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Dario Spirito

Publications and source records attributed to Dario Spirito.

At least 19 recordsLinked to original sources
arXiv2026

The reciprocal complement of a surface

We study the reciprocal complement $\mathcal{R}(D)$ of a two-dimensional finitely generated $K$-algebra $D$ by linking it with the properties of a surface with coordinate ring $D$. We give several sufficient criteria to have $\dim\mathcal{R}(D)=2$, and we use them to show several explicit examples; in particular, we determine the dimension of $\mathcal{R}(D)$ when $D$ is the quotient of $K[X,Y,Z]$ by an irreducible polynomial of degree $2$. We also study the integral closure of the localizations of $\mathcal{R}(K[X,Y])$.

math.AC↗
arXiv2026

Groups of invertible ideals of one-dimensional Pr\"ufer domains as groups of integer-valued functions

Let $G$ be a one-dimensional $\ell$-subgroup of the group $\mathcal{F}(X,\mathbb{Z})$ of integer-valued functions on a set $X$. We show that $G$ is free under some hypothesis on the spectrum of $G$ and on its quotient groups at the prime ideals. We translate this result in the context of the study of freeness of the group $\mathrm{Inv}(D)$ of invertible ideals of a Pr\"ufer domain $D$: in particular, we introduce the class of \emph{dd-domains} as the class of Pr\"ufer domains having a set $X$ that is dense in $\mathrm{Spec}(D)$ (with respect to the inverse topology) and whose localizations are DVRs. This class is exactly the class of Pr\"ufer domains for which $\mathrm{Inv}(D)$ is isomorphic (as an $\ell$-group) to a subgroup of $\mathcal{F}(X,\mathbb{Z})$.

math.AC↗
arXiv2025

Smooth sequences and overring operators

We introduce smooth sequences of integral domains as well-ordered ascending chains that behave well at limit ordinals. Subsequently, we use this notion to give some conditions on the freeness of kernels of extension maps between groups of invertible ideals of Prüfer domains. We also define overring operators to construct smooth sequences in a recursive way.

math.AC↗
arXiv2025

The reciprocal complement of a curve

We give a geometric interpretation of the reciprocal complement of an integral domain $D$ in the case $D$ is a one-dimensional finitely generated algebra over an algebraically closed field.

math.AC↗
arXiv2024

Residual functions and divisorial ideals

We define a \emph{residual function} on a topological space $X$ as a function $f:X\longrightarrow\mathbb{Z}$ such that $f^{-1}(0)$ contains an open dense set, and we use this notion to study the freeness of the group of divisorial ideals on a Prüfer domain.

math.AC↗
arXiv2024

Radical factorization in higher dimension

We generalize the theory of radical factorization from almost Dedekind domain to strongly discrete Prüfer domains; we show that, for a fixed subset $X$ of maximal ideals, the finitely generated ideals with $\mathcal{V}(I)\subseteq X$ have radical factorization if and only if $X$ contains no critical maximal ideals with respect to $X$. We use these notions to prove that in the group $\mathrm{Inv}(D)$ of the invertible ideals of a strongly discrete Prüfer domains is often free: in particular, we show it when the spectrum of $D$ is Noetherian or when $D$ is a ring of integer-valued polynomials on a subset over a Dedekind domain.

math.AC↗
arXiv2024

A realization theorem for almost Dedekind domains

An integral domain $D$ is called an SP-domain if every ideal is a product of radical ideals. Such domains are always almost Dedekind domains, but not every almost Dedekind domain is an SP-domain. The SP-rank of $D$ provides a natural measure of the deviation of $D$ from being an SP-domain. In the present paper we show that every ordinal number $α$ can be realized as the SP-rank of an almost Dedekind domain.

math.AC↗
arXiv2024

The ramification tree and almost Dedekind domains of prescribed SP-rank

Given a valuation $v$ with quotient field $K$ and a sequence $\mathcal{K} :K_0\subseteq K_1\subseteq\cdots$ of finite extensions of $K$, we construct a weighted tree $\mathcal{T}(v,\mathcal{K})$ encoding information about the ramification of $v$ in the extensions $K_i$; conversely, we show that a weighted tree $\mathcal{T}$ can be expressed as $\mathcal{T}(v,\mathcal{K})$ under some mild hypothesis on $v$ or on $\mathcal{T}$. We use this construction to construct, for every countable successor ordinal number $α$, an almost Dedekind domain $D$, integral over $V$ (the valuation domain of $v$) whose SP-rank is $α$. Subsequently, we extend this result to countable limit ordinal numbers by considering integral extensions of Dedekind domains with countably many maximal ideals.

