Localizations of integer-valued polynomials and of their Picard group

TL;DR

This paper establishes criteria for the localization behavior of integer-valued polynomial rings and analyzes the structure of their Picard groups, especially in relation to Jaffard and pre-Jaffard families, with applications to almost Dedekind domains.

Contribution

It provides a necessary and sufficient condition for integer-valued polynomial rings to behave well under localization and explores the Picard group structure in relation to specific families of domains.

Findings

Picard group decomposes as a direct sum over Jaffard families.

Conditions identified for Picard group isomorphisms in pre-Jaffard families.

Results apply to almost Dedekind domains with specific topological properties.

Abstract

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.