---
tag: ESPG0BJERZ
kind: concept
name: Regular scheme
msc: [14A15, 13H05]
specializes: [EK261G1XP9]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_scheme
same_as:
  wikidata: Q18393924
  wikipedia: Regular scheme
---

## Summary

A regular scheme is a locally Noetherian [[scheme|5A35TYZD02]] whose local rings are all [[regular local rings|63XXTF6B5M]].
It is the algebraic notion of a space without singularities that works over any base, including $\Z$.
Over a perfect field, a scheme of finite type is regular exactly when it is smooth.

## Explanations

### L1 human

At a smooth point of a surface, two coordinates are enough to describe everything nearby.
At a singular point, such as the tip of a cone, more are needed than the dimension.
Regularity says exactly this, in algebra: near each point, the number of local generators equals the dimension.
The plane is regular, and so is $\operatorname{Spec} \Z$, but the cusp $y^2 = x^3$ is not regular at the origin.