Localizations of integer-valued polynomials and of their Picard group

2022/05/23 by Dario Spirito, Spirito, Dario
Mathematics · #Advanced Topology and Set Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Meromorphic and Entire Functions #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2205.11312

openalex publication_date 2022/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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) behave in relation to Jaffard, weak Jaffard and pre-Jaffard families; in particular, we show that P(D)≃\bigoplusP(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.

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