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In algebraic geometry, divisors are a generalization of codimension one subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil). Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields. Cartier divisors and Weil divisors are parallel notions. Weil divisors are codimension one objects, while Cartier divisors are locally described by a single equation. On non-singular varieties, these two are identical, but when the variety has singular points, the two can differ. An example of a surface on which the two concepts differ is a ''cone'', i.e. a singular quadric. At the (unique) singular point, the vertex of the cone, a single line drawn on the cone is a Weil divisor, but is not a Cartier divisor (since it is not locally principal). The ''divisor'' appellation is part of the history of the subject, going back to the Dedekind–Weber work which in effect showed the relevance of Dedekind domains to the case of algebraic curves.〔Section VI.6 of .〕 In that case the free abelian group on the points of the curve is closely related to the fractional ideal theory. An algebraic cycle is a higher-dimensional generalization of a divisor; by definition, a Weil divisor is a cycle of codimension one. ==Divisors in a Riemann surface== A Riemann surface is a 1-dimensional complex manifold, so its codimension 1 submanifolds are 0-dimensional. The divisors of a Riemann surface are the elements of the free abelian group over the points of the surface. Equivalently, a divisor is a finite linear combination of points of the surface with integer coefficients. The degree of a divisor is the sum of its coefficients. We define the divisor of a meromorphic function ''f'' as : where ''R''(''f'') is the set of all zeroes and poles of ''f'', and ''sν'' is given by : A divisor that is the divisor of a meromorphic function is called principal. On a compact Riemann surface, a meromorphic function has as many poles as zeroes, and therefore on such surfaces the degree of a principal divisor is 0. Since the divisor of a product is the sum of the divisors, the set of principal divisors is a subgroup of the group of divisors. Two divisors that differ by a principal divisor are called linearly equivalent. We define the divisor of a meromorphic 1-form similarly. Since the space of meromorphic 1-forms is a 1-dimensional vector space over the field of meromorphic functions, any two meromorphic 1-forms yield linearly equivalent divisors. The equivalence class of these divisors is called the canonical divisor (usually denoted ''K''). The Riemann–Roch theorem is an important relation between the divisors of a Riemann surface and its topology. ==Weil divisor== 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「Divisor (algebraic geometry)」の詳細全文を読む スポンサード リンク
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