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The same statement holds if ''G'' is a ] and ''N'' is a ''closed'' normal subgroup. The same statement holds if ''G'' is a ] and ''N'' is a ''closed'' normal subgroup.


The spectral sequence is an instance of the more general ] of the composition of two derived functors. Indeed, ''H''<sup>&lowast;</sup>(''G'', -) is the ] of (&minus;)<sup>''G''</sup> (i.e. taking ''G''-invariants) and the composition of the functors (&minus;)<sup>''N''</sup> and (&minus;)<sup>''G/N''</sup> is exactly (&minus;)<sup>''G''</sup>. The spectral sequence is an instance of the more general ] of the composition of two derived functors. Indeed, ''H''<sup>∗</sup>(''G'', -) is the ] of (&minus;)<sup>''G''</sup> (i.e. taking ''G''-invariants) and the composition of the functors (&minus;)<sup>''N''</sup> and (&minus;)<sup>''G/N''</sup> is exactly (&minus;)<sup>''G''</sup>.


==Reference== ==References==
* {{Citation | last1=Lyndon | first1=Roger B. | title=The cohomology theory of group extensions | year=1948 | journal=] | issn=0012-7094 | volume=15 | issue=1 | pages=271–292}} * {{Citation | last1=Lyndon | first1=Roger B. | title=The cohomology theory of group extensions | year=1948 | journal= | issn=0012-7094 | volume=15 | issue=1 | pages=271–292}}
* {{Citation | last1=Hochschild | first1=G. | last2=Serre | first2=Jean-Pierre | author2-link=en:Jean-Pierre Serre | title=Cohomology of group extensions | id={{MathSciNet | id = 0052438}} | year=1953 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=74 | pages=110–134}} * {{Citation | last1=Hochschild | first1=G. | last2=Serre | first2=Jean-Pierre | author2-link=en:Jean-Pierre Serre | title=Cohomology of group extensions | id={{MathSciNet | id = 0052438}} | year=1953 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=74 | pages=110–134}}
* {{Citation | last1=Neukirch | first1=Jürgen | last2=Schmidt | first2=Alexander | last3=Wingberg | first3=Kay | title=Cohomology of Number Fields | publisher=] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-3-540-66671-4 | id={{MathSciNet | id = 1737196}} | year=2000 | volume=323}} * {{Citation | last1=Neukirch | first1=Jürgen | last2=Schmidt | first2=Alexander | last3=Wingberg | first3=Kay | title=Cohomology of Number Fields | publisher= | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-3-540-66671-4 | id={{MathSciNet | id = 1737196}} | year=2000 | volume=323}}


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Revision as of 16:47, 5 December 2007

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 ) ) ⟹ H p + q ( G , A ) . {\displaystyle 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 (−) (i.e. taking G-invariants) and the composition of the functors (−) and (−) is exactly (−).

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)

Springer-Verlag Duke Mathematical Journal

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