arXiv subjects

Dario Spirito

Publications and source records attributed to Dario Spirito.

At least 37 records · Page 2Linked to original sources

arXiv2021

Metrizability of spaces of valuation domains associated to pseudo-convergent sequences

Let $V$ be a valuation domain of rank one with quotient field $K$. We study the set of extensions of $V$ to the field of rational functions $K(X)$ induced by pseudo-convergent sequences of $K$ from a topological point of view, endowing this set either with the Zariski or with the constructible topology. In particular, we consider the two subspaces induced by sequences with a prescribed breadth or with a prescribed pseudo-limit. We give some necessary conditions for the Zariski space to be metrizable (under the constructible topology) in terms of the value group and the residue field of $V$.

math.AC↗

arXiv2021

Isolated points of the Zariski space

Let $D$ be an integral domain and $L$ be a field containing $D$. We study the isolated points of the Zariski space $\mathrm{Zar}(L|D)$, with respect to the constructible topology. In particular, we completely characterize when $L$ (as a point) is isolated and, under the hypothesis that $L$ is the quotient field of $D$, when a valuation domain of dimension $1$ is isolated; as a consequence, we find all isolated points of $\mathrm{Zar}(D)$ when $D$ is a Noetherian domain. We also show that if $V$ is a valuation domain and $L$ is transcendental over $V$ then the set of extensions of $V$ to $L$ has no isolated points.

math.AC↗

arXiv2021

Extending valuations to the field of rational functions using pseudo-monotone sequences

Let $V$ be a valuation domain with quotient field $K$. We show how to describe all extensions of $V$ to $K(X)$ when the $V$-adic completion $\widehat{K}$ is algebraically closed, generalizing a similar result obtained by Ostrowski in the case of one-dimensional valuation domains. This is accomplished by realizing such extensions by means of pseudo-monotone sequences, a generalization of pseudo-convergent sequences introduced by Chabert. We also show that the valuation rings associated to pseudo-convergent and pseudo-divergent sequences (two classes of pseudo-monotone sequences) roughly correspond, respectively, to the closed and the open balls of $K$ in the topology induced by $V$.

math.RA↗

arXiv2021

The derived sequence of a pre-Jaffard family

We introduce the concept of \emph{pre-Jaffard family}, a generalization of Jaffard families obtained by substituting the locally finite hypothesis with a much weaker compactness hypothesis. From any such family, we construct a sequence of overrings of the starting domain that allows to decompose stable semistar operations and singular length functions in more cases than what is allowed by Jaffard families. We also apply the concept to one-dimensional domains, unifying the treatment of sharp and dull degree of a Prüfer domain.

math.AC↗

arXiv2020

Asymptotic for the number of star operations on one-dimensional Noetherian domains

We study the set of star operations on local Noetherian domains $D$ of dimension $1$ such that the conductor $(D:T)$ (where $T$ is the integral closure of $D$) is equal to the maximal ideal of $D$. We reduce this problem to the study of a class of closure operations (more precisely, multiplicative operations) in a finite extension $k\subseteq B$, where $k$ is a field, and then we study how the cardinality of this set of closures vary as the size of $k$ varies while the structure of $B$ remains fixed.

math.AC↗

arXiv2020

The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences

Let $V$ be a valuation domain with quotient field $K$. Given a pseudo-convergent sequence $E$ in $K$, we study two constructions associating to $E$ a valuation domain of $K(X)$ lying over $V$, especially when $V$ has rank one. The first one has been introduced by Ostrowski, the second one more recently by Loper and Werner. We describe the main properties of these valuation domains, and we give a notion of equivalence on the set of pseudo-convergent sequences of $K$ characterizing when the associated valuation domains are equal. Then, we analyze the topological properties of the Zariski-Riemann spaces formed by these valuation domains.

math.AC↗

arXiv2020

Radicals of principal ideals and the class group of a Dedekind domain

For a Dedekind domain $D$, let $\mathcal{P}(D)$ be the set of ideals of $D$ that are radical of a principal ideal. We show that, if $D,D'$ are Dedekind domains and there is an order isomorphism between $\mathcal{P}(D)$ and $\mathcal{P}(D')$, then the rank of the class groups of $D$ and $D'$ is the same.

math.AC↗

arXiv2019

The Golomb space is topologically rigid

The $Golomb$ $space$ $\mathbb N_τ$ is the set $\mathbb N$ of positive integers endowed with the topology $τ$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space $\mathbb N_τ$ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by the first author at Mathoverflow in 2017.

math.GN↗

arXiv2019

Suprema in spectral spaces and the constructible closure

Given an arbitrary spectral space $X$, we endow it with its specialization order $\leq$ and we study the interplay between suprema of subsets of $(X,\leq)$ and the constructible topology. More precisely, we investigate about when the supremum of a set $Y\subseteq X$ exists and belongs to the constructible closure of $Y$. We apply such results to algebraic lattices of sets and to closure operations on them, proving density properties of some distinguished spaces of rings and ideals. Furthermore, we provide topological characterizations of some class of domains in terms of topological properties of their ideals.

