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


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

and there is a spectral sequence of homological type


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

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.

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


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

This group is a central extension


 * $$0 \to \Z \to G \to \Z \oplus \Z \to 0$$

with center $$\Z$$ 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


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

Cohomology of wreath products
For a group G, the wreath product is an extension


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

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


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

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.