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 precise statement is as follows:

Let "G" be a finite group, "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:

: $H^p(G/N,\; H^q(N,A))\; implies\; H^\{p+q\}(G,A).,$

The same statement holds if "G" is a profinite group and "N" is a "closed" normal subgroup.

The associated five-term exact sequence is the usual inflation-restriction exact sequence::0 → "H"¹("G"/"N", "A"^"N") → "H"¹("G", "A") → "H"¹("N", "A")^"G"/"N" → "H"²("G"/"N", "A"^"N") →"H"²("G", "A")

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".

References

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* | year=1953 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=74 | pages=110–134
* | year=2000 | volume=323