Coherence conditions in flat regular pullbacks

TL;DR

This paper explores how four coherence-like properties behave in regular conductor pullback rings, providing necessary and sufficient conditions for these properties to hold, with applications to rings of integer-valued polynomials.

Contribution

It establishes precise criteria for when regular conductor pullbacks exhibit various coherence-like properties, extending understanding of their algebraic structure.

Findings

Characterizes when pullback rings are finite conductor, coherent, GCD, or quasi-coherent.

Provides exact conditions for integer-valued polynomial rings to have these properties.

Enhances understanding of coherence conditions in algebraic constructions.

Abstract

We investigate the behavior of four coherent-like conditions in regular conductor squares. In particular, we find necessary and sufficient conditions in order that a pullback ring be a finite conductor ring, a coherent ring, a generalized GCD ring, or quasi-coherent ring. As an application of these results, we are able to determine exactly when the ring of integer-valued polynomials determined by a finite subset possesses one of the four coherent-like properties.