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 $G$ 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

:$H^p(G/N,H^q(N,A)) \Longrightarrow H^(G,A)$

and there is a spectral sequence of homological type

:$H_p(G/N,H_q(N,A)) \Longrightarrow H_(G,A)$,
where the arrow '$\Longrightarrow$' 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

:$\left ( \begin 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end \right ), \ a, b, c \in \mathbb Z.$

This group is a central extension

:$0 \to \mathbb \to G \to \mathbb \oplus \mathbb \to 0$

with center $\mathbb$ 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

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

Cohomology of wreath products

For a group ''G'', the wreath product is an extension

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

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.

Properties

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

:$0 \to H^1(G/N,A^N) \to H^1(G,A) \to H^1(N,A)^{G/N} \to H^2(G/N,A^N) \to H^2(G,A).$

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., Theorem 8^bis.12

References

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{{DEFAULTSORT:Lyndon-Hochschild-Serre spectral sequence
Spectral sequences
Group theory