On the Krull dimensions of rings of integer-valued polynomials and polynomial rings

Abstract

It was conjectured that every integral domain $D$ satisfies $\dim \mathrm{Int}(D)\le \dim D[X]$. We disprove this conjecture and, more strongly, classify all pairs for which both dimensions are finite. Writing $m=\dim D[X]$ and $n=\dim \mathrm{Int}(D)$, the pairs that occur are precisely $\{(1,1),(2,2)\}\cup \{(m,n)\in {\mathbb{Z}}_{>0}^2:m\ge 3,n\ge m-1\}$.