Affine Schemes and Basic Constructions

Abstract

The objective of the chapter is to introduce the notion of schemes and their morphisms, in particular, to explain the construction of affine schemes. If A is a ring, the set of its prime ideals is viewed as a point set, called the prime spectrum of A; the latter is denoted by Spec A. Any element f∈A can be viewed as a function on Spec A. Just let \(f(\mathfrak{p})\), for any prime ideal \(\mathfrak{p} \subset A\), be the residue class of f in \(A/\mathfrak{p}\). Furthermore, write D(f) for the subset of all points \(\mathfrak{p} \in \mathrm{Spec}\,A\) such that \(f(\mathfrak{p})\) is not the zero class. Then D(f) is called a basic open subset in Spec A, while the topology generated by all these sets is referred to as the Zariski topology on Spec A. There is a remarkable fact to be discovered: for any element f∈A, the point set D(f)⊂Spec A can canonically be identified with the prime spectrum Spec A f of the localization of A by f. In particular, the natural localization morphism A⟶A f can be interpreted as the process of restricting functions living on Spec A and given by elements in A, to functions on the basic open subset D(f). In order to handle local functions on Spec A in convenient terms, the notion of sheaf is introduced. This way Spec A is viewed as a ringed space, i.e. as a topological space together with a sheaf of functions on it. Such a pair is called an affine scheme. More general schemes occur as ringed spaces that look locally like affine schemes.