Affine ringed spaces and Serre's criterion

Abstract

We study the notion of affine ringed space, see its meaning in topological, differentiable and algebro-geometric contexts and show how to reduce the affineness of a ringed space to that of a ringed finite space. Then, we characterize schematic finite spaces and affine schematic spaces in terms of combinatorial data. Finally, we prove Serre's criterion of affineness for schematic finite spaces. This yields, in particular, Serre's criterion of affineness on schemes.