On \(D\)-algebras between \(D[X]\) and \(\operatorname{Int}(D)\) (Q6556320)

Let \(D\) be an integral domain with quotient field \(K\), and \(\text{Int}(D) = \{ f \in K[X] \mid f(D) \subseteq D \}\) the ring of integer-valued polynomials on \(D\). The goal of the authors is to give conditions on \(D\) such that every \(D\)-algebra \(\mathbb B\) with \(D[X] \subset \mathbb B \subset \text{Int}(D)\) is locally free.\N\NHaving gathered local results in Section 1, the authors prove in Theorem 2.3 that the desired property holds for any \textit{locally essential} domain. The main idea is to show that in this case any \(\mathbb B\) has locally a \textit{regular} basis. In Section 3 the results are extended to several variables.