---
tag: 93RQRG9R1P
kind: statement
role: theorem
name: Serre's theorem on regular local rings
msc: [13H05, 13D05]
uses: [MZ02BKPHJA, 0670T59HGE, NJ3APQ8BHH, 63XXTF6B5M, GWA0T1WKCN, MR9QCH1PNA, MVGBS3J1P7]
about: [63XXTF6B5M, GWA0T1WKCN]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_local_ring
---

## Statement

Let $(R, \mathfrak{m}, k)$ be a commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] with [[residue field|NJ3APQ8BHH]] $k$.
The following are equivalent:

1. $R$ is a [[regular local ring|63XXTF6B5M]].
2. $R$ has finite [[global dimension|GWA0T1WKCN]].
3. The [[projective dimension|MR9QCH1PNA]] $\operatorname{pd}_R k$ is finite.

When they hold, $\operatorname{gl.dim} R = \operatorname{pd}_R k = \dim R$, where $\dim R$ is the [[Krull dimension|MVGBS3J1P7]].

## Remarks

If $R$ is regular, then $\mathfrak{m}$ is generated by a regular sequence of length $\dim R$, and its [[Koszul complex|YZY11W08SD]] is a free resolution of $k$ of that length.
For a Noetherian local ring, the global dimension equals $\operatorname{pd}_R k$, which gives the implication from 1 to 2.
The converse is the deep part: a finite resolution of $k$ forces the minimal number of generators of $\mathfrak{m}$ to equal $\dim R$.
Example: $R = K[x]/(x^2)$ is not regular, and indeed $\cdots \to R \xrightarrow{x} R \xrightarrow{x} R \to K \to 0$ is a minimal resolution of $K$ that never stops.
A major consequence is that every localization of a regular local ring at a prime ideal is again regular.

## Explanations

### L1 human

Regular local rings are the algebraic version of smooth points of a space.
The theorem says that smoothness can be detected by homological algebra alone: a local ring is regular exactly when every module has a finite resolution.
It is enough to test one module, the residue field.

### L2 human

The theorem was proved by Serre and, independently, by Auslander and Buchsbaum.
The homological criterion gives a short proof that a localization of a regular local ring is regular, since finite global dimension passes to localizations.
It also gives [[Hilbert's syzygy theorem|R0DSZPRG6T]] in the local setting, and, combined with the [[Auslander–Buchsbaum formula|SQB74J5E8Q]], it controls the resolutions of all finitely generated modules over a regular local ring.