Controlling the dimensions of formal fibers of a unique factorization domain at the height one prime ideals

Abstract

Let $T$ be a complete local (Noetherian) equidimensional ring with maximal ideal $\mathfrak{m} $ such that the Krull dimension of $T$ is at least two and the depth of $T$ is at least two. Suppose that no integer of $T$ is a zerodivisor and that $|T|=|T/\mathfrak{m} |$. Let $d$ and $t$ be integers such that $1\leq d \leq \dim T-1$, $0 \leq t \leq \dim T - 1$ and $d - 1 \leq t$. Assume that, for every $\mathfrak{p} \in Ass T$, $ht \mathfrak{p} \leq d-1$ and that if $z$ is a regular element of $T$ and $Q \in Ass (T/zT)$, then $ht Q \leq d$. We construct a local unique factorization domain $A$ such that the completion of $A$ is $T$ and such that the dimension of the formal fiber ring at every height one prime ideal of $A$ is $d - 1$ and the dimension of the formal fiber ring of $A$ at $(0)$ is $t$.