-)^{G}}$ (i.e., taking G-invariants) and the composition of the functors ${\displaystyle (-)^{N}}$ and ${\displaystyle (-)^{G/N}}$ is exactly ${\displaystyle (-)^{G}}$.

A similar spectral sequence exists for group homology, as opposed to group cohomology, as well.^[4]