The Local Picard Group of a Ring Extension

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