The local Picard group of a ring extension

TL;DR

This paper introduces the local Picard group for ring extensions and explores its decomposition properties, especially for polynomial and integer-valued polynomial rings, under certain algebraic conditions.

Contribution

It defines the local Picard group for D-algebras and establishes its decomposition as a direct sum over Jaffard families, extending to pre-Jaffard families under specific hypotheses.

Findings

Decomposition of LPic(R,D) as a direct sum over Jaffard families.

Identification of conditions for the isomorphism to hold for pre-Jaffard families.

Application to polynomial rings and integer-valued polynomial rings.

Abstract

Given an integral domain $D$ and a $D$-algebra $R$, we introduce the local Picard group $\mathrm{LPic}(R,D)$ as the quotient between the Picard group $\mathrm{Pic}(R)$ and the canonical image of $\mathrm{Pic}(D)$ in $\mathrm{Pic}(R)$, and its subgroup $\mathrm{LPic}_u(R,D)$ generated by the the integral ideals of $R$ that are unitary with respect to $D$. We show that, when $D\subseteq R$ is a ring extension that satisfies certain properties (for example, when $R$ is the ring of polynomial $D[X]$ or the ring of integer-valued polynomials $\mathrm{Int}(D)$), it is possible to decompose $\mathrm{LPic}(R,D)$ as the direct sum $\bigoplus\mathrm{LPic}(RT,T)$, where $T$ ranges in a Jaffard family of $D$. We also study under what hypothesis this isomorphism holds for pre-Jaffard families of $D$.