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 Int(D) and the quotient group (Formula presented.) behave in relation to Jaffard, weak Jaffard, and pre-Jaffard families; in particular, we show that (Formula presented.) 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.