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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 Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities

In mathematics, an algebraic variety is essentially a (finite or infinite) set of points where a polynomial (in one or more variables) attains, or a set of such polynomials all attain, a value of zero. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) algebraic geometry. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with

The word "variety" is employed in the sense of a mathematical manifold, for which, in Romance languages, cognates of the word "variety" are used. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be The Romance languages (sometimes referred to as Romanic languages, or Neolatin languages) are a branch of the Indo-European language family comprising all Cognates in Linguistics are words that have a common origin They may occur within a language such as shirt and skirt as two English words descended from

Historically, the fundamental theorem of algebra established a link between algebra and geometry by saying that a polynomial in one variable over the complex numbers is determined by the set of its roots, which is an inherently geometric object. In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Building on this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and subsets of affine space. Hilbert's Nullstellensatz ( German: "theorem of zeros" is a theorem in Algebraic geometry, a branch of Mathematics, that relates Algebraic In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean 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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those

Contents

Formal definitions

Algebraic varieties can be classed into four kinds: affine varieties, quasi-affine varieties, projective varieties, and quasi-projective varieties. In Mathematics, a quasiprojective variety in Algebraic geometry is a locally closed subset of a Projective variety, i In Mathematics, a quasiprojective variety in Algebraic geometry is a locally closed subset of a Projective variety, i There also exists the more general notion of an abstract algebraic variety. In Algebraic geometry, an abstract algebraic variety is an Algebraic variety that is defined intrinsically that is without an embedding into another variety

Affine varieties

Let k be an algebraically closed field and let An be an affine n-space over k. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. The polynomials f in the ring k[x1, . . . , xn] can be viewed as k-valued functions on An by evaluating f at the points in An. For each subset S of 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. In Mathematics, an algebraic set over a field K is the set of solutions in K n ( n -tuples of elements of 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 the field uses this definition, the 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. In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic In Topology and related branches of Mathematics, a closed set is a set whose complement is open.

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. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

Projective varieties

Let Pn be a projective n-space over k. In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which Let f \in k[x_0, ..., x_n] be a Homogeneous polynomial of degree d. In Mathematics, a homogeneous polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same It is not well-defined to evaluate f on points in Pn in homogeneous coordinates. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations However, because f is homogeneous, f(λx0, . . . , λxn) = λdf(x0, . . . , xn), so it does make sense to ask whether f 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. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

Basic results

Discussion and generalizations

The basic definitions and facts above enable one to do classical algebraic geometry. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with To be able to do more — for example, to deal with varieties over fields that are not algebraically closed — some foundational changes are required. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients 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. In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. 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. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of 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. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. e. , quotients of polynomial algebras by prime ideals. In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers

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. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Ring theory, a branch of Mathematics, the radical of an ideal is a kind of completion of the ideal.

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. In Mathematics, in particular in Algebraic geometry, a complete algebraic variety is an Algebraic variety X, such that for any variety In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one 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. In Mathematics, sheaf cohomology is the aspect of Sheaf theory, concerned with sheaves of Abelian groups that applies Homological algebra to 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. In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that A nilpotent in a field must be 0: these if allowed in co-ordinate rings aren't seen as co-ordinate functions.

From the categorical point of view, nilpotents must be allowed, in order to have finite limits of varieties (to get fiber products). In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, a branch of Mathematics, a pullback (also called a fibered product or Cartesian square) is the limit of a 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. In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany

There are further generalizations called stacks and algebraic spaces. In Algebraic geometry, a branch of Mathematics, an algebraic stack is a concept introduced to generalize algebraic varieties, schemes, and In Mathematics, an algebraic space is a generalization of the schemes of Algebraic geometry introduced by Michael Artin for use in Deformation

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. In Graph theory, an isomorphism of graphs G and H is a Bijection between the vertex sets of G and H In Complex analysis, see Holomorphic function. In Mathematics, a regular function in the sense of Algebraic geometry The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function This article is about the Identity Map software design pattern

Algebraic manifolds

Main article: Algebraic manifold

An algebraic manifold is an algebraic variety which is also a m-dimensional manifold, and hence every sufficiently small local patch is isomorphic to km. An algebraic manifold is an Algebraic variety which is also a Manifold. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Equivalently, the variety is smooth (free from singular points). In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, a singular point of an Algebraic variety V is a point P that is 'special' (so singular in the geometric sense that V When k is the real numbers, R, algebraic manifolds are called Nash manifolds. In Mathematics, the real numbers may be described informally in several different ways In Real algebraic geometry, a Nash function on an open semialgebraicsubset U of R n is an Analytic function f 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. In Mathematics, a projective algebraic manifold is a Complex manifold which is a submanifold of a Complex projective space which is determined by the zeros The Riemann sphere is one example. In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that

See also

References

This article incorporates material from Isomorphism of varieties on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

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