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| In [[mathematics]], the '''Herbrand quotient''' is a [[quotient]] of orders of [[Group cohomology|cohomology]] groups of a [[cyclic group]]. It was invented by [[Jacques Herbrand]]. It has an important application in [[class field theory]].
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| ==Definition==
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| If ''G'' is a finite cyclic group acting on a [[G-module|''G''-module]] ''A'', then the cohomology groups ''H''<sup>''n''</sup>(''G'',''A'') have period 2 for ''n''≥1; in other words
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| :''H''<sup>''n''</sup>(''G'',''A'') = ''H''<sup>''n''+2</sup>(''G'',''A''),
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| an [[isomorphism]] induced by [[cup product]] with a generator of ''H''<sup>''2''</sup>(''G'','''Z'''). (If instead we use the [[Tate cohomology group]]s then the periodicity extends down to ''n''=0.)
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| A '''Herbrand module''' is an ''A'' for which the cohomology groups are finite. In this case, the '''Herbrand quotient''' ''h''(''G'',''A'') is defined to be the quotient
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| :''h''(''G'',''A'') = |''H''<sup>''2''</sup>(''G'',''A'')|/|''H''<sup>''1''</sup>(''G'',''A'')|
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| of the order of the even and odd cohomology groups.
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| ===Alternative definition===
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| The quotient may be defined for a pair of [[endomorphism]]s of an [[Abelian group]], ''f'' and ''g'', which satisfy the condition ''fg'' = ''gf'' = 0. Their Herbrand quotient ''q''(''f'',''g'') is defined as
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| :<math> q(f,g) = \frac{|\mathrm{ker} f:\mathrm{im} g|}{|\mathrm{ker} g:\mathrm{im} f|} </math>
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| if the two [[index of a group|indices]] are finite. If ''G'' is a cyclic group with generator γ acting on an Abelian group ''A'', then we recover the previous definition by taking ''f'' = 1 - γ and ''g'' = 1 + γ + γ<sup>2</sup> + ... .
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| ==Properties==
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| *The Herbrand quotient is [[multiplicative function|multiplicative]] on [[short exact sequence]]s.<ref name=C245>Cohen (2007) p.245</ref> In other words, if
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| :0 → ''A'' → ''B'' → ''C'' → 0
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| is exact, and any two of the quotients are defined, then so is the third and<ref name=S134>Serre (1979) p.134</ref>
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| :''h''(''G'',''B'') = ''h''(''G'',''A'')''h''(''G'',''C'')
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| *If ''A'' is finite then ''h''(''G'',''A'') = 1.<ref name=S134/>
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| *For ''A'' is a submodule of the ''G''-module ''B'' of finite index, if either quotient is defined then so is the other and they are equal:<ref name=C245/> more generally, if there is a ''G''-morphism ''A'' → ''B'' with finite kernel and cokernel then the same holds.<ref name=S134/>
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| *If '''Z''' is the integers with ''G'' acting trivially, then ''h''(''G'','''Z''') = |''G''|
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| *If ''A'' is a finitely generated ''G''-module, then the Herbrand quotient ''h''(''A'') depends only on the complex ''G''-module '''C'''⊗''A'' (and so can be read off from the character of this complex representation of ''G'').
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| These properties mean that the Herbrand quotient is usually relatively easy to calculate, and is often much easier to calculate than the orders of either of the individual cohomology groups.
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| ==See also==
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| *[[Class formation]]
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| ==References==
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| {{reflist}}
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| * {{cite book | first1=M.F. | last1=Atiyah | author1-link=M. F. Atiyah | first2=C.T.C. | last2=Wall | author2-link=C. T. C. Wall | chapter=Cohomology of Groups | editor1-first=J.W.S. | editor1-last=Cassels | editor1-link=J. W. S. Cassels | editor2-first=Albrecht | editor2-last=Fröhlich | editor2-link=Albrecht Fröhlich | title=Algebraic Number Theory | year=1967 | publisher=Academic Press | zbl=0153.07403 }} See section 8.
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| * {{cite book | first1=Emil | last1=Artin | author1-link=Emil Artin | first2=John | last2=Tate |author2-link=John Tate | title=Class Field Theory | publisher=AMS Chelsea | year=2009 | isbn=0-8218-4426-1 | zbl=1179.11040 | page=5 }}
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| * {{cite book |last=Cohen |first=Henri |authorlink= Henri Cohen (number theorist) |year=2007 |title=Number Theory – Volume I: Tools and Diophantine Equations|isbn= 978-0-387-49922-2|publisher=[[Springer-Verlag]]|series=[[Graduate Texts in Mathematics]]|volume=239| zbl=1119.11001 | pages=242–248}}
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| * {{cite book | first=Gerald J. | last=Janusz | title=Algebraic number fields | series=Pure and Applied Mathematics | volume=55 | publisher=Academic Press | year=1973 | page=142 | zbl=0307.12001 }}
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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 | pages=120–121 }}
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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 }}
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| [[Category:Algebraic number theory]]
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| [[Category:Abelian group theory]]
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