A Counterexample to Problem 19 on Integer-valued Polynomial Rings

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.The proof presented in this note was completed by Rethlas, a natural-language automated reasoning system; the author was responsible for reviewing and checking the argument.