Conductivity near the percolation threshold: Difference between revisions
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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]]. | |||
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: | |||
::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'') | |||
: | |||
In this sequence, there are maps | |||
* ''inflation'' ''H''<sup> 1</sup>(''G''/''N'', ''A''<sup>''N''</sup>) → ''H''<sup> 1</sup>(''G'', ''A'') | |||
* ''restriction'' ''H''<sup> 1</sup>(''G'', ''A'') → ''H''<sup> 1</sup>(''N'', ''A'')<sup>''G''/''N''</sup> | |||
* ''transgression'' ''H''<sup> 1</sup>(''N'', ''A'')<sup>''G''/''N''</sup> → ''H''<sup> 2</sup>(''G''/''N'', ''A''<sup>''N''</sup>) | |||
* ''inflation'' ''H''<sup> 2</sup>(''G''/''N'', ''A''<sup>''N''</sup>) →''H''<sup> 2</sup>(''G'', ''A'') | |||
The inflation and restriction are defined for general ''n'': | |||
* ''inflation'' ''H''<sup>''n''</sup>(''G''/''N'', ''A''<sup>''N''</sup>) → ''H''<sup>''n''</sup>(''G'', ''A'') | |||
* ''restriction'' ''H''<sup>''n''</sup>(''G'', ''A'') → ''H''<sup>''n''</sup>(''N'', ''A'')<sup>''G''/''N''</sup> | |||
The transgression is defined for general ''n'' | |||
* ''transgression'' ''H''<sup>''n''</sup>(''N'', ''A'')<sup>''G''/''N''</sup> → ''H''<sup>''n''+1</sup>(''G''/''N'', ''A''<sup>''N''</sup>) | |||
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> | |||
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> | |||
==References== | |||
{{reflist}} | |||
* {{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 }} | |||
* {{cite book | page=282 | title=Handbook of Algebra, Volume 1 | first=Michiel | last=Hazewinkel | publisher=Elsevier | year=1995 | isbn=0444822127 }} | |||
* {{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 }} | |||
* {{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 }} | |||
* {{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 }} | |||
* {{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 }} | |||
[[Category:Homological algebra]] | |||
{{algebra-stub}} | |||
Revision as of 13:51, 21 January 2014
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.
Specifically, let G be a 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 group of A. The quotient group G/N acts on AN = { a A : na = a for all n N}. Then the inflation-restriction exact sequence is:
- 0 → H 1(G/N, AN) → H 1(G, A) → H 1(N, A)G/N → H 2(G/N, AN) →H 2(G, A)
In this sequence, there are maps
- inflation H 1(G/N, AN) → H 1(G, A)
- restriction H 1(G, A) → H 1(N, A)G/N
- transgression H 1(N, A)G/N → H 2(G/N, AN)
- inflation H 2(G/N, AN) →H 2(G, A)
The inflation and restriction are defined for general n:
- inflation Hn(G/N, AN) → Hn(G, A)
- restriction Hn(G, A) → Hn(N, A)G/N
The transgression is defined for general n
- transgression Hn(N, A)G/N → Hn+1(G/N, AN)
only if Hi(N, A)G/N = 0 for i ≤ n-1.[1]
The sequence for general n may be deduced from the case n=1 by dimension-shifting or from the Lyndon–Hochschild–Serre spectral sequence.[2]
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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