Note on integer-valued polynomials on a residually cofinite subset
Let \(D\) be an integral domain with quotient field \(K\), \(E\) a subset of \(K\) and \(X\) an indeterminate over \(K\). The ring \(\mathrm{Int}(E, D) := \{ f\in K[X] : f (E)\subseteq D\}\), of integer-valued polynomials on \(E\) with respect to \(D\), is known to be a \(D\)-algebra. In this paper, the author studied the faithful flatness, the local freeness and the Krull dimension of integer-valued polynomials on a residually cofinite subset over locally essential domains. The main resut stated that if \(D=\displaystyle\bigcap_{p\in \mathcal{P}} D_{p}\), where \(\mathcal{P}\subseteq\mathrm{Spec}(D)\), is a locally essential domain and \(E\subseteq D\) is residually cofinite with \(D\), then: \((1)\) \(\mathrm{Int}(E, D)\) is a faithfully flat \(D\)-module. \((2)\) Assume that \(E\) is infinite and \(D_{p}\) is a \(DVR\), for each \(p\in \mathcal{P}\) with \(D/p\) finite, then \(\mathrm{Int}(E, D)\) is a locally free \(D\)-module. In particular, if the domain \(D\) is either Krull-type or strong Mori, then \(\dim(\mathrm{Int}(E, D)) = \dim(D[X])\).
scientific article; zbMATH DE number 7375625