Note on the Krull dimension of rings of integer-valued polynomials
M. M. Chems-Eddin, B. Feryouch, A. Tamoussit
TL;DR
The paper investigates the Krull dimension of rings of integer-valued polynomials $\mathrm{Int}(E,D)$ and its extension $\mathrm{Int}_B(E,D)$ across various integral domains $D$, subsets $E$, and overrings $B$. It extends theories of Cahen–Chabert and Tartarone by proving that for a divided prime $p$ with infinite residue field and suitable $E$, $\mathrm{Int}(E,D)_p = D_p[X]$ and $\mathrm{Int}(E,D)/p[X] \cong \mathrm{Int}(E/p, D/p)$, yielding $\dim(\mathrm{Int}(E,D)) = \dim(D_p[X]) + \dim(\mathrm{Int}(E/p, D/p)) - 1$. Moreover, if $\mathrm{Int}(E,D) \subseteq B[X]$ for a pair $D \subset B$ sharing a finite quotient $D/I$, then $\dim(\mathrm{Int}(E,D)) = \dim(B[X])$, and the authors derive corollaries for finite-character domains, locally PVDs, almost Krull and $t$-dimensional one cases, plus results for $\mathrm{Int}_B(E,D)$ with dimension bounds and PVD/Jaffard-type conclusions. The work also develops local-to-global dimension bounds via overrings and valuative dimensions, and includes concrete PVD/Jaffard-type examples to illustrate the theory. These results offer a framework for constructing non-Noetherian rings with prescribed Krull dimension via rings of integer-valued polynomials.
Abstract
Let $D$ be an integral domain with quotient field $K,$ $E$ a subset of $K$ and $X$ an indeterminate over $K$. The set $\mathrm{Int}(E,D):=\{f\in K[X];\; f(E)\subseteq D\}$, of integer-valued polynomials on $E$ over $D$, is known to be an integral domain. The purpose of this note is to calculate the Krull dimension of $\mathrm{Int}(E,D)$ across various classes of integral domains $D$ and specific subsets $E$ of $D$. We further extend our study to the ring $\mathrm{Int}_B(E,D):=\{f\in B[X];\; f(E)\subseteq D\},$ where $B$ is an integral domain containing $D$