The Local Picard Group of a Ring Extension

Given an integral domain D and a D-algebra R, we introduce the local Picard group LPic(R,D) as the quotient between the Picard group Pic(R) and the canonical image of Pic(D) in Pic(R), and its subgroup LPicu(R,D) generated by the the integral ideals of R that are unitary with respect to D. We show that, when D⊆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 Int(D)), it is possible to decompose LPic(R,D) as the direct sum ⨁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.