On the methods to construct UFD counterexamples to a cancellation problem

TL;DR

This paper discusses methods for constructing unique factorization domain (UFD) counterexamples to the cancellation problem, demonstrating their generality and potential for creating numerous similar examples over algebraically closed fields.

Contribution

It introduces a general framework for constructing UFD counterexamples to the cancellation problem, expanding on previous specific examples and enabling broader application.

Findings

Methods are very general and adaptable.

Constructed UFD examples over algebraically closed fields.

Framework allows creation of many new counterexamples.

Abstract

In a previous paper, the author together with prof. dr. Finston constructed a class of UFDs A_{n,m} where n,m\in \N^*. These rings are all stably equivalent (A_{n,m}[T]\cong A_{p,q}[T] for all n,m,p,q) but are only isomorphic themselves if (n,m)=(p,q). These examples are the first UFD examples over a characteristically closed field satisfying this behavior. In this paper, we describe the methods used in this article, and show that they are very general, enabling the reader to construct many more such examples, based on the same principles.