## Journal of Commutative Algebra

- J. Commut. Algebra
- Volume 10, Number 4 (2018), 475-498.

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

Sarah M. Fleming, Lena Ji, S. Loepp, Peter M. McDonald, Nina Pande, and David Schwein

#### 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 $t$.

#### Article information

**Source**

J. Commut. Algebra, Volume 10, Number 4 (2018), 475-498.

**Dates**

First available in Project Euclid: 16 December 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.jca/1544950826

**Digital Object Identifier**

doi:10.1216/JCA-2018-10-4-475

**Mathematical Reviews number (MathSciNet)**

MR3892144

**Zentralblatt MATH identifier**

07003224

**Subjects**

Primary: 13B35: Completion [See also 13J10] 13J10: Complete rings, completion [See also 13B35]

**Keywords**

Completions of local rings formal fibers

#### Citation

Fleming, Sarah M.; Loepp, Lena Ji, S.; McDonald, Peter M.; Pande, Nina; Schwein, David. Controlling the dimensions of formal fibers of a unique factorization domain at the height one prime ideals. J. Commut. Algebra 10 (2018), no. 4, 475--498. doi:10.1216/JCA-2018-10-4-475. https://projecteuclid.org/euclid.jca/1544950826