Formal Fibers of Unique Factorization Domains

Abstract

Let $\left( T,\,M \right)$ be a complete local (Noetherian) ring such that $\dim\,T\,\ge \,2$ and $\left| T \right|\,=\,\left| T/M \right|$ and let ${{\left\{ {{p}_{i}} \right\}}_{i\in \Im }}$ be a collection of elements of $T$ indexed by a set $\mathcal{J}$ so that $\left| \mathcal{J} \right|\,<\,\left| T \right|$. For each $i\,\in \,\mathcal{J}$, let ${{C}_{i}}:=\left\{ {{Q}_{i1}},...,{{Q}_{i{{n}_{i}}}} \right\}$ be a set of nonmaximal prime ideals containing ${{p}_{i}}$ such that the ${{Q}_{ij}}$ are incomparable and ${{p}_{i}}\in {{Q}_{jk}}$ if and only if $i\,=\,j$. We provide necessary and sufficient conditions so that $T$ is the $\mathbf{m}$-adic completion of a local unique factorization domain $\left( A,\,\mathbf{m} \right)$, and for each $i\,\in \,\mathcal{J}$, there exists a unit ${{t}_{i}}$ of $T$ so that ${{p}_{i}}{{t}_{i}}\in A$ and ${{C}_{i}}$ is the set of prime ideals $Q$ of $T$ that are maximal with respect to the condition that $Q\cap A={{p}_{i}}{{t}_{i}}A$.

We then use this result to construct a (nonexcellent) unique factorization domain containing many ideals for which tight closure and completion do not commute. As another application, we construct a unique factorization domain $A$ most of whose formal fibers are geometrically regular.