nLab
algebraic variety

Contents

Idea

Classically, an algebraic variety is a ringed space which looks locally like the geometric locus of zeros of a set of polynomial equations in finitely many variables (equipped with a sheaf of regular functions which are locally quotients of polynomials).

Historically, there were several formalisms of various schools including the Italian school of algebraic geometry in the early 20th century (Veronese, Castelnuovo, Severi, …), the American school between the two wars (Oscar Zariski?), Andre Weil, the abstract varieties of Jean-Pierre Serre and finally the scheme language of the Grothendieck school. One should note that in the case of (esp. projective) varieties over complex numbers there is an additional possibility to work using complex-analytic tools and complex topology.

Definition

Given an algebraically closed field kk, an algebraic kk-variety usually means either quasiprojective varity or abstract variety (in the sense of Serre). Quasiprojective unify affine, quasiaffine, projective and embedded quasiprojective kk-varieties.

  • an embedded affine kk-variety (or an affine algebraic set) is a set of zeros of a locus of common zeros of a set of polynomial equations in the affine space A k n\mathbf{A}^n_k. By the Hilbert Nullstellensatz there is a more invariant definition. Affine kk-varieties are maximal spectra (= sets of maximal ideals) of finitely generated noetherian (commutative unital) kk-algebras without nilpotents with the Zariski topology; the algebra can be recovered as the coordinate ring of the variety; this correspondence is an equivalence of categories, if the morphisms are properly defined.

    Affine varietes can be embedded as closed subvarieties into an affine space (in the sense of algebraic geometry). As topological spaces affine varieties are noetherian.

  • Projective kk-varieties are obtained in a similar way from graded kk-algebras, or, in embedded incarnation, as loci of zeros of a set of homogeneous polynomials in projective space P k n\mathbf{P}^n_k.

  • Embedded quasiaffine kk-varieties are Zariski-open subspaces of affine kk-varieties.

  • Embedded quasiprojective kk-varieties are Zariski-open subspaces of projective kk-varieties. We can remove the embedding by equipping them with the shaef of regular functions and therefore considering them as locally ringed spaces. In the category of locally ringed spaces, projective, affine, and quasiaffine varieties are (isomorphic to) special cases of quasiprojective. Alternatively, we can put all 4 classes without sheaves into a category, by defining regular maps directly, and we get an isomorphic category of varieties.

    In fact, by noticing that the affine kk-space is Zariski open in a projective space of the same dimension, we see that the quasiprojective case includes all others.

Morphisms between varieties are so-called regular maps?.

Sometimes a smooth algebraic variety may also be called algebraic manifold.

An abstract kk-prevariety in the sense of Serre is a locally ringed space which is locally isomorphic to affine kk-variety. The category of kk-prevarieties has a product which is obtained by locally gluing products in the category of affine kk-varieties. This enables defining a diagonal XXXX\to X\to X; a prevariety is separated, or an abstract kk-variety if the diagonal is closed in Zariski topology (which is, of course, not a product of Zariski topologies of factors).

Properties

Relation to schemes

There is an equivalence of categories between the category of integral schemes of finite type over SpeckSpec\,k, where kk is an algebraically closed field, and the category of (irreducible) algebraic kk-varieties.

Of course, given a variety the corresponding scheme and variety have different sets of points; the points in common are the closed points of the scheme. The remaining points are the generic points of subvarieties. Generic points were often used, without proper foundations, in other language, already in the works of the Italian school.

Some modern algebraic geometers mean, by varieties, objects of certain slightly bigger categories of relative SS-schemes of finite type (where SS is not necessarily SpeckSpec\,k for kk a field); typically they are required to be separated reduced SS-schemes of finite type.

References

An amusing discussion on the differences between schemes and varieties can be found at Secret blogging seminar: algebraic geometry without prime ideals.

Revised on November 24, 2013 05:59:31 by Urs Schreiber (89.204.137.79)