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.

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 {Z} .

This group is a central extension

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

with center $\mathbb {Z}$ 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[1]

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.[2]

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.[3]

References

Knudson, Kevin (2001). Homology of Linear Groups. Progress in Mathematics. Vol. 193. Basel: Birkhäuser Verlag. doi:10.1007/978-3-0348-8338-2. ISBN 3-7643-6415-7. MR 1807154. Example A.2.4
Nakaoka, Minoru (1960), "Decomposition Theorem for Homology Groups of Symmetric Groups", Annals of Mathematics, Second Series, 71 (1): 16–42, doi:10.2307/1969878, JSTOR 1969878, for a brief summary see section 2 of Carlson, Jon F.; Henn, Hans-Werner (1995), "Depth and the cohomology of wreath products", Manuscripta Mathematica, 87 (2): 145–151, CiteSeerX 10.1.1.540.1310, doi:10.1007/BF02570466, S2CID 27212941
McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, vol. 58 (2nd ed.), Cambridge University Press, ISBN 978-0-521-56759-6, MR 1793722, Theorem 8^bis.12

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, JSTOR 1990851, MR 0052438
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001

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[1] Knudson, Kevin (2001). Homology of Linear Groups. Progress in Mathematics. Vol. 193. Basel: Birkhäuser Verlag. doi:10.1007/978-3-0348-8338-2. ISBN 3-7643-6415-7. MR 1807154. Example A.2.4

[2] Nakaoka, Minoru (1960), "Decomposition Theorem for Homology Groups of Symmetric Groups", Annals of Mathematics, Second Series, 71 (1): 16–42, doi:10.2307/1969878, JSTOR 1969878, for a brief summary see section 2 of Carlson, Jon F.; Henn, Hans-Werner (1995), "Depth and the cohomology of wreath products", Manuscripta Mathematica, 87 (2): 145–151, CiteSeerX 10.1.1.540.1310, doi:10.1007/BF02570466, S2CID 27212941

[3] McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, vol. 58 (2nd ed.), Cambridge University Press, ISBN 978-0-521-56759-6, MR 1793722, Theorem 8^bis.12