Affine schemes and topological closures in the Zariski–Riemann space of valuation rings

Abstract

Let F be a field, let D be a subring of F, and let X be the Zariski–Riemann space of valuation rings containing D and having quotient field F. We consider the Zariski, inverse and patch topologies on X when viewed as a projective limit of projective integral schemes having function field contained in F, and we characterize the locally ringed subspaces of X that are affine schemes.