Localizations of integer‐valued polynomials and of their Picard group

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 Int(𝐷) and the quotient group (𝐷) ∶= Pic(Int(𝐷))∕Pic(𝐷) behave in relation to Jaffard, weak Jaffard, and pre-Jaffard families; in particular, we show that (𝐷) ≃ ⨁ (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.