Algebraic variety

This article is about algebraic varieties. For the term "a variety of algebras", and an explanation of the difference between a variety of algebras and an algebraic variety, see variety (universal algebra).

In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) algebraic geometry.

The word "variety" is employed in the sense of a mathematical manifold, for which, in Romance languages, cognates of the word "variety" are used.

Historically, the fundamental theorem of algebra established a link between algebra and geometry by saying that a monic polynomial in one variable over the complex numbers is determined by the set of its roots, which can be considered a geometric object. Building on this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and subsets of affine space. Using the Nullstellensatz and related results, we are able to capture the geometric notion of a variety in algebraic terms as well as bring geometry to bear on questions of ring theory.

Contents

Formal definitions

Algebraic varieties can be classed into four kinds: affine varieties, quasi-affine varieties, projective varieties, and quasi-projective varieties. There also exists the more general notion of an abstract algebraic variety.

Affine varieties

Let k be an algebraically closed field and let An be an affine n-space over k. The polynomials ƒ in the ring k[x1, ..., xn] can be viewed as k-valued functions on An by evaluating ƒ at the points in An. For each finitely generated ideal S in k[x1, ..., xn], define the zero-locus of S to be the set of points in An on which the functions in S vanish:

Z(S) = \{x \in \mathbb A^n \mid f(x) = 0 \mbox{ for all } f\in S\}.

A subset V of An is called an affine algebraic set if V = Z(S) for some S. A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets. An irreducible affine algebraic set is called an affine variety. Not all the literature on algebraic varieties uses this definition; a more relaxed definition that calls any affine algebraic set an affine algebraic variety occurs in several of the basic books on the topic.

Affine varieties can be given a natural topology, called the Zariski topology, by declaring all algebraic sets to be closed.

Given a subset V of An, let I(V) be the ideal of all functions vanishing on V:

I(V) = \{f \in k[x_1,\cdots,x_n] \mid f(x) = 0 \mbox{ for all } x\in V\}.

For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal.

Projective varieties

Let Pn be a projective n-space over k. Let f \in k[x_0, \dots, x_n] be a homogeneous polynomial of degree d. It is not well-defined to evaluate ƒ on points in Pn in homogeneous coordinates. However, because ƒ is homogeneous, f(λx0, ..., λxn) = λdf(x0, ..., xn), so it does make sense to ask whether ƒ vanishes at a point [x0 : ... : xn]. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish:

Z(S) = \{x \in \mathbb P^n \mid f(x) = 0 \mbox{ for all } f\in S\}.

A subset V of Pn is called an projective algebraic set if V = Z(S) for some S. An irreducible projective algebraic set is called a projective variety.

Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.

Given a subset V of Pn, let I(V) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

Examples

Affine algebraic variety

Example 1

Let k be the field of complex numbers C. Let A2 be a two dimensional affine space over C. The polynomials ƒ in the ring k[x, y] can be viewed as complex valued functions on A2 by evaluating ƒ at the points in A2. Let subset S of k[x, y] contain a single element ƒ(x, y):

f(x, y) = x+y-1=0. \,

The zero-locus of ƒ(x, y) the set of points in A2 on which this function vanishes: it is the set of all pairs of complex numbers (x,y) such that y = 1 − x, commonly known as a line. This is the set Z(ƒ):

Z(f) = \{ (x,1-x) \in \mathbb{C}^2 \}.

Thus the subset V = Z(ƒ) of A2 is an algebraic set. The set V is not an empty set. And it is irreducible as it cannot be written as the union of two proper algebraic subsets. Thus it is an affine algebraic variety.

Example 2

Let again k be the field of complex numbers C. Let A2 be a two dimensional affine space over C. The polynomials g in the ring k[x, y] can be viewed as complex valued functions on A2 by evaluating g at the points in A2. Let subset S of k[x, y] contain a single element g(x, y):

g(x, y) = x^2 + y^2 - 1 = 0. \,

The zero-locus of g(x, y) the set of points in A2 on which this function vanishes, that is the set of points (x,y) such that xx + yy = 1, commonly known as a circle.

Basic results

Discussion and generalizations

The basic definitions and facts above enable one to do classical algebraic geometry. To be able to do more — for example, to deal with varieties over fields that are not algebraically closed — some foundational changes are required. The current notion of a variety is considerably more abstract than the one above, though equivalent in the case of varieties over algebraically closed fields. An abstract algebraic variety is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings. A scheme is a locally ringed space such that every point has a neighbourhood, which, as a locally ringed space, is isomorphic to a spectrum of a ring. Basically, a variety is a scheme whose structure sheaf is a sheaf of K-algebras with the property that the rings R that occur above are all domains and are all finitely generated K-algebras, i.e., quotients of polynomial algebras by prime ideals.

This definition works over any field K. It allows you to glue affine varieties (along common open sets) without worrying whether the resulting object can be put into some projective space. This also leads to problems since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled. These are usually not considered varieties, and we get rid of them by requiring the schemes underlying a variety to be separated. (There is strictly speaking also a third condition, namely, that in the definition above one needs only finitely many affine patches.)

Some modern researchers also remove the restriction on a variety having integral domain affine charts, and when speaking of a variety simply mean that the affine charts have trivial nilradical.

A complete variety is a variety such that any map from an open subset of a nonsingular curve into it can be extended uniquely to the whole curve. Every projective variety is complete, but not vice versa.

These varieties have been called 'varieties in the sense of Serre', since Serre's foundational paper FAC on sheaf cohomology was written for them. They remain typical objects to start studying in algebraic geometry, even if more general objects are also used in an auxiliary way.

One way that leads to generalisations is to allow reducible algebraic sets (and fields K that aren't algebraically closed), so the rings R may not be integral domains. This is not a big step technically. More serious is to allow nilpotents in the sheaf of rings. A nilpotent in a field must be 0: these if allowed in coordinate rings aren't seen as coordinate functions.

From the categorical point of view, nilpotents must be allowed, in order to have finite limits of varieties (to get fiber products). Geometrically this says that fibres of good mappings may have 'infinitesimal' structure. In the theory of schemes of Grothendieck these points are all reconciled: but the general scheme is far from having the immediate geometric content of a variety.

There are further generalizations called stacks and algebraic spaces.

Isomorphism of algebraic varieties

Let V1 and V2 be algebraic varieties. We say that V1 and V2 are isomorphic, and write V1 ≅ V2, if there are regular maps φ : V1 → V2 and ψ : V2 → V1 such that the compositions ψ ° φ and φ ° ψ are the identity maps on V1 and V2 respectively.

Algebraic manifolds

An algebraic manifold is an algebraic variety which is also an m-dimensional manifold, and hence every sufficiently small local patch is isomorphic to km. Equivalently, the variety is smooth (free from singular points). When k is the real numbers, R, algebraic manifolds are called Nash manifolds. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions. Projective algebraic manifolds are an equivalent definition for projective varieties. The Riemann sphere is one example.

See also

References

This article incorporates material from Isomorphism of varieties on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.