Formal Fibers of Unique Factorization Domains

http://dx.doi.org/10.4153/CJM-2010-014-6
Canad. J. Math. 62(2010), 721-736

  Published:2009-12-04
 Printed: Aug 2010

Let $(T,M)$ be a complete local (Noetherian) ring such that $\dim T\geq 2$ and $|T|=|T/M|$ and let $\{p_i\} _{i \in \mathcal I}$ be a collection of elements of T indexed by a set $\mathcal I$ so that $|\mathcal I | < |T|$. For each $i \in \mathcal{I}$, let $C_i$:={$Q_{i1}$,$\dots$,$Q_{in_i}$} 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 ${\bf m}$-adic completion of a local unique factorization domain $(A, {\bf m})$, and for each $i \in \mathcal I$, 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.