Dimension of an algebraic variety
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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 Ln 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 xi are replace by linearly independent linear combinations of them. The dimension of V is
- The maximal length of the chains of distinct nonempty subvarieties.
- The Krull dimension 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.
- 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.
- 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.
Dimension of a projective algebraic set
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
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 is replaced by the product of the variables in it: 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.
For an algebraic set defined over the reals (that is defined by polynomials with real coefficients), it may occurs that the dimension of the set of its real points differs from its dimension. For example, the algebraic surface of equation 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 of a real algebraic set is the dimension (as algebraic set) of its Zariski closure. It is equal to the maximum of the dimensions of the manifolds contained in the set of the real points of the algebraic set.
The real dimension is more difficult to compute than the algebraic dimension, and, to date, there is no available software to compute it.