# Smooth scheme

In algebraic geometry, a smooth scheme X of dimension n over an algebraically closed field k is an algebraic scheme[1] that is regular and has dimension n. More generally, an algebraic scheme over a field k is said to be smooth if ${\displaystyle X\times _{k}{\overline {k}}}$ is smooth for any algebraic closure ${\displaystyle {\overline {k}}}$ of k.

If k is perfect, then an algebraic scheme over k is smooth if and only if it is regular.

There is also a notion of a "smooth morphism" between schemes, and the above definition coincides with it. That is, an algebraic scheme X over k is smooth of dimension n if and only if ${\displaystyle X\to \operatorname {Spec} k}$ is smooth of relative dimension n.

## Properties

A smooth scheme is connected if and only if it is irreducible. A connected smooth scheme is normal.{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

## Generic smoothness

A scheme X is said to be generically smooth of dimension n over k if X contains an open dense subset that is smooth of dimension n over k. Any integral scheme over a perfect field (in particular an algebraically closed field) is generically smooth.

## Examples

Examples of smooth schemes are:

## Notes

1. By "algebraic scheme" we mean a scheme of finite type over a field.