The local Picard group of a ring extension

2023/06/27 by Spirito, Dario · 2 citations
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2306.15439

Abstract

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 LPic_u(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 \bigoplusLPic(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.

Cited by

• Smooth sequences and overring operators
• Line bundles and exact sequences for the ideal class group and the Picard group

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