---
tag: EK261G1XP9
kind: concept
name: Normal scheme
msc: [14A15, 13B22]
specializes: [ZXSN8D4MCB]
sources:
  - url: https://en.wikipedia.org/wiki/Normal_scheme
same_as:
  wikidata: Q7051829
  wikipedia: Normal scheme
---

## Summary

A normal scheme is a [[scheme|5A35TYZD02]] whose local rings are [[integrally closed domains|H20P600HQS]].
Normality is a mild regularity condition.
Over an algebraically closed field, a normal curve is smooth, and a normal variety is smooth outside a closed set of codimension at least $2$.
Every integral scheme has a normal model, its normalization.

## Explanations

### L1 human

Normal schemes have no singularities of the simplest kinds, such as a curve crossing itself or a cusp.
The test is algebraic: a fraction that satisfies a monic polynomial equation over the local ring must already lie in the ring.
On the cusp $y^2 = x^3$, the fraction $t = y/x$ satisfies $t^2 = x$ but is not a polynomial function on the cusp, so the cusp is not normal.
The plane and $\operatorname{Spec} \Z$ are normal.