In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space.
More formally, an affine algebraic set is the set of the common zeros over an algebraically closed field of some family of polynomials in the polynomial ring An affine variety is an affine algebraic set which is not the union of two smaller algebraic sets; algebraically, this means that (the radical of) the ideal generated by the defining polynomials is prime. One-dimensional affine varieties are called affine algebraic curves, while two-dimensional ones are affine algebraic surfaces.
Some texts use the term variety for any algebraic set, and irreducible variety for any algebraic set whose defining ideal is prime (affine variety in the above sense).
In some contexts (see, for example, Hilbert's Nullstellensatz), it is useful to distinguish the field in which the coefficients are considered, from the algebraically closed field (containing ) over which the common zeros are considered (that is, the points of the affine algebraic set are in ). In this case, the variety is said defined over , and the points of the variety that belong to are said -rational or rational over . In the common case where is the field of real numbers, a -rational point is called a real point. When the field is not specified, a rational point is a point that is rational over the rational numbers. For example, Fermat's Last Theorem asserts that the affine algebraic variety (it is a curve) defined by has no rational points for any integer greater than two.
Introduction An affine algebraic set is the set of solutions in an algebraically closed field of a system of polynomial equations with coefficients in . More precisely, if are polynomials with coefficients in , they define an affine algebraic set
An affine (algebraic) variety is an affine algebraic set that is not the union of two proper affine algebraic subsets. Such an affine algebraic set is often said to be irreducible.
If is an affine algebraic set, and is the ideal of all polynomials that are zero on , then the quotient ring is called the of (alternative notations: , , ). The ideal is radical, so the coordinate ring is a reduced ring, and, if is an (irreducible) affine variety, then is prime, so the coordinate ring is an integral domain. The elements of the coordinate ring can be thought of as polynomial functions on and are also called the regular functions or the polynomial functions on the variety. They form the ring of regular functions on the variety, or, simply, the ring of the variety; in more technical terms (see ), it is the space of global sections of the structure sheaf of .
If (with ) is an element of its value at is defined by evaluation of any coset representative at x: A member of the coordinate ring is well defined as a polynomial function because if and only if if and only if for all When there is no risk of ambiguity or confusion, it is common to not distinguish between elements of and their residue classes in and to denote them using the same name (e.g., writing both and ).
The dimension of a variety is an integer associated to every variety, and even to every algebraic set, whose importance relies on the large number of its equivalent definitions (see Dimension of an algebraic variety).
Examples
The complement of a hypersurface in an affine variety (that is for some polynomial ) is affine. Its defining equations are obtained by saturating by the defining ideal of . The coordinate ring is thus the localization . For instance, for and , is isomorphic to the hypersurface in kn+1.Hartshorne, Chapter I, Lemma 4.2 In particular, (the affine line with the origin removed) is affine, isomorphic to the curve in (see ). On the other hand, (the affine plane with the origin removed) is not an affine variety (compare this to Hartogs' extension theorem in complex analysis). See . The subvarieties of codimension one in the affine space are exactly the hypersurfaces, that is the varieties defined by a single polynomial. The normalization of an irreducible affine variety is affine; the coordinate ring of the normalization is the integral closure of the coordinate ring of the variety. (Similarly, the normalization of a projective variety is a projective variety.)
Rational points
For an affine variety over an algebraically closed field , and a subfield of , a -rational point of is a point That is, a point of whose coordinates are elements of . The collection of -rational points of an affine variety is often denoted Often, if the base field is the complex numbers , points that are -rational (where is the real numbers) are called real points of the variety, and -rational points ( the rational numbers) are often simply called rational points.
For instance, is a -rational and an -rational point of the variety as it is in and all its coordinates are integers. The point is a real point of that is not -rational, and is a point of that is not -rational. This variety is called a circle, because the set of its -rational points is the unit circle. It has infinitely many -rational points that are the points
where is a rational number.
The circle is an example of an algebraic curve of degree two that has no -rational point. This can be deduced from the fact that, modulo , the sum of two squares cannot be .
It can be proved that an algebraic curve of degree two with a -rational point has infinitely many other -rational points; each such point is the second intersection point of the curve and a line with a rational slope passing through the rational point.
The complex variety has no -rational points, but has many complex points.
If is an affine variety in defined over the complex numbers , the -rational points of can be drawn on a piece of paper or by graphing software. The figure on the right shows the -rational points of
Singular points and tangent space Let be an affine variety defined by the polynomials and be a point of .
