Maps on Divisor Class Groups Induced by Ring Homomorphisms of Finite Flat Dimension

TL;DR

This paper develops a general criterion for when ring homomorphisms induce maps on divisor class groups, extending classical cases to those with finite flat dimension, and explores the kernels of these induced maps.

Contribution

It introduces a broad criterion for induced divisor class group maps via ring homomorphisms with finite flat dimension, generalizing previous special cases.

Findings

Criterion applies to homomorphisms with finite flat dimension

Extension of Spiroff's work on kernels of induced maps

Generalizes classical flat or surjective cases

Abstract

Let f: A\to B be a ring homomorphism between Noetherian normal integral domains. We establish a general criterion for f to induce a homomorphism Cl(f): Cl(A)\to Cl(B) on divisor class groups. For instance, this criterion applies whenever f has finite flat dimension; this special case generalizes the more classical situations where f is flat or is surjective with kernel generated by an A-regular element. We extend some of Spiroff's work on the kernels of induced maps to this more general setting.