Manifolds and Varieties via Sheaves

Abstract

In rough terms, a manifold is a “space” that looks locally like Euclidean space. An algebraic variety can be defined similarly as a “space” that looks locally like the zero set of a collection of polynomials. Point set topology alone would not be sufficient to capture this notion of space. These examples come with distinguished classes of functions (C ^∞functions in the first case, and polynomials in the second), and we want these classes to be preserved under the above local identifications. Sheaf theory provides a natural language in which to make these ideas precise. A sheaf on a topological space X is essentially a distinguished class of functions, or things that behave like functions, on open subsets of X. The main requirement is that the condition to be distinguished be local, which means that it can be checked in a neighborhood of every point of X. For a sheaf of rings, we have an additional requirement, that the distinguished functions on $U \subseteq X$ should form a commutative ring. With these definitions, the somewhat vague idea of a space can be replaced by the precise notion of a concrete ringed space, which consists of a topological space together with a sheaf of rings of functions. Both manifolds and varieties are concrete ringed spaces.