Algebraic variety

In algebraic geometry, the algebraic variety was the classical object of study. An affine algebraic variety was an irreducible algebraic set in some affine space, over an algebraically closed field K. It therefore was given by a co-ordinate ring that was an integral domain, a quotient of a polynomial ring over K by a prime ideal. A projective algebraic variety was the closure in projective space of an affine variety. These definitions were good enough to support the geometry of 1950.

In foundational changes forced on algebraic geometry for technical reasons, there were some more abstract definitions made. An abstract algebraic variety would be a particular kind of locally ringed space, namely such that every point has a neighbourhood, as ringed space, of type Spec(R) (spectrum of a ring) with R the co-ordinate ring of an affine algebraic variety of the kind discussed in the first paragraph. This definition had the big advantage of allowing varieties which were complete, in the sense of algebraic geometry, but not given as projective varieties (which are complete, but now that became an intrinsic concept). For example an abelian variety could be defined as a connected group object in the category of complete varieties.

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 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 schemess of Grothendieck these points are all reconciled: but the general scheme is far from having the immediate geometric content of a variety.