---
tag: 286G0P1B69
kind: concept
name: Smooth morphism of rings
msc: [13B40, 14B25, 13D10]
specializes: [8KPDNVKX7B]
sources:
  - url: https://en.wikipedia.org/wiki/Formally_smooth_map
---

## Summary

A smooth morphism of rings is a map $R \to A$ of commutative rings that makes $A$ a [[finitely presented|764GZHTGV8]] $R$-algebra and is [[formally smooth|8KPDNVKX7B]].
It is the algebraic counterpart of a submersion of manifolds: the fibres are nonsingular and vary nicely.
A smooth algebra is flat over the base, and its module of [[Kähler differentials|7HCFARHG7D]] is finitely generated and projective.
Over a field, smooth algebras are the coordinate rings of nonsingular varieties, in a sense that is stable under extension of the field.

## Explanations

### L1 human

A polynomial ring $R[x_1, \ldots, x_n]$ is the model of a smooth algebra: it is like flat space over $R$.
A smooth algebra looks, near each point and to first order, like such a polynomial ring.
The circle $x^2 + y^2 = 1$ over the real numbers gives a smooth algebra $\R[x, y]/(x^2 + y^2 - 1)$.
The cusp $y^2 = x^3$ does not, because of its sharp point at the origin.
Smoothness is the property that lets one extend solutions step by step, as in Hensel's lemma.