Conductivity near the percolation threshold

2014-01-21T11:51:10Z

141.108.4.130:

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 ''AN = { a <math>\in</math> A : na = a '' for all '' n <math>\in</math> N}''. Then the inflation-restriction exact sequence is:

::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'')
:

In this sequence, there are maps
* ''inflation'' ''H'' 1(''G''/''N'', ''A''''N'') → ''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'', ''A''''N'')
* ''inflation'' ''H'' 2(''G''/''N'', ''A''''N'') →''H'' 2(''G'', ''A'')

The inflation and restriction are defined for general ''n'':
* ''inflation'' ''H''''n''(''G''/''N'', ''A''''N'') → ''H''''n''(''G'', ''A'')
* ''restriction'' ''H''''n''(''G'', ''A'') → ''H''''n''(''N'', ''A'')''G''/''N''

The transgression is defined for general ''n''
* ''transgression'' ''H''''n''(''N'', ''A'')''G''/''N'' → ''H''''n''+1(''G''/''N'', ''A''''N'')
only if ''H''''i''(''N'', ''A'')''G''/''N'' = 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]]
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Conductivity near the percolation threshold