Dimension of an algebraic variety

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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[edit]

Let K be a field, and LK be an algebraically closed extension. An affine algebraic set V is the set of the common zeros in Ln of the elements of an ideal I in a polynomial ring R=K[x_1, \ldots, x_n]. 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 xi are replaced by linearly independent linear combinations of them. The dimension of V is

  • The maximal length d of the chains V_0\subset V_1\subset \ldots \subset V_d 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.

This is the transcription of the preceding definition in the language of commutative algebra, the Krull dimension being the maximal length of the chains p_0\subset p_1\subset \ldots \subset p_d 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

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.

This is the algebraic analogue to the fact that a connected manifold has a constant dimension.

This definition is not intrinsic as it apply only to algebraic sets that are explicitly embedded in an affine or projective space.

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.

This allows, through a Gröbner basis computation to compute the dimension of the algebraic set defined by a given system of polynomial equations

This allows to prove easily that the dimension is invariant under birational equivalence.

Dimension of a projective algebraic set[edit]

Let V be a projective algebraic set defined as the set of the common zeros of a homogeneous ideal I in a polynomial ring R=K[x_0, x_1, \ldots, x_n] over a field K, and let A=R/I be the graded algebra of the polynomials over V.

All the definitions of the previous section apply, with the change that, when A or I appear explicitly in the definition, the value of the dimension must be reduced by one. For example, the dimension of V is one less than the Krull dimension of A.

Computation of the dimension[edit]

Given a system of polynomial equations, it may be difficult to compute the dimension of the algebraic set that it defines.

Without further information on the system, there is only one practical method that consists to compute a Gröbner basis and to deduce the degree of the denominator of the Hilbert series of the ideal generated by the equations.

The second step, which is usually the fastest, may be accelerated in the following way: Firstly, the Gröbner basis is replaced by the list of its leading monomials (this is already done for the computation of the Hilbert series). Then each monomial like {x_1}^{e_1}\cdots {x_n}^{e_n} is replaced by the product of the variables in it: x_1^{\min (e_1,1)}\cdots x_n^{\min(e_n,1)}. Then the dimension is the maximal size of a subset S of the variables, such that none of these products of variables depends only on the variables in S.

This algorithm is implemented in several computer algebra systems. For example in Maple, this is the function Groebner[HilbertDimension].

Real dimension[edit]

The dimension of a set of real points, typically a semialgebraic set, is the dimension of its Zariski closure. For an algebraic set defined over the reals (that is defined by polynomials with real coefficients), it may occur that the dimension of the set of its real points differs from its dimension. For example, the algebraic surface of equation x^2+y^2+z^2=0 is an algebraic variety of dimension two, which has only one real point (0, 0, 0), and thus the real dimension zero.

The real dimension is more difficult to compute than the algebraic dimension, and, to date, there is no available software to compute it.

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