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
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
. Some are restricted to algebraic varieties while others apply also to any
algebraic set
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
. Some are intrinsic, as independent of any embedding of the variety into an
affine
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine comb ...
or
projective space, while other are related to such an embedding.
Dimension of an affine algebraic set
Let be a
field, and be an algebraically closed extension. An
affine algebraic set
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine com ...
is the set of the common
zeros in of the elements of an ideal in a polynomial ring
Let
be the algebra of the polynomial functions over . The dimension of is any of the following integers. It does not change if is enlarged, if is replaced by another algebraically closed extension of and if 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 it does not change if the are replaced by linearly independent linear combinations of them. The dimension of is
* ''The maximal length''
of the chains
''of distinct nonempty (irreducible) subvarieties of .''
This definition generalizes a property of the dimension of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
or a
vector space. It is thus probably the definition that gives the easiest intuitive description of the notion.
* ''The
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally th ...
of the coordinate ring .''
This is the transcription of the preceding definition in the language of
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, the Krull dimension being the maximal length of the chains
of
prime ideals of .
* ''The maximal Krull dimension of the
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
s at the points of ''.
This definition shows that the dimension is a ''local property if
is irreducible.'' If
is irreducible, it turns out that all the local rings at closed points have the same Krull dimension (see ).
* ''If is a variety, the Krull dimension of the local ring at any point of ''
This rephrases the previous definition into a more geometric language.
* ''The maximal dimension of the
tangent vector spaces at the non
singular points of ''.
This relates the dimension of a variety to that of a
differentiable manifold. More precisely, if if defined over the reals, then the set of its real regular points, if it is not empty, is a differentiable manifold that has the same dimension as a variety and as a manifold.
* ''If is a variety, the dimension of the
tangent vector space at any non
singular point of ''.
This is the algebraic analogue to the fact that a connected
manifold has a constant dimension. This can also be deduced from the result stated below the third definition, and the fact that the dimension of the tangent space is equal to the Krull dimension at any non-singular point (see
Zariski tangent space
In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point ''P'' on an algebraic variety ''V'' (and more generally). It does not use differential calculus, being based directly on abstract algebra, an ...
).
* ''The number of
hyperplanes or
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
s in
general position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...
which are needed to have an intersection with 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 commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection.
Definitions
Fo ...
in the coordinate ring ''.
This the algebraic translation of the preceding definition.
* ''The difference between and the maximal length of the regular sequences contained in ''.
This is the algebraic translation of the fact that the intersection of general hypersurfaces is an algebraic set of dimension .
* ''The degree of the
Hilbert polynomial of ''.
* ''The degree of the denominator of the
Hilbert series of ''.
This allows, through a
Gröbner basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbn ...
computation to compute the dimension of the algebraic set defined by a given
system of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some field .
A ''solution'' of a polynomial system is a set of values for the ...
.
* ''The dimension of the simplicial complex whose
Stanley-Reisner ring is
where
is the
radical of any initial ideal of I.''
Taking initial ideals preserves Hilbert polynomial/series, and taking radicals preserves the dimension.
* ''If is a prime ideal (i.e. is an algebraic variety), the
transcendence degree
In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...
over of the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of ''.
This allows to prove easily that the dimension is invariant under
birational equivalence
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational f ...
.
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