|
In algebraic geometry, the Chow ring (named after W. L. Chow) of a smooth algebraic variety over any field is an algebro-geometric analog of the cohomology ring of a complex variety considered as a topological space. The elements of the Chow ring are formed out of actual subvarieties (so-called algebraic cycles), and the multiplicative structure is derived from the intersection of subvarieties. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general. ==Rational equivalence and Chow groups== For what follows, define a variety over a field ''k'' to be an integral scheme of finite type over ''k''. For any scheme ''X'' of finite type over ''k'', an algebraic cycle on ''X'' means a finite linear combination of subvarieties of ''X'' with integer coefficients. (Here and below, subvarieties are understood to be closed in ''X'', unless stated otherwise.) For a natural number ''i'', the group ''Z''''i''(''X'') of ''i''-dimensional cycles (or ''i''-cycles, for short) on ''X'' is the free abelian group on the set of ''i''-dimensional subvarieties of ''X''. For a variety ''W'' of dimension ''i'' + 1 and any rational function ''f'' on ''W'' which is not identically zero, the divisor of ''f'' is the ''i''-cycle : where the sum runs over all ''i''-dimensional subvarieties ''Z'' of ''W'' and the integer ord''Z''(''f'') denotes the order of vanishing of ''f'' along ''Z''. (Thus ord''Z''(''f'') is negative if ''f'' has a pole along ''Z''.) The definition of the order of vanishing requires some care for ''W'' singular.〔Fulton. Intersection Theory, section 1.2 and Appendix A.3.〕 For a scheme ''X'' of finite type over ''k'', the group of ''i''-cycles rationally equivalent to zero is the subgroup of ''Z''''i''(''X'') generated by the cycles (''f'') for all (''i''+1)-dimensional subvarieties ''W'' of ''X'' and all nonzero rational functions ''f'' on ''W''. The Chow group ''CH''''i''(X) of ''i''-dimensional cycles on ''X'' is the quotient group of ''Z''''i''(''X'') by the subgroup of cycles rationally equivalent to zero. Sometimes one writes () for the class of a subvariety ''Z'' in the Chow group. For example, when ''X'' is a variety of dimension ''n'', the Chow group ''CH''''n''-1(X) is the divisor class group of ''X''. When ''X'' is smooth over ''k'', this is isomorphic to the Picard group of line bundles on ''X''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Chow ring」の詳細全文を読む スポンサード リンク
|