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| In mathematics, the '''inflation-restriction exact sequence''' is an [[exact sequence]] occurring in [[group cohomology]] and is a special case of the [[five-term exact sequence]] arising from the study of [[spectral sequences]].
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| Specifically, let ''G'' be a [[group (mathematics)|group]], ''N'' a [[normal subgroup]], and ''A'' an [[abelian group]] which is equipped with an action of ''G'', i.e., a [[homomorphism]] from ''G'' to the [[automorphism|automorphism group]] of ''A''. The quotient group ''G/N'' acts on ''A<sup>N</sup> = { a <math>\in</math> A : na = a '' for all '' n <math>\in</math> N}''. Then the inflation-restriction exact sequence is:
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| ::0 → ''H''<sup> 1</sup>(''G''/''N'', ''A''<sup>''N''</sup>) → ''H''<sup> 1</sup>(''G'', ''A'') → ''H''<sup> 1</sup>(''N'', ''A'')<sup>''G''/''N''</sup> → ''H''<sup> 2</sup>(''G''/''N'', ''A''<sup>''N''</sup>) →''H''<sup> 2</sup>(''G'', ''A'')
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| In this sequence, there are maps
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| * ''inflation'' ''H''<sup> 1</sup>(''G''/''N'', ''A''<sup>''N''</sup>) → ''H''<sup> 1</sup>(''G'', ''A'')
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| * ''restriction'' ''H''<sup> 1</sup>(''G'', ''A'') → ''H''<sup> 1</sup>(''N'', ''A'')<sup>''G''/''N''</sup>
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| * ''transgression'' ''H''<sup> 1</sup>(''N'', ''A'')<sup>''G''/''N''</sup> → ''H''<sup> 2</sup>(''G''/''N'', ''A''<sup>''N''</sup>)
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| * ''inflation'' ''H''<sup> 2</sup>(''G''/''N'', ''A''<sup>''N''</sup>) →''H''<sup> 2</sup>(''G'', ''A'')
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| The inflation and restriction are defined for general ''n'':
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| * ''inflation'' ''H''<sup>''n''</sup>(''G''/''N'', ''A''<sup>''N''</sup>) → ''H''<sup>''n''</sup>(''G'', ''A'')
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| * ''restriction'' ''H''<sup>''n''</sup>(''G'', ''A'') → ''H''<sup>''n''</sup>(''N'', ''A'')<sup>''G''/''N''</sup>
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| The transgression is defined for general ''n''
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| * ''transgression'' ''H''<sup>''n''</sup>(''N'', ''A'')<sup>''G''/''N''</sup> → ''H''<sup>''n''+1</sup>(''G''/''N'', ''A''<sup>''N''</sup>)
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| only if ''H''<sup>''i''</sup>(''N'', ''A'')<sup>''G''/''N''</sup> = 0 for ''i'' ≤ ''n''-1.<ref name=GS67>Gille & Szamuely (2006) p.67</ref>
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| The sequence for general ''n'' may be deduced from the case ''n''=1 by dimension-shifting or from the [[Lyndon–Hochschild–Serre spectral sequence]].<ref name=GS68>Gille & Szamuely (2006) p.68</ref>
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| ==References==
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| {{reflist}}
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| * {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }}
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| * {{cite book | page=282 | title=Handbook of Algebra, Volume 1 | first=Michiel | last=Hazewinkel | publisher=Elsevier | year=1995 | isbn=0444822127 }}
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| * {{cite book | first=Helmut | last=Koch | title=Algebraic Number Theory | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-63003-1 | zbl=0819.11044 | series=Encycl. Math. Sci. | volume=62 | edition=2nd printing of 1st }}
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| * {{cite book | pages=112–113 | title=Cohomology of Number Fields | volume=323 | series=Grundlehren der Mathematischen Wissenschaften | first1=Jürgen | last1=Neukirch | authorlink1=Jürgen Neukirch | first2=Alexander | last2=Schmidt | first3=Kay | last3=Wingberg | edition=2nd | publisher=[[Springer-Verlag]] | year=2008 | isbn=3-540-37888-X | zbl=1136.11001 }}
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| * {{cite book | page=214 | title=The Solution of The K(GV) Problem | volume=4 | series=Advanced Texts in Mathematics| first=Peter | last=Schmid | publisher=Imperial College Press | year=2007 | isbn=1860949703 }}
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| * {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Local fields | others=Translated from the French by Marvin Jay Greenberg | series=[[Graduate Texts in Mathematics]] | volume=67 | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 | pages=117–118 }}
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| [[Category:Homological algebra]]
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| {{algebra-stub}}
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