Ultrafilter and Constructible Topologies on Spaces of Valuation Domains

Finocchiaro, Carmelo Antonio; Fontana, Marco; Loper, K. Alan

doi:10.1080/00927872.2011.651760

Cited by 13 publications

(12 citation statements)

References 18 publications

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“…We give next one of the main results in [9] which, for the case of the Zariski topology, was already proved in [24, Corollary 2.2, Proposition 2.7 and Corollary 2.9]. More precisely, Theorem 3.11.…”

Section: The Ultrafilter Topologymentioning

confidence: 79%

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

Finocchiaro

Fontana

Loper

2014

Commutative Algebra

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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.

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Section: The Ultrafilter Topologymentioning

confidence: 79%

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

Finocchiaro

Fontana

Loper

2014

Commutative Algebra

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“…569-577], [32, p. 367] and [31, p. 263]. 3 The powers of the maximal ideal of a regular local ring R define a rank one discrete valuation ring denoted ord R . If dim R = d, then the residue field of ord R is a pure transcendental extension of the residue field of R of transcendence degree d − 1.…”

Section: Notation and Terminologymentioning

confidence: 99%

“…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].…”

Section: The Patch Topology Of Q * (D)mentioning

confidence: 99%

Directed unions of local quadratic transforms of a regular local ring

2017

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Let D be a 2-dimensional regular local ring and let Q(D) denote the quadratic tree of 2-dimensional regular local overrings of D. We explore the topology of the tree Q(D) and the family R(D) of rings obtained as intersections of rings in Q(D). If A is a finite intersection of rings in Q(D), then A is Noetherian and the structure of A is well understood. However, other rings in R(D) need not be Noetherian. The two main goals of this paper are to examine topological properties of the quadratic tree Q(D), and to examine the structure of rings in the set R(D).

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“…The following easy result will provide a class of integral domains for which the equality Proof. By Proposition 2.3(2) and [7,Remark 3.2], it is enough to show that K = A U (= {x ∈ K | B x ∩ Σ ∈ U }), for every nontrivial ultrafilter U on Σ. By contradiction, assume that there exists an element x 0 ∈ K \A U .…”

Section: 2mentioning

confidence: 99%

The constructible topology on spaces of valuation domains

Finocchiaro¹,

Fontana²,

Loper³

2013

Trans. Amer. Math. Soc.

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We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on an arbitrary spectral space and we observe that this topology coincides with the constructible topology. If K is a field and A a subring of K, we show that the space Zar(K|A) of all valuation domains, having K as the quotient field and containing A, (endowed with the Zariski topology) is a spectral space by giving in this general setting the explicit construction of a ring whose Zariski spectrum is homeomorphic to Zar(K|A). We extend results regarding spectral topologies on the spaces of all valuation domains and apply the theory developed to study representations of integrally closed domains as intersections of valuation overrings. As a very particular case, we prove that two collections of valuation domains of K with the same ultrafilter closure represent, as an intersection, the same integrally closed domain. © 2013 American Mathematical Society

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Ultrafilter and Constructible Topologies on Spaces of Valuation Domains

Cited by 13 publications

References 18 publications

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

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

Directed unions of local quadratic transforms of a regular local ring

The constructible topology on spaces of valuation domains

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