Lyndon–Hochschild–Serre spectral sequence

Short description: A spectral sequence relating the group cohomology of a normal subgroup N and the quotient group G/N

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

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

and there is a spectral sequence of homological type

[math]\displaystyle{ H_p(G/N,H_q(N,A)) \Longrightarrow H_{p+q}(G,A) }[/math],

where the arrow '[math]\displaystyle{ \Longrightarrow }[/math]' means convergence of spectral sequences.

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

[math]\displaystyle{ \left ( \begin{array}{ccc} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array} \right ), \ a, b, c \in \Z. }[/math]

This group is a central extension

[math]\displaystyle{ 0 \to \Z \to G \to \Z \oplus \Z \to 0 }[/math]

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

[math]\displaystyle{ H_i (G, \Z) = \left \{ \begin{array}{cc} \Z & i=0, 3 \\ \Z \oplus \Z & i=1,2 \\ 0 & i\gt 3. \end{array} \right. }[/math]

Cohomology of wreath products

For a group G, the wreath product is an extension

[math]\displaystyle{ 1 \to G^p \to G \wr \Z / p \to \Z / p \to 1. }[/math]

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

[math]\displaystyle{ H^r(\Z/p, H^s(G^p, k)) \Rightarrow H^{r+s}(G \wr \Z/p, k), }[/math]

is known to degenerate at the [math]\displaystyle{ E_2 }[/math]-page.

Properties

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

[math]\displaystyle{ 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). }[/math]

Generalizations

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

A similar spectral sequence exists for group homology, as opposed to group cohomology, as well.

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