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Herbrand quotient : ウィキペディア英語版
Herbrand quotient
In mathematics, the Herbrand quotient is a quotient of orders of cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory.
==Definition==
If ''G'' is a finite cyclic group acting on a ''G''-module ''A'', then the cohomology groups ''H''''n''(''G'',''A'') have period 2 for ''n''≥1; in other words
:''H''''n''(''G'',''A'') = ''H''''n''+2(''G'',''A''),
an isomorphism induced by cup product with a generator of ''H''''2''(''G'',Z). (If instead we use the Tate cohomology groups then the periodicity extends down to ''n''=0.)
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
:''h''(''G'',''A'') = |''H''''2''(''G'',''A'')|/|''H''''1''(''G'',''A'')|
of the order of the even and odd cohomology groups.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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