---
date: 2022-01-15 21:49
modification date: Monday 24th January 2022 14:59:51
title: regular ring
aliases: ["regular ring", "regular", "regular local ring", "regular rings", "regular local rings"]

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Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #CA #AG/basics 
- Refs:
	- #todo/add-references
- Links:
	- [Commutative Algebra resources](Projects/2022%20Advanced%20Qual%20Projects/Commutative%20Algebra/000%20Resources.md)

---

# regular ring

Idea: regular local rings correspond to [smooth points](smooth%20points.md) on [schemes](schemes). In general, regular rings $\subseteq$ [Gorenstein](Unsorted/Gorenstein.md) rings $\subseteq$ [Cohen-Macaulay](Unsorted/Cohen-Macaulay.md) rings.

# Regular local rings

Let $R$ be a local Noetherian ring and let $n$ be the minimal number of generators of its maximal ideal $\mfm_R$. By [Krull's intersection theorem](Unsorted/Krull's%20intersection%20theorem.md), 
$$n\geq \krulldim(R),$$so define $R$ to be  **regular** iff this is an equality.
Note that a minimal set of generators is a [[regular system]].

Equivalently, $R$ is regular iff its [residue fields](residue%20field) $\kappa(R) \da R/\mfm$ satisfy
$$\dim_{\kappa(R)} \mfm/\mfm^2 = \krulldim(R).$$

Equivalently, $R$ is regular iff $R$ has finite [[global dimension]].

# Regular rings

An arbitrary commutative ring $R$ is **regular** iff the prime localizations $R\plocalize{\mfp}$ are regular local rings (as above) for every $\mfp \in \spec R$.

# Examples

- Regular rings:
	- Fields are regular local.
	- Formal power series rings $k\fps{t_1, t_2,\cdots, t_m}$ in finitely many indeterminates over a field are regular local.
	- Every [DVR](DVR.md) is regular local.
	- The [p-adic integers](Unsorted/p-adic.md) $\ZZpadic$ are DVRs and thus regular local.
	- Formal power series rings $A\fps{x_0, \cdots, x_n}$ over a regular local ring are again regular local.
	- Any localization of a regular ring.
- Non-regular rings:
	- $k[x]/\gens{x^2}$ #todo/exericse

# Results

- See [Serre's criterion](Serre's%20criterion) for regularity.
	- [Localizations](Unsorted/localization%20of%20rings.md) and [completions](completion) of regular local rings are regular.
- Consequence for [schemes](Unsorted/scheme.md): a point $x\in X\in\Sch\slice k$ is a **smooth point** iff the [stalk](stalk) $\OO_{X, x}$ is a regular local ring.