Dimension of an algebraic variety: Difference between revisions

Content deleted Content added

Inline

Revision as of 14:14, 28 January 2014

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 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 x_i are replace 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.

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 $p_{0}\subset p_{1}\subset \ldots \subset p_{d}$ 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

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.

Dimension of a projective algebraic set

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

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

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 $x^{2}+y^{2}+z^{2}=0$ 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 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.

External links

"Algebraic function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

@@ Line 1: / Line 1: @@
+{{unreferenced|date=February 2013}}
-In [[mathematics]], the '''dimension''' of an [[algebraic variety]] ''V'' in [[algebraic geometry]] is defined, informally speaking, as the number of independent [[rational function]]s that exist on ''V''.
+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 space|affine]] or [[projective space]], while other are related to such an embedding.
-For example, an [[algebraic curve]] has by definition dimension 1. That means that any two rational functions ''F'' and ''G'' on it must satisfy some [[polynomial]] relation
+== Dimension of an affine algebraic set ==
-:''P''(''F'',''G'') = 0.
+Let ''K'' be a [[field (mathematics)|field]], and ''L'' ⊇ ''K'' be an  algebraically closed extension. An [[affine algebraic set]] ''V'' is the set of the common [[zero of a function|zeros]] in ''L''<sup>''n''</sup> of the elements of an ideal ''I'' in a polynomial ring <math>R=K[x_1, \ldots, x_n].</math> 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 of an ideal|radical]]). The dimension is also independent of the choice of coordinates; in other words is does not change if the ''x''<sub>''i''</sub> are replace by linearly independent linear combinations of them. The dimension of ''V'' is
-This implies that ''F'' and ''G'' are constrained to take related values (up to some finite freedom of choice): they cannot be truly independent.
+* ''The maximal length'' <math>d</math> of the chains <math>V_0\subset V_1\subset \ldots \subset V_d</math> ''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 <math>p_0\subset p_1\subset \ldots \subset p_d</math> of [[prime ideal]]s of ''A''.
+* ''The maximal Krull dimension of the [[local ring]]s 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 [[Singular point of an algebraic variety|regular point]] of V''
+This shows that the dimension is constant on a variety
+* ''The maximal dimension of the [[tangent space|tangent vector space]]s at the non [[singular point of an algebraic variety|singular point]]s 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 space|tangent vector space]] at any non [[singular point of an algebraic variety|singular point]] of V''.
+This is the algebraic analogue to the fact that a connected [[manifold]] has a constant dimension.
+* ''The number of [[hyperplane]]s or [[hypersurface]]s 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]].
+== Dimension of a projective algebraic set ==
-== Formal definition ==
-For an [[algebraic variety]] ''V'' over a field ''K'', the '''dimension''' of ''V'' is the [[transcendence degree]] over ''K'' of the [[function field]] ''K(V)'' of all [[rational function]]s on ''V'', with values in ''K''.
+Let ''V'' be a [[projective algebraic set]] defined as the set of the common zeros of a homogeneous ideal ''I'' in a polynomial ring <math>R=K[x_0, x_1, \ldots, x_n]</math> 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''.
-For the function field even to be defined, ''V'' here must be an irreducible algebraic set; in which case the function field (for an affine variety) is just the [[field of fractions]] of the coordinate ring of ''V''. Using polynomial equations, it is easy to define sets that have 'mixed dimension': a union of a curve and a plane in space, for example. These fail to be irreducible.
+== Computation of the dimension ==
-==References==
+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 <math>{x_1}^{e_1}\cdots {x_n}^{e_n}</math> is replaced by the product of the variables in it: <math>x_1^{\min (e_1,1)}\cdots x_n^{\min(e_n,1)}.</math> 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 system]]s. For example in [[Maple (software)|Maple]], this is the function ''Groebner[HilbertDimension]''.
+== Real dimension ==
+{{see also|Complex dimension}}
+For an algebraic set defined over the [[real number|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 <math>x^2+y^2+z^2=0</math> 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 [[manifold]]s 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.
+== See also ==
+*[[Dimension theory (algebra)]]
+==External links==
 *{{Springer|id=a/a011490|title=Algebraic function}}
 [[Category:Algebraic varieties]]
 [[Category:Dimension]]
+[[Category:Computer algebra]]
-[[pt:Dimensão de uma variedade algébrica]]