The Noetherian property in rings of integer-valued polynomials

Abstract:

Let $D$ be a Noetherian domain, $D\prime$ its integral closure, and $\operatorname {Int}(D)$ its ring of integer-valued polynomials in a single variable. It is shown that, if $D\prime$ has a maximal ideal $M\prime$ of height one for which $D\prime /M\prime$ is a finite field, then $\operatorname {Int}(D)$ is not Noetherian; indeed, if $M\prime$ is the only maximal ideal of $D\prime$ lying over $M\prime \cap D$, then not even $\operatorname {Spec}(\operatorname {Int}(D))$ is Noetherian. On the other hand, if every height-one maximal ideal of $D\prime$ has infinite residue field, then a sufficient condition for $\operatorname {Int}(D)$ to be Noetherian is that the global transform of $D$ is a finitely generated $D$-module.