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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

The precise statement is as follows:

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

Hp(G/N,Hq(N,A))→Hp+q(G,A)displaystyle H^p(G/N,H^q(N,A))to H^p+q(G,A)

and there is a spectral sequence of homological type

Hp(G/N,Hq(N,A))→Hp+q(G,A)displaystyle H_p(G/N,H_q(N,A))to H_p+q(G,A).

The same statement holds if G is a profinite group, N is a closed normal subgroup and H* denotes the continuous cohomology.

Example: Cohomology 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

(1ab01c001), a,b,c∈Z.displaystyle left(beginarrayccc1&a&b\0&1&c\0&0&1endarrayright), a,b,cin mathbb Z .

This group is a central extension

0→Z→G→Z⊕Z→0displaystyle 0to mathbb Z to Gto mathbb Z oplus mathbb Z to 0

with center Zdisplaystyle mathbb Z 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^[1]

Hi(G,Z)={Zi=0,3Z⊕Zi=1,20i>3.{displaystyle H_i(G,mathbb Z )=leftbeginarrayccmathbb Z &i=0,3\mathbb Z oplus mathbb Z &i=1,2\0&i>3.endarrayright.

Example: Cohomology of wreath products

For a group G, the wreath product is an extension

1→Gp→G≀Z/p→Z/p→1.displaystyle 1to G^pto Gwr mathbb Z /pto mathbb Z /pto 1.

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

Hr(Z/p,Hs(Gp,k))⇒Hr+s(G≀Z/p,k),displaystyle H^r(mathbb Z /p,H^s(G^p,k))Rightarrow H^r+s(Gwr mathbb Z /p,k),

is known to degenerate at the E2displaystyle E_2-page.^[2]

Properties

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

0→H1(G/N,AN)→H1(G,A)→H1(N,A)G/N→H2(G/N,AN)→H2(G,A).displaystyle 0to H^1(G/N,A^N)to H^1(G,A)to H^1(N,A)^G/Nto 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

• 
  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, 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001