Non-compact subsets of the Zariski space of an integral domain

Let $V$ be a minimal valuation overring of an integral domain $D$ and let $\mathrm{Zar}(D)$ be the Zariski space of the valuation overrings of $D$. Starting from a result in the theory of semistar operations, we prove a criterion under which the set $\mathrm{Zar}(D)\setminus\{V\}$ is not compact. We then use it to prove that, in many cases, $\mathrm{Zar}(D)$ is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings.Comment: To appear in the Illinois Journal of Mathematic

Calculating the density of solutions of equations related to the P\'olya-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 $\theta_uu_q(n)+\theta_ww_q(n)+\theta_2\frac{n(n+1)}{2}+\theta_1n+\theta_0\equiv 0\bmod d$, where $d,q\geq 2$ are fixed integers, $\theta_u,\theta_w,\theta_2,\theta_1,\theta_0$ 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 $\theta_u=\theta_2=0$ and for the case $\theta_u=1$, $\theta_w=\theta_2=\theta_1=0$ and either $d|q$ or $d$ and $q$ are coprime.Comment: to appear in Acta Arithmetic

Towards a classification of stable semistar operations on a Pr\"ufer domain

We study stable semistar operations defined over a Pr\"ufer domain, showing that, if every ideal of a Pr\"ufer domain $R$ has only finitely many minimal primes, every such closure can be described through semistar operations defined on valuation overrings of $R$.Comment: to appear in Communications in Algebr

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 $V$.Comment: all comments are welcome!

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.Comment: any comment is welcome! Trans. Amer. Math. Soc. 373 (2020), no. 11, 7959-799

A topological version of Hilbert's Nullstellensatz

We prove that the space of radical ideals of a ring $R$, endowed with the hull-kernel topology, is a spectral space, and that it is canonically homeomorphic to the space of the nonempty Zariski closed subspaces of Spec$(R)$, endowed with a Zariski-like topology.Comment: J. Algebra (to appear

Topological properties of semigroup primes of a commutative ring

A semigroup prime of a commutative ring $R$ is a prime ideal of the semigroup $(R,\cdot)$. One of the purposes of this paper is to study, from a topological point of view, the space \scal(R) of prime semigroups of $R$. We show that, under a natural topology introduced by B. Olberding in 2010, \scal(R) is a spectral space (after Hochster), spectral extension of \Spec(R), and that the assignment R\mapsto\scal(R) induces a contravariant functor. We then relate -- in the case $R$ is an integral domain -- the topology on \scal(R) with the Zariski topology on the set of overrings of $R$. Furthermore, we investigate the relationship between \scal(R) and the space $\boldsymbol{\mathcal{X}}(R)$ consisting of all nonempty inverse-closed subspaces of \spec(R), which has been introduced and studied in C.A. Finocchiaro, M. Fontana and D. Spirito, "The space of inverse-closed subsets of a spectral space is spectral" (submitted). In this context, we show that \scal( R) is a spectral retract of $\boldsymbol{\mathcal{X}}(R)$ and we characterize when \scal( R) is canonically homeomorphic to $\boldsymbol{\mathcal{X}}(R)$, both in general and when \spec(R) is a Noetherian space. In particular, we obtain that, when $R$ is a B\'ezout domain, \scal( R) is canonically homeomorphic both to $\boldsymbol{\mathcal{X}}(R)$ and to the space \overr(R) of the overrings of $R$ (endowed with the Zariski topology). Finally, we compare the space $\boldsymbol{\mathcal{X}}(R)$ with the space \scal(R(T)) of semigroup primes of the Nagata ring $R(T)$, providing a canonical spectral embedding \xcal(R)\hookrightarrow\scal(R(T)) which makes \xcal(R) a spectral retract of \scal(R(T)).Comment: 21 page

The local Picard group of a ring extension

Given an integral domain $D$ and a $D$-algebra $R$, we introduce the local Picard group $\mathrm{LPic}(R,D)$ as the quotient between the Picard group $\mathrm{Pic}(R)$ and the canonical image of $\mathrm{Pic}(D)$ in $\mathrm{Pic}(R)$, and its subgroup $\mathrm{LPic}_u(R,D)$ generated by the the integral ideals of $R$ that are unitary with respect to $D$. We show that, when $D\subseteq R$ is a ring extension that satisfies certain properties (for example, when $R$ is the ring of polynomial $D[X]$ or the ring of integer-valued polynomials $\mathrm{Int}(D)$), it is possible to decompose $\mathrm{LPic}(R,D)$ as the direct sum $\bigoplus\mathrm{LPic}(RT,T)$, where $T$ ranges in a Jaffard family of $D$. We also study under what hypothesis this isomorphism holds for pre-Jaffard families of $D$