3 - Algebraic varieties

Summary

In the broadest terms, a manifold means a topological space equipped with a sheaf of rings which is locally isomorphic to a given ringed space or spaces, its ‘local models’. For differentiable manifolds and complex manifolds, respectively, the local models are open sets in ℝⁿ and ℂⁿ, together with the sheaves of differentiable and holomorphic functions on these spaces. Algebraic manifolds, or varieties, are defined analogously. We first fix a finitely generated field extension K of the ground field k, and then take as our local models the spectra Spm R of rings having K as their field of fractions. In other words, an algebraic variety is obtained by gluing together ringed affine varieties possessing the same algebraic function field. This chapter explains these notions of affine varieties, their sheaves of rings and their gluings.

We begin by defining the n-dimensional affine space Aⁿ over the complex numbers ℂ as the set ℂⁿ equipped with the Zariski topology and elementary sheaf of rings O assigning to a basic open set D(f) ⊂ ℂⁿ the ring of rational functions ℂ[x₁, …, x_n, 1/f(x)]. These constructions are easily generalised from Aⁿto the set Spm R of maximal ideals in any finitely generated algebra R over any algebraically closed field k. One calls Spm R an affine variety, and a morphism Spm R → Spm S is the same thing as a k-algebra homomorphism S → R. An algebraic variety is then a ringed topological space obtained by gluing together affine varieties with a common function field, and in good cases are separated: the most important examples are projective varieties (Section 3.2). Many properties of affine varieties can be defined for general algebraic varieties using a covering by affine charts.

Section 3.3 explains categories and functors in elementary terms, and how an algebraic variety X determines in a natural way a functor X from the category of algebras over the ground field k to the category of sets.