A Counterexample to Problem 19 on Integer-valued Polynomial Rings

TL;DR

This paper provides a counterexample showing that the ring of integer-valued polynomials on a certain domain is not flat or free, answering a longstanding open problem negatively.

Contribution

It constructs an explicit one-dimensional Noetherian local domain over a finite field where the integer-valued polynomial ring is not flat, challenging previous assumptions.

Findings

The ring of integer-valued polynomials on the constructed domain is not flat as a module.

The counterexample demonstrates that such rings need not be free.

An application of Elliott's flatness criterion is used to establish the counterexample.

Abstract

We give a negative answer to Problem 19 of Cahen, Fontana, Frisch, and Glaz concerning the flatness and freeness of rings of integer-valued polynomials. We construct an explicit one-dimensional Noetherian local domain D over the field with two elements and prove that the ring of integer-valued polynomials on D is not flat as a D-module. The argument shows that a certain polynomial is integer-valued on D with values in the integral closure T of D, but does not belong to the product of T with the ring of integer-valued polynomials on D. An application of Elliott's flatness criterion then yields the counterexample. In particular, the ring of integer-valued polynomials on an arbitrary integral domain need not be free.