Unique Factorization of Power Series Rings over Q-UFDs of Dimension at Most 3
Abstract
Let R be a Noetherian unique factorization domain containing ℚ, with dim R≤3. We prove that R[[t]] is a unique factorization domain if and only if two conditions hold: every two-generated parameter ideal q in every height-two localization satisfies q2̅⊆ q, and every height-three localization has depth three. No excellence or completion hypothesis is imposed. The main local result is that A[[t]] is not a UFD whenever A is a local ℚ-UFD of dimension at least three and depth two. We also prove that the UFD property of R[[t]] passes to every coefficient localization in dimension at most three, give an all-dimensional criterion when the completions of the maximal localizations satisfy Serre's condition (S2), and obtain an arbitrary-characteristic criterion for excellent UFDs of dimension at most three.
Related
- Unique factorization results for generalized power series
- Bouvier's Conjecture and Dimension Sequences of Unique Factorization Domains
- The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring
- Spherical Completeness, Coherence, and GCD Properties of Formal Power Series and Witt Vector Rings
- The prime spectrum of rings of integer-coefficient power series on the disk
BibTeX
@misc{paran2026unique,
title = {{Unique Factorization of Power Series Rings over Q-UFDs of Dimension at Most 3}},
author = {Elad Paran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24551},
eprint = {2610.24551},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24551},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unique Factorization of Power Series Rings over Q-UFDs of Dimension at Most 3
AU - Elad Paran
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24551
UR - https://arxiv.org/abs/2610.24551
AB - Let R be a Noetherian unique factorization domain containing ℚ, with dim R≤3. We prove that R[[t]] is a unique factorization domain if and only if two conditions hold: every two-generated parameter ideal q in every height-two localization satisfies q^2̅⊆ q, and every height-three localization has depth three. No excellence or completion hypothesis is imposed. The main local result is that A[[t]] is not a UFD whenever A is a local ℚ-UFD of dimension at least three and depth two. We also prove that the UFD property of R[[t]] passes to every coefficient localization in dimension at most three, give an all-dimensional criterion when the completions of the maximal localizations satisfy Serre's condition (S_2), and obtain an arbitrary-characteristic criterion for excellent UFDs of dimension at most three.
ER -