Localizations of integer‐valued polynomials and of their Picard group

Dario Spirito

Mathematische Nachrichten, 2023, vol. 296, issue 11, 5242-5258

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 P(D):=Pic(Int(D))/Pic(D)$\mathcal {P}(D):=\mathrm{Pic}(\mathrm{Int}(D))/\mathrm{Pic}(D)$ behave in relation to Jaffard, weak Jaffard, and pre‐Jaffard families; in particular, we show that P(D)≃⨁P(T)$\mathcal {P}(D)\simeq \bigoplus \mathcal {P}(T)$ 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.

Date: 2023
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