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In Mathematics , an algebraic variety is essentially a set of common zeroes of a set of Polynomial s. Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) Algebraic Geometry . 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. Building on this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of Polynomial Ring s 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 . FORMAL DEFINITIONS Affine varieties Let ''k'' be an Algebraically Closed Field and let A''n'' be '''affine ''n''-space''' over ''k''. The polynomials ''f'' in the ring ''k'' ..., ''x''''n'' can be viewed as ''k''-valued functions on A''n'' by evaluating ''f'' at the points in A''n''. For each subset ''S'' of ''k'' ..., ''x''''n'' , define the zero-locus of ''S'' to be the set of points in A''n'' on which the functions in ''S'' vanish: : A subset ''V'' of A''n'' 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'''. 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 A''n'', let ''I''(''V'') be the Ideal of all functions vanishing on ''V'': : For any affine algebraic set ''V'', the coordinate ring of ''V'' is the quotient of the polynomial ring by this ideal. Projective varieties Let P''n'' be '''projective ''n''-space''' over ''k''. Homogeneous Polynomial s in ''k'' ''x''1, ..., ''x''''n'' can be viewed as ''k''-valued functions on P''n'' by evaluating them on Homogeneous Coordinates . The homogeneity of the polynomial ensures that this construction is well-defined. For each set ''S'' of homogeneous polynomials, define the zero-locus of ''S'' to be the set of points in P''n'' on which the functions in ''S'' vanish: : A subset ''V'' of P''n'' 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 P''n'', 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. 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 Algebra s by Prime Ideal s. 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 Nilpotent s in the sheaf of rings. 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 Product s). Geometrically this says that fibres of good mappings may have 'infinitesimal' structure. In the theory of Scheme s 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 Stack s and Algebraic Space s. ALGEBRAIC MANIFOLDS An Algebraic manifold is an algebraic variety which is also a ''m''-dimensional Manifold , and hence every suficiently small local patch is isomorphic to ''k''''m''. Equiviently the variety is Smooth , (free from Singular Points ). When ''k'' is the Real Number s '''R''', algebraic manifolds are called Nash Manifold s. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions. Projective Algebraic Manifold are an equivalent definition for projective varieties. The Riemann Sphere is one example. SEE ALSO
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