math.AC↗
arXiv2023

Free groups of ideals

We study the freeness of the group $\mathrm{Inv}(D)$ of invertible ideals of an integral domain $D$, and the freeness of some related groups of (fractional) ideals. We study the relation between $\mathrm{Inv}(D)$ and $\mathrm{Inv}(D_P)$, in particular in the locally finite case, and we analyze in more detail the case where $D$ is Noetherian (obtaining a characterization of when $\mathrm{Inv}(D)$ is free for one-dimensional analytically unramified Noetherian domains) and where $D$ is Prüfer.

math.AC↗
arXiv2023

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 $D$.

math.AC↗
arXiv2023

Boundness in almost Dedekind domains

We study different form of boundness for ideals of almost Dedekind domains, generalizing the notions of critical ideals, radical factorization, and SP-domains. We show that every almost Dedekind domain has at least one noncritical maximal ideals and, indeed, the set of noncritical maximal ideals is dense in the maximal space, with respect to the constructible topology; as a consequence, we show that every almost Dedekind domain is SP-scattered, and in particular that the group $\mathrm{Inv}(D)$ of invertible ideals of an almost Dedekind domain $D$ is always free. If $D$ is an almost Dedekind domain with nonzero Jacobson radical, we also show that there is at least one element whose ideal function is bounded.

math.AC↗
arXiv2022

The Golomb topology of polynomial rings, II

We study the interplay of the Golomb topology and the algebraic structure in polynomial rings $K[X]$ over a field $K$. In particular, we focus on infinite fields $K$ of positive characteristic such that the set of irreducible polynomials of $K[X]$ is dense in the Golomb space $G(K[X])$. We show that, in this case, the characteristic of $K$ is a topological invariant, and that any self-homeomorphism of $G(K[X])$ is the composition of multiplication by a unit and a ring automorphism of $K[X]$.

math.AC↗
arXiv2022

Radical semistar operations

We introduce and study the set of radical stable operations of an integral domain $D$. We show that their set is a complete lattice that is the join-completion of the set of spectral semistar operations, and we characterize when every radical operation is spectral (under the hypothesis that $D$ is rad-colon coherent). When $D$ is a Prüfer domain such that every set of minimal prime ideals is scattered, we completely classify stable semistar operations.

math.AC↗
arXiv2022

Localizations of integer-valued polynomials and of their Picard group

We prove a necessary and sufficient criterion for the ring of integer-valued polynomials to behave well under localization. Then, we study how the Picard group of $\mathrm{Int}(D)$ and the quotient group $\mathcal{P}(D):=\mathrm{Pic}(\mathrm{Int}(D))/\mathrm{Pic}(D)$ behave in relation to Jaffard, weak Jaffard and pre-Jaffard families; in particular, we show that $\mathcal{P}(D)\simeq\bigoplus\mathcal{P}(T)$ when $T$ ranges in a Jaffard family of $D$, and study when similar isomorphisms hold when $T$ ranges in a pre-Jaffard family. In particular, we show that the previous isomorphism holds when $D$ is an almost Dedekind domain such that the ring integer-valued polynomials behave well under localization and such that the maximal space of $D$ is scattered with respect to the inverse topology.

math.AC↗
arXiv2022

The polynomial closure is not topological

We characterize the polynomial closure of a pseudo-convergent sequence in a valuation domain $V$ of arbitrary rank, and then we use this result to show that the polynomial closure is never topological when $V$ has rank at least $2$.

math.AC↗
arXiv2022

Almost Dedekind domains without radical factorization

We study almost Dedekind domains with respect to the failure of ideals to have radical factorization, that is, we study how to measure how far an almost Dedekind domain is from being an SP-domain. To do so, we consider the maximal space $\mathcal{M}=\mathrm{Max}(R)$ of an almost Dedekind domain $R$, interpreting its (fractional) ideals as maps from $\mathcal{M}$ to $\mathbb{Z}$, and looking at the continuity of these maps when $\mathcal{M}$ is endowed with the inverse topology and $\mathbb{Z}$ with the discrete topology. We generalize the concept of critical ideals by introducing a well-ordered chain of closed subsets of $\mathcal{M}$ (of which the set of critical ideals is the first step) and use it to define the class of \emph{SP-scattered domains}, which includes the almost Dedekind domains such that $\mathcal{M}$ is scattered and, in particular, the almost Dedekind domains such that $\mathcal{M}$ is countable. We show that for this class of rings the group $\mathrm{Inv}(R)$ is free by expressing it as a direct sum of groups of continuous maps, and that, for every length function $\ell$ on $R$ and every ideal $I$ of $R$, the length of $R/I$ is equal to the length of $R/\mathrm{rad}(I)$.

math.AC↗