math.GN↗

arXiv2019

The Golomb topology of polynomial rings

We study properties of the Golomb topology on polynomial rings over fields, in particular trying to determine conditions under which two such spaces are not homeomorphic. We show that if $K$ is an algebraic extension of a finite field and $K'$ is a field of the same characteristic, then the Golomb spaces of $K[X]$ and $K'[X]$ are homeomorphic if and only if $K$ and $K'$ are isomorphic.

math.GN↗

arXiv2019

Multiplicative closure operations on ring extensions

Let $A\subseteq B$ be a ring extension and $\mathcal{G}$ be a set of $A$-submodules of $B$. We introduce a class of closure operations on $\mathcal{G}$ (which we call \emph{multiplicative operations on $(A,B,\mathcal{G})$}) that generalizes the classes of star, semistar and semiprime operations. We study how the set $\mathrm{Mult}(A,B,\mathcal{G})$ of these closure operations vary when $A$, $B$ or $\mathcal{G}$ vary, and how $\mathrm{Mult}(A,B,\mathcal{G})$ behave under ring homomorphisms. As an application, we show how to reduce the study of star operations on analytically unramified one-dimensional Noetherian domains to the study of closures on finite extensions of Artinian rings.

math.AC↗

arXiv2019

The Golomb topology on a Dedekind domain and the group of units of its quotients

We study the Golomb spaces of Dedekind domains with torsion class group. In particular, we show that a homeomorphism between two such spaces sends prime ideals into prime ideals and preserves the $P$-adic topology on $R\setminus P$. Under certain hypothesis, we show that we can associate to a prime ideal $P$ of $R$ a partially ordered set, constructed from some subgroups of the group of units of $R/P^n$, which is invariant under homeomorphisms, and use this result to show that the unique self-homeomorphisms of the Golomb space of $\mathbb{Z}$ are the identity and the multiplication by $-1$. We also show that the Golomb space of any Dedekind domain contained in the algebraic closure of $\mathbb{Q}$ is not homeomorphic to the Golomb space of $\mathbb{Z}$.

math.GN↗

arXiv2018

When the Zariski space is a Noetherian space

We characterize when the Zariski space $\mathrm{Zar}(K|D)$ (where $D$ is an integral domain, $K$ is a field containing $D$ and $D$ is integrally closed in $K$) and the set $\mathrm{Zar_{min}}(L|D)$ of its minimal elements are Noetherian spaces.

math.AC↗

arXiv2018

Decomposition and classification of length functions

We study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.

math.AC↗

arXiv2018

Star operations on Kunz domains

We study star operations on Kunz domains, a class of analytically irreducible, residually rational domains associated to pseudo-symmetric numerical semigroups, and we use them to refute a conjecture of Houston, Mimouni and Park. We also find an estimate for the number of star operations in a particular case, and a precise counting in a sub-case.

math.AC↗

arXiv2018

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

Given an arbitrary spectral space $X$, we consider the set ${\boldsymbol{\mathcal{X}}}(X)$ of all nonempty subsets of $X$ that are closed with respect to the inverse topology. We introduce a Zariski-like topology on ${\boldsymbol{\mathcal{X}}}(X)$ and, after observing that it coincides the upper Vietoris topology, we prove that ${\boldsymbol{\mathcal{X}}}(X)$ is itself a spectral space, that this construction is functorial, and that ${\boldsymbol{\mathcal{X}}}(X)$ provides an extension of $X$ in a more `complete' spectral space. Among the applications, we show that, starting from an integral domain $D$, ${\boldsymbol{\mathcal{X}}}(\mathrm{Spec}(D))$ is homeomorphic to the (spectral) space of all the stable semistar operations of finite type on $D$.

math.GN↗

arXiv2018

Topological properties of localizations, flat overrings and sublocalizations

We study the set of localizations of an integral domain from a topological point of view, showing that it is always a spectral space and characterizing when it is a proconstructible subspace of the space of all overrings. We then study the same problems in the case of quotient rings, flat overrings and sublocalizations.

math.AC↗

arXiv2018

Calculating the density of solutions of equations related to the Pólya-Ostrowski group through Markov chains

Motivated by a problem in the theory of integer-valued polynomials, we investigate the natural density of the solutions of equations of the form $θ_uu_q(n)+θ_ww_q(n)+θ_2\frac{n(n+1)}{2}+θ_1n+θ_0\equiv 0\bmod d$, where $d,q\geq 2$ are fixed integers, $θ_u,θ_w,θ_2,θ_1,θ_0$ are parameters and $u_q$ and $w_q$ are functions related to the $q$-adic valuations of the numbers between 1 and $n$. We show that the number of solutions of this equation in $[0,N)$ satisfies a recurrence relation, with which we can associate to any pair $(d,q)$ a stochastic matrix and a Markov chain. Using this interpretation, we calculate the density for the case $θ_u=θ_2=0$ and for the case $θ_u=1$, $θ_w=θ_2=θ_1=0$ and either $d|q$ or $d$ and $q$ are coprime.

math.NT↗