---
tag: ZHBREAF3MJ
kind: definition
name: Smooth morphism of rings
msc: [13B40, 14B25, 13D10]
defines: 286G0P1B69
uses: [0310REJJY8, DPBJ04JMH8, 764GZHTGV8, 8KPDNVKX7B, 7HCFARHG7D, 73RYYEVDKA, 0SDC934R52, MVGBS3J1P7]
sources:
  - url: https://en.wikipedia.org/wiki/Formally_smooth_map
---

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $A$ be a commutative [[$R$-algebra|DPBJ04JMH8]].
Then $A$ is *smooth* over $R$, and the structure map $R \to A$ is a *smooth morphism*, if both conditions hold:

1. $A$ is a [[finitely presented|764GZHTGV8]] $R$-algebra.
2. $A$ is [[formally smooth|8KPDNVKX7B]] over $R$.

## Remarks

If $A$ is smooth over $R$, then $A$ is [[flat|73RYYEVDKA]] over $R$, and the module of [[Kähler differentials|7HCFARHG7D]] $\Omega_{A/R}$ is a finitely generated projective $A$-module.

Example: the [[polynomial ring|0SDC934R52]] $A = R[x_1, \ldots, x_n]$ is smooth over $R$.
It is finitely presented, with no relations, and it is formally smooth, because a lift is obtained by lifting the images of the variables.

Non-example: let $k$ be a field and $A = k[x_1, x_2, x_3, \ldots]$ in infinitely many variables.
It is formally smooth over $k$ by the same argument, but it is not finitely generated, so clause 1 fails.

Another non-example: $A = k[x, y]/(y^2 - x^3)$ is finitely presented over $k$ but not smooth.
Here $\Omega_{A/k}$ is generated by $dx, dy$ with the one relation $2y \, dy - 3x^2 \, dx = 0$.
At the maximal ideal $\mathfrak m = (x, y)$, this relation vanishes modulo $\mathfrak m$, so $\Omega_{A/k} \otimes_A A/\mathfrak m$ has dimension $2$.
A smooth algebra over a field has differentials that are locally free of rank equal to the local [[Krull dimension|MVGBS3J1P7]], which is $1$ here.