---
date: 2022-04-05 23:42
modification date: Monday 18th April 2022 11:55:23
title: "regular scheme"
aliases: [regular scheme, regular, regular in codimension one, regularity]
---

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

---
- Tags: 
	- #AG/basics 
- Refs:
	- #todo/add-references
- Links:
	- [smooth scheme](Unsorted/smooth%20scheme.md)
	- [closed point](Unsorted/special%20fiber.md)
	- [Cohen-Macaulay](Unsorted/Cohen-Macaulay.md)
	- [Noetherian scheme](Unsorted/Noetherian%20scheme.md)


---

# regular scheme

A scheme $X$ is **regular** iff every local ring $\OO_{X, x}$ is a [[regular ring]]. 

Note that it is sufficient to check this at closed points $x\in\abs{X}$ by [[Serre's criterion]].

Used in the definition of [smoothness](Unsorted/smooth%20scheme.md).

Idea: Dimension 1 regular Noetherian local rings are like germs of smooth curves.

## Examples

[smooth](Unsorted/smooth%20scheme.md) implies regular but not conversely: let $k$ be an imperfect field of positive odd prime characteristic $p$ and let $a\in k \sm k^p$ be an element which is not a $p$th power.
Then the curve $X = V(y^2=x^p-a) \subseteq \AA^2\slice k$ is a [[Dedekind scheme]] but its base change $X_{\kbar} = V(y^2=z^p)$ where $z\da x-a^{1\over p}$ is not regular at the origin.


# regular in codimension one

- A scheme $X$ is **regular in codimension 1** iff every local ring $R$ of dimension 1 is regular (so $\dim_k \mfm_R/\mfm_R^2 = 1$ for the unique $\mfm_R \in \mspec R$).
	- Often translates to smooth in codimension 1.

- Technical condition needed to make sense of [Weil divisors](Unsorted/Weil%20divisor.md).