On the geometry of Prüfer intersections of valuation rings

Olberding, Bruce

doi:10.2140/pjm.2015.273.353

Cited by 8 publications

(43 citation statements)

References 28 publications

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“…For recent investigations on topological spaces of overrings of an integral domain see, for instance, [18], [19], [47], [48], [49], [50]. It is known that:…”

Section: Semistar Operationsmentioning

confidence: 99%

Topological properties of semigroup primes of a commutative ring

2017

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A semigroup prime of a commutative ring R is a prime ideal of the semigroup (R, Â·). One of the purposes of this paper is to study, from a topological point of view, the space S(R) of prime semigroups of R. We show that, under a natural topology introduced by B. Olberding in 2010, S(R) is a spectral space (after Hochster), spectral extension of Spec(R) , and that the assignment Râ\u86¦ S(R) induces a contravariant functor. We then relateâ\u80\u94in the case R is an integral domainâ\u80\u94the topology on S(R) with the Zariski topology on the set of overrings of R. Furthermore, we investigate the relationship between S(R) and the space X(R) consisting of all nonempty inverse-closed subspaces of Spec(R) , which has been introduced and studied in Finocchiaro et al. (submitted). In this context, we show that S(R) is a spectral retract of X(R) and we characterize when S(R) is canonically homeomorphic to X(R) , both in general and when Spec(R) is a Noetherian space. In particular, we obtain that, when R is a BÃ©zout domain, S(R) is canonically homeomorphic both to X(R) and to the space Overr(R) of the overrings of R (endowed with the Zariski topology). Finally, we compare the space X(R) with the space S(R(T)) of semigroup primes of the Nagata ring R(T), providing a canonical spectral embedding X(R) â\u86ª S(R(T)) which makes X(R) a spectral retract of S(R(T))

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“…For recent investigations on topological spaces of overrings of an integral domain see, for instance, [18], [19], [47], [48], [49], [50]. It is known that:…”

Section: Semistar Operationsmentioning

confidence: 99%

Topological properties of semigroup primes of a commutative ring

2017

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“…Different authors have investigated this problem: for example, Gilmer and Roquette gave explicit construction of Prüfer domains constructed as intersection of valuation domains, or, which is the same thing, as the integral closure of some subring (see [12] and [24], respectively). Recently, Olberding gave a geometric Email address: gperugin@math.unipd.it (Giulio Peruginelli) to appear in J. Algebra criterion on a subset Z of the Zariski-Riemann space of all the valuation domains of a field in order for the holomorphy ring V ∈Z V to be a Prüfer domain; this criterion is given in terms of projective morphisms of Z, considered as a locally ringed space, into the projective line (see [19]). In [20] Olberding gave a sufficient condition on a family of rank one valuation domains which satisfies certain assumptions so that the intersection of the elements of the family is a Prüfer domain.…”

Section: Introductionmentioning

confidence: 99%

Prüfer intersection of valuation domains of a field of rational functions

Peruginelli

2018

Journal of Algebra

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Let V be a rank one valuation domain with quotient field K. We characterize the subsets S of V for which the ring of integer-valued polynomials Int(S,The characterization is obtained by means of the notion of pseudomonotone sequence and pseudo-limit in the sense of Chabert, which generalize the classical notions of pseudo-convergent sequence and pseudo-limit by Ostrowski and Kaplansky, respectively. We show that Int(S, V ) is Prüfer if and only if no element of the algebraic closure K of K is a pseudo-limit of a pseudo-monotone sequence contained in S, with respect to some extension of V to K. This result expands a recent result by Loper and Werner.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Olberding

2018

Journal of Pure and Applied Algebra

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Abstract. Let F be a field. For each nonempty subset X of the Zariski-Riemann space of valuation rings of F , let A(X) = V ∈X V and J(X) = V ∈X MV , where MV denotes the maximal ideal of V . We examine connections between topological features of X and the algebraic structure of the ring A(X). We show that if J(X) = 0 and A(X) is a completely integrally closed local ring that is not a valuation ring of F , then there is a subspace Y of the space of valuation rings of F that is perfect in the patch topology such that A(X) = A(Y ). If any countable subset of points is removed from Y , then the resulting set remains a representation of A(X). Additionally, if F is a countable field, the set Y can be chosen homeomorphic to the Cantor set. We apply these results to study properties of the ring A(X) with specific focus on topological conditions that guarantee A(X) is a Prüfer domain, a feature that is reflected in the Zariski-Riemann space when viewed as a locally ringed space. We also classify the rings A(X) where X has finitely many patch limit points, thus giving a topological generalization of the class of Krull domains, one that also includes interesting Prüfer domains. To illustrate the latter, we show how an intersection of valuation rings arising naturally in the study of local quadratic transformations of a regular local ring can be described using these techniques.

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On the geometry of Prüfer intersections of valuation rings

Cited by 8 publications

References 28 publications

Topological properties of semigroup primes of a commutative ring

Topological properties of semigroup primes of a commutative ring

Prüfer intersection of valuation domains of a field of rational functions

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

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