---
date: 2022-01-15 21:49
modification date: Wednesday 9th February 2022 19:03:54
title: smooth scheme
aliases: [smooth morphism, smooth, smooth scheme, smoothable]

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


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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [scheme](Unsorted/scheme.md)

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# smooth scheme

## For general schemes over a field

A scheme $X\in \Sch\slice k$  is **smooth** over $k$ iff 

- The base change $X_{\kbar}$ is [regular](Unsorted/regular%20scheme.md)
- The base changes $X_{L}$ are regular for every (arbitrary) extension $L/k$
- The base changes $X_{L}$ are regular for every finite extension $L/k$.
- There exists a [perfect](Unsorted/perfect%20field.md) extension $L/k$ such that the base change $X_{L}$ is regular.
- $X\to \spec k$ satisfies the [[infinitesimal criterion for smoothness]].

Note that smoothness over a field implies [regularity](Unsorted/regular%20scheme.md), but the converse is false if $k$ is not perfect.

## Geometric defiition, for finite type schemes

Let $X\in \Sch^\ft\slice k$, then there is a [closed immersion](closed%20immersion.md) $X\injects \AA^N_{/k}$ and $X = V(f_1, \cdots, f_m)$ for some $f_i\in \kx{N}$ . 
Say $X$ is **smooth of dimension $n$** iff 
	- There exist neighborhoods of each point $U_x\ni x$ with $\dim U_x \geq n$, and
	- $\rank Df \geq N-n$ everywhere, where $Df = \tv{ \dd{f_i}{x_j}}$ is the matrix of partial derivatives.
	- Equivalently, the dimension of the [Zariski tangent space](Zariski tangent space.md) at every point is *equal* to $n$.
	- At singular points, the dimension of the tangent space *increases*.

## For derived schemes

![](attachments/Pasted%20image%2020220420100946.png)


Recall that if $X$ and $S$ are smooth, then $\pi : X \to  S$ is smooth if and only the differential is everywhere surjective.

![](attachments/Pasted%20image%2020220209190353.png)

# Smoothable schemes
![](attachments/Pasted%20image%2020220411140920.png)

# Notes

- If a variety $X\in \Var\slice k$ is smooth then any [coherent sheaf](coherent%20sheaf) has a finite resolution by [locally free](locally%20free) sheaves of [finite type](finite%20type.md) and the subcategory of [perfect complexes](perfect%20complexes.md) coincides with the entire [bounded derived category](bounded%20derived%20category) $\derivedcat{\Coh(X)}^b$.