The Jacobian matrix of at is the matrix of the partial derivatives
The point is regular if the rank of equals the codimension of , and singular otherwise.
If is regular, the tangent space to at is the affine subspace of defined by the linear equations
If the point is singular, the affine subspace defined by these equations is also called a tangent space by some authors, while other authors say that there is no tangent space at a singular point.. A more intrinsic definition which does not use coordinates is given by Zariski tangent space.
The Zariski topology
The affine algebraic sets of kn form the closed sets of a topology on kn, called the Zariski topology. This follows from the fact that and (in fact, a countable intersection of affine algebraic sets is an affine algebraic set).
The Zariski topology can also be described by way of basic open sets, where Zariski-open sets are countable unions of sets of the form for These basic open sets are the complements in kn of the closed sets zero loci of a single polynomial. If k is Noetherian (for instance, if k is a field or a principal ideal domain), then every ideal of k is finitely-generated, so every open set is a finite union of basic open sets.
If V is an affine subvariety of kn the Zariski topology on V is simply the subspace topology inherited from the Zariski topology on kn.
Geometry–algebra correspondence The geometric structure of an affine algebraic set is linked in a deep way to the algebraic structure of its coordinate ring. Let I and J be ideals of k[V], the coordinate ring of an affine algebraic set V. For affine algebraic subset let (abbreviated to I(W) when clear from context) be the ideal of all elements of (i.e., polynomial functions ) that vanish on all points of W. For ideal I, let (abbreviated to V(I) when clear from context) be the set of all points of V on which all vanish. Finally, let denote the radical of the ideal I, the set of elements f for which some power of f is in I. We have the following mild generalization of the Nullstellensatz:
If and we recover the usual statement of the Nullstellensatz for affine space.
Radical ideals (ideals that are their own radical) of k[V] correspond to algebraic subsets of V. Indeed, for radical ideals I and J, if and only if Hence V(I) = V(J) if and only if I = J. Furthermore, the function taking an affine algebraic set W and returning I(W), the set of all functions that also vanish on all points of W, is the inverse of the function assigning an algebraic set to a radical ideal, by the Nullstellensatz. Hence the correspondence between affine algebraic sets and radical ideals is a bijection. The coordinate ring of an affine algebraic set is reduced (nilpotent-free), as an ideal I in a ring R is radical if and only if the quotient ring R/I is reduced.
Prime ideals of the coordinate ring correspond to affine subvarieties. An affine algebraic set V(I) can be written as the union of two other algebraic sets if and only if I = JK for proper ideals J and K not equal to I (in which case ). This is the case if and only if I is not prime. Affine subvarieties are precisely those whose coordinate ring is an integral domain. This is because an ideal is prime if and only if the quotient of the ring by the ideal is an integral domain.
Maximal ideals of k[V] correspond to points of V. If I and J are radical ideals, then if and only if As maximal ideals are radical, maximal ideals correspond to minimal algebraic sets (those that contain no proper algebraic subsets), which are points in V. If V is an affine algebraic set with coordinate ring this correspondence becomes explicit through the map where denotes the image in the quotient algebra R of the polynomial An algebraic subset is a point if and only if the coordinate ring of the subset is a field, as the quotient of a ring by a maximal ideal is a field.
The following table summarizes this correspondence, for algebraic subsets of an affine variety and ideals of the corresponding coordinate ring: Type of algebraic subset W Type of ideal I(W) Type of coordinate ring k[W] affine algebraic subset radical ideal reduced ring affine subvariety prime ideal integral domain point maximal ideal field
Products of affine varieties A product of affine varieties can be defined using the isomorphism then embedding the product in this new affine space. Let and have coordinate rings and respectively, so that their product has coordinate ring . Let be an algebraic subset of and an algebraic subset of Then each is a polynomial in , and each is in . The product of and is defined as the algebraic set in The product is irreducible if each , is irreducible.This is because, over an algebraically closed field, the tensor product of integral domains is an integral domain; see integral domain#Properties.
The Zariski topology on is not the topological product of the Zariski topologies on the two spaces. Indeed, the product topology is generated by products of the basic open sets and Hence, polynomials that are in but cannot be obtained as a product of a polynomial in with a polynomial in will define algebraic sets that are closed in the Zariski topology on but not in the product topology.
Morphisms of affine varieties
A morphism, or regular map, of affine varieties is a function between affine varieties that is polynomial in each coordinate: more precisely, for affine varieties and , a morphism from to is a map of the form where for each These are the morphisms in the category of affine varieties.