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

26 pages; to appear in Journal of Pure and Applied Algebra