Motivating (sheaves And) Schemes

This note is not intended to introduce schemes, but rather to accompany a presentation of their definition (such as [1, §II.1-2]) as an explanation of why they are defined the way they are. Compared to a classical notion such as that of a projective variety, schemes differ in several key aspects: • Schemes are constructed locally, using the machinery of sheaves. • Rather than the usual notion of a projective variety, in which one considers only irreducible varieties, and treats two varieties to be the same if they have the same points (say, over an algebraically closed field), schemes consider objects which are reducible, and more substantively, non-reduced; i.e., rings of functions are allowed to have nilpotents. • Points of schemes do not correspond to points on varieties in some projective space, and in particular include “generic points”, corresponding to entire subvarieties. • Schemes allow arbitrary “rings of functions”, rather than finitely-generated k-algebras. The last point requires little justification, as it allows one to treat at once both curves over fields and rings of integers. For instance, the number-theoretic fact that every ideal in a ring of integers can be uniquely factored as a product of prime ideals, and the geometric fact that every line bundle on a curve can be realized as the line bundle associated to a divisor (that is, a collection of points with multiplicities), are both manifestations of the general relationship in scheme theory between Weil divisors and Cartier divisors. Way cool. That leaves the first three points, which we will discuss in turn, attempting to explain why it is natural and reasonable to take this approach.