Sheaves of Modules

An important concept in topology is a vector bundle, which is, roughly, a locally trivial parametric family of vector spaces indexed by a topological space. It is possible to think of a vector bundle as its total space with extra structure, or the sheaf of its sections. Both points of view still exist in schemes, but the sheaf-theoretical point of view is more fundamental, and reveals some additional features. In particular, taking kernels a cokernels, one gets the abelian category of coherent sheaves. Coherent sheaves are more general than algebraic vector bundles, including, for example, sheaves of ideals, which correspond, for Noetherian schemes, to closed subschemes. A particularly important application of sheaves of ideals is the theory of blow-ups, a construction which allows us, for example, to replace a point with a subscheme of codimension 1, while not disturbing (and, in fact, often even improving) smoothness.