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.

On the topology of valuation-theoretic representations of integrally closed domains

“…, x n ∈ F . For more on the inverse topology in the context of the Zariski-Riemann space of a field, see [7,28,27].…”

“…In Section 4 we apply the results of Section 3 to develop sufficient topological conditions for an intersection of valuation rings to be a Prüfer domain; that is, a domain A for which each localization of A at a maximal ideal is a valuation domain. In general the question of whether an intersection of valuation rings is a Prüfer domain is very subtle; see [11,20,26,28,29,32,33] and their references for various approaches to this question. While Prüfer domains have been thoroughly investigated in Multiplicative Ideal Theory (see for example [8,10,19]), our interest here lies more in the point of view that these rings form the "coordinate rings" of sets X in Zar(F ) that are non-degenerate in the sense that every localization of A(X) lies in X.…”

Directed unions of local quadratic transforms of a regular local ring

“…As follows in Remark 4.4, properties of the Zariski topology, which are the focus of the next section, can be derived from the patch topology, so our approach in this section is to focus on the patch limit points of subsets of Q * (D) and use this description in the next section to describe properties of the Zariski topology of Q * (D). The patch topology is a common tool for studying the Zariski-Riemann space of valuation rings of a field; see for example [3,4,17,25,26,27,28].…”

The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications

“…In the context of spectral spaces, the co-compact topology of X is called the inverse topology of X, and plays a crucial rôle in Hochster's study of spectral spaces; it owes its name to the fact that the order canonically associated to the inverse topology coincides with the reverse order of that induced by the spectral topology. Subsets of a spectral space that are closed in the inverse topology are strictly related to the study of representations of integrally closed domains as intersections of collections of valuation domains (see also [33], [34], and [35]), and they represent a way to classify several distinguished classes of semistar operations of finite type: it was shown in [12] and [11] that complete (or, e.a.b.) semistar operations (respectively, stable semistar operations -definitions recalled later) correspond to the subsets of Zar(D) (respectively, Spec(D)) that are closed in the inverse topology.…”