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. In fact, the associated five term exact sequence is the usual inflation-restriction exact sequence.

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 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

• Lyndon, Roger B. (1948), The cohomology theory of group extensions, vol. 15, pp. 271–292, ISSN 0012-7094
• Hochschild, G.; Serre, Jean-Pierre (1953), "Cohomology of group extensions", Transactions of the American Mathematical Society, 74: 110–134, ISSN 0002-9947, MR0052438 {{citation}}: Check |author2-link= value (help)
• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin, New York, ISBN 978-3-540-66671-4, MR1737196{{citation}}: CS1 maint: location missing publisher (link)

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