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Algebraic variety is a central object of study in algebraic geometry. Classically, an algebraic variety was defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions of ''algebraic variety'' generalize this concept in several different ways while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions provide that algebraic variety is irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The concept of algebraic variety is similar to that of manifold. The difference between them is that an algebraic variety may have singular points, while a manifold will not. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry. ==Introduction and definitions== An ''affine variety'' over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way. The most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety in the 1950s. ===Affine varieties=== (詳細はalgebraically closed field and let be an affine ''n''-space over . The polynomials in the ring can be viewed as -valued functions on by evaluating at the points in , i.e. by choosing values in ''A'' for each ''xi''. For each set ''S'' of polynomials in , define the zero-locus ''Z''(''S'') to be the set of points in on which the functions in ''S'' simultaneously vanish, that is to say : A subset ''V'' of is called an affine algebraic set if ''V'' = ''Z''(''S'') for some ''S''. A nonempty affine algebraic set ''V'' is called irreducible if it cannot be written as the union of two proper algebraic subsets. An irreducible affine algebraic set is also called an affine variety. (Many authors use the phrase ''affine variety'' to refer to any affine algebraic set, irreducible or not〔Hartshorne, p.xv, notes that his choice is not conventional; see for example, Harris, p.3〕) Affine varieties can be given a natural topology by declaring the closed sets to be precisely the affine algebraic sets. This topology is called the Zariski topology. Given a subset ''V'' of , we define ''I''(''V'') to be the ideal of all polynomial functions vanishing on ''V'': : For any affine algebraic set ''V'', the coordinate ring or structure ring of ''V'' is the quotient of the polynomial ring by this ideal. 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「algebraic variety」の詳細全文を読む スポンサード リンク
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