Some Closure Operations in Zariski-Riemann Spaces of Valuation Domains: A Survey

Abstract

In this survey we present several results concerning various topologies that were introduced in recent years on spaces of valuation domains. Spaces of valuation domainsThe motivations for studying from a topological point of view spaces of valuation domains come from various directions and, historically, mainly from Zariski's work on the reduction of singularities of an algebraic surface and a three-dimensional variety and, more generally, for establishing new foundations of algebraic geometry by algebraic means (see [45], [46], [47] and [48]). Other important applications with algebro-geometric flavour are due to Nagata [33] [34], Temkin [43], Temkin and Tyomkin [44]. Further motivations come from rigid algebraic geometry started by J. Tate [42] (see also the papers by Fresnel and van der Put [18], Huber and Knebusch [27], Fujiwara and Kato [19]), and from real algebraic geometry (see for instance Schwartz [39] and Huber [26]). For a deeper insight on these topics see [27].In the following, we want to present some recent results in the literature concerning various topologies on collections of valuation domains.Let K be a field, A an arbitrary subring of K and let qf(A) denote the quotient field of A. SetWhen A is the prime subring of K, we will simply denote by Zar(K) the space Zar(K|A). Recall that O. Zariski in [46] introduced a topological structure on the set Z := Zar(K|A) by taking, as a basis for the open sets, the subsets B F := B Z F := {V ∈ Z | V ⊇ F }, for F varying in the family of all finite subsets of K (see also [48, Chapter VI, §17, page 110]). When no confusion can arise, we will simply denote by B x the basic open set B {x} of Z. This topology is what that is now called the Zariski topology on Z Acknowledgments. During the preparation of this paper, the first two authors were partially supported by a research grant PRIN-MiUR.

Topological Aspects of Irredundant Intersections of Ideals and Valuation Rings

“…, x n ∈ F. With this topology, the same topology as in Example 2.2(8), X is a spectral space and is termed the Zariski-Riemann space of F/A. For some recent articles emphasizing a topological approach to the Zariski-Riemann space, see [8,9,10,11,12,13,42,44,45].…”

Realizations of pairs and Oka families in tensor triangulated categories

“…In fact, if X is a topologically noetherian scheme, using the fact that there is a homeomorphism X Spec(D perf (X )) (see [1,Corollary 5.6]), we see that the theorem gives us a correspondence between (radical) thick tensor ideals in Spec(D perf (X )) and closed subspaces of X in the inverse topology. We can build on this idea in two ways: first, we can think about the constructible topology on X , because the constructible topology on a spectral space always coincides with the constructible topology on its inverse (see, for instance, [18,Corollary 4.8]). Secondly, we can think about the closed irreducible subspaces of the scheme X in inverse topology.…”