---
tag: 1HD3EK5392
kind: statement
role: theorem
name: Regular local rings are UFDs
msc: [13H05, 13F15]
uses: [63XXTF6B5M, 0W4X9G6263, V3ZBMRKC08, 93RQRG9R1P, GGEMJF449D]
about: [63XXTF6B5M, 0W4X9G6263]
sources:
  - url: https://en.wikipedia.org/wiki/Auslander%E2%80%93Buchsbaum_theorem
same_as:
  wikidata: Q4822774
  wikipedia: Auslander–Buchsbaum theorem
---

## Statement

Let $R$ be a [[regular local ring|63XXTF6B5M]].
Then $R$ is a [[unique factorization domain|0W4X9G6263]].

## Remarks

This is the Auslander–Buchsbaum theorem.
Proof idea: $R$ is a domain by [[regular local rings are domains|V3ZBMRKC08]]; one argues by induction on $\dim R$.
Pick $x \in \mathfrak m \setminus \mathfrak m^2$; then $x$ is prime, since $R/xR$ is regular, hence a domain.
By [[Nagata's criterion|GGEMJF449D]], it is enough to show that $R[1/x]$ is a UFD.
Its localizations at primes are regular of smaller dimension, hence UFDs by induction.
So each height one prime of $R[1/x]$ is locally principal.
Finite free resolutions, which exist by [[Serre's theorem|93RQRG9R1P]], then show that it is principal.

Example: the power series ring $k[\![x_1, \ldots, x_n]\!]$ over a field $k$ and the localizations $k[x_1, \ldots, x_n]_{\mathfrak p}$ of a polynomial ring at prime ideals are UFDs.

The regularity hypothesis matters.
For a field $k$, the local ring $R = k[x, y, z]_{(x, y, z)}/(xy - z^2)$ is a domain of dimension $2$ and embedding dimension $3$.
It is not a UFD, since $xy = z \cdot z$ are two factorizations of the same element into irreducible elements that are not associate.

## Explanations

### L1 human

At a smooth point of a variety, every function factors uniquely into irreducible pieces, just as integers factor uniquely into primes.
At a singular point this can fail: on the cone $xy = z^2$ the function $z^2$ factors in two different ways.

### L2 human

The proof sketched above relies on homological algebra, through Serre's theorem and finite free resolutions.
In geometry it means that on a smooth variety every hypersurface through a point is defined locally by one equation, so Weil divisors and Cartier divisors agree.
It is used with the [[divisor class group|WN8PK6RQZR]]: the class group of a regular local ring is trivial.