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In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are intrinsic, as independent of any embedding of the variety into an affine or projective space, while other are related to such an embedding. == Dimension of an affine algebraic set == Let ''K'' be a field, and ''L'' ⊇ ''K'' be an algebraically closed extension. An affine algebraic set ''V'' is the set of the common zeros in ''L''''n'' of the elements of an ideal ''I'' in a polynomial ring Let ''A''=''R''/''I'' be the algebra of the polynomials over ''V''. The dimension of ''V'' is any of the following integers. It does not change if ''K'' is enlarged, if ''L'' is replaced by another algebraically closed extension of ''K'' and if ''I'' is replaced by another ideal having the same zeros (that is having the same radical). The dimension is also independent of the choice of coordinates; in other words is does not change if the ''x''''i'' are replaced by linearly independent linear combinations of them. The dimension of ''V'' is * ''The maximal length'' of the chains ''of distinct nonempty subvarieties.'' This definition generalizes a property of the dimension of a Euclidean space or a vector space. It is thus probably the definition that gives the easiest intuitive description of the notion. * ''The Krull dimension of A.'' This is the transcription of the preceding definition in the language of commutative algebra, the Krull dimension being the maximal length of the chains of prime ideals of ''A''. * ''The maximal Krull dimension of the local rings at the points of V''. This definition shows that the dimension is a ''local property''. * ''If V is a variety, the Krull dimension of the local ring at any regular point of V'' This shows that the dimension is constant on a variety * ''The maximal dimension of the tangent vector spaces at the non singular points of V''. This relies the dimension of a variety to that of a differentiable manifold. More precisely, if ''V'' if defined over the reals, then the set of its real regular points is a differentiable manifold that has the same dimension as variety and as a manifold. * ''If V is a variety, the dimension of the tangent vector space at any non singular point of V''. This is the algebraic analogue to the fact that a connected manifold has a constant dimension. * ''The number of hyperplanes or hypersurfaces in general position which are needed to have an intersection with V which is reduced to a nonzero finite number of points.'' This definition is not intrinsic as it apply only to algebraic sets that are explicitly embedded in an affine or projective space. * ''The maximal length of a regular sequence in A''. This the algebraic translation of the preceding definition. * ''The difference between n and the maximal length of the regular sequences contained in I''. This is the algebraic translation of the fact that the intersection of ''n''-''d'' hypersurfaces is, in general, an algebraic set of dimension ''d''. * ''The degree of the Hilbert polynomial of A''. * ''The degree of the denominator of the Hilbert series of A''. This allows, through a Gröbner basis computation to compute the dimension of the algebraic set defined by a given system of polynomial equations * ''If I is a prime ideal (i.e. V is an algebraic variety), the transcendence degree over K of the field of fractions of A''. This allows to prove easily that the dimension is invariant under birational equivalence. 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「Dimension of an algebraic variety」の詳細全文を読む スポンサード リンク
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