- This article is about algebraic varieties. For varieties of algebras, and an explanation of the difference, see variety (universal algebra).
In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. An affine algebraic variety is defined to be an irreducible algebraic set in some affine space, over an algebraically closed field K. An algebraic set in K^n is defined as the set of zeros of a finite collection of multivariable polynomials with coefficients from K. It is called irreducible if the ideal of polynomials vanishing on the set is a prime ideal. The quotient of the polynomial ring by this ideal is the coordinate ring of the affine algebraic variety. This is an integral domain since it is a quotient of a ring by a prime ideal. A projective algebraic variety is the closure in projective space of an affine variety. These basic definitions allow you to do a lot of classical algebraic geometry.
To be able to do more, especially to be able to think clearly and succintly about varieties over non-algebraically closed fields some foundational changes are required. The current notion of a variety is considerable 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. 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 modulo prime ideals.
This definition works over any field K. It allows you to glue affine varieties 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.)
Here are some interesting subclasses of varieties. A projective variety is a variety which admits an embedding into projective space. 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 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. 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.
See also
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