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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク affine variety ： ウィキペディア英語版 affine variety In algebraic geometry, an affine variety over an algebraically closed field ''k'' is the zero-locus in the affine ''n''-space $k^n$ of some finite family of polynomials of ''n'' variables with coefficients in ''k'' that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. A Zariski open subvariety of an affine variety is called a quasi-affine variety. If ''X'' is an affine variety defined by a prime ideal ''I'', then the quotient ring :$k(\backslash ldots,\; x\_n)/I$ is called the coordinate ring of ''X''. This ring is precisely the set of all regular functions on ''X''; in other words, it is the space of global sections of the structure sheaf of ''X''. A theorem of Serre gives a cohomological characterization of an affine variety; it says an algebraic variety is affine if and only if :$H^i(X,\; F)\; =\; 0$ for any $i\; >\; 0$ and any quasi-coherent sheaf ''F'' on ''X''. (cf. Cartan's theorem B.) This makes the cohomological study of an affine variety non-existent, in a sharp contrast to the projective case in which cohomology groups of line bundles are of central interest. An affine variety plays a role of a local chart for algebraic varieties; that is to say, general algebraic varieties such as projective varieties are obtained by gluing affine varieties. Linear structures that are attached to varieties are also (trivially) affine varieties; e.g., tangent spaces, fibers of algebraic vector bundles. An affine variety is, up to an equivalence of categories a special case of an affine scheme, which is precisely the spectrum of a ring. In complex geometry, an affine variety is an analog of a Stein manifold. == Introduction == The most concrete point of view to describe an affine algebraic variety is that it is the set of solutions in an algebraically closed field ''k'' of a system of polynomial equations with coefficients in ''k''. More precisely, if $f\_1,\; \backslash ldots,\; f\_k$ are polynomials with coefficients in ''k'', they define an affine variety (or affine algebraic set) :$V(f\_1,\backslash ldots,\; f\_k)\; =\; \backslash left\backslash .$ By Hilbert's Nullstellensatz, the points of the variety are in one to one correspondence with the maximal ideals of its ''coordinate ring'', the ''k''-algebra $R=k(\backslash ldots,\; x\_n)/\backslash langle\; f\_1,\; \backslash ldots,\; f\_k\backslash rangle,$ through the map $(a\_1,\backslash ldots,\; a\_n)\; \backslash mapsto\; \backslash langle\; \backslash overline,\; \backslash ldots,\; \backslash overline\backslash rangle,$ where $\backslash overline$ denotes the image in the quotient algebra ''R'' of the polynomial $x\_i-a\_i.$ In scheme theory, this correspondence has been extended to prime ideals to define the affine scheme $\backslash operatorname(R),$ which may be identified to the variety, through an equivalence of categories. The elements of the coordinate ring ''R'' are also called the ''regular functions'' or the ''polynomial functions'' on the variety. They form the ''ring of the regular functions'' on the variety, or, simply, the ''ring of the variety''. In fact an element $\backslash overline\backslash in\; R$ is the image of a polynomial $f\backslash in\; k(x\_n),$ which defines a function from ''k''''n'' into ''k''; The restriction of ''f'' to the variety does not depend on the choice of $f$ among the polynomials mapped on $\backslash overline$ by the quotient. The dimension of a variety is an integer associated to every variety, and even to every algebraic set, whose importance relies on the large number of its equivalent definitions (see Dimension of an algebraic variety). 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「affine variety」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.