---
tag: EXKGG9X6J0
kind: concept
name: Smooth morphism
msc: [14A15, 14B25]
specializes: [5M6KS8F47R]
sources:
  - url: https://en.wikipedia.org/wiki/Smooth_morphism
same_as:
  wikidata: Q7546425
  wikipedia: Smooth morphism
---

## Summary

A smooth morphism of schemes is a flat morphism, locally of finite presentation, whose fibres are smooth schemes over their residue fields.
It is the algebraic analogue of a submersion of differentiable manifolds.
A variety over a field is smooth exactly when its structure morphism is smooth, and smooth morphisms are stable under composition and base change.

## Explanations

### L1 human

A smooth morphism is a family of smooth spaces that fit together without jumps.
Near each point, it looks like the projection of a product of the target with an affine space, up to an étale change of coordinates.
The projection $\mathbb{A}^2 \to \mathbb{A}^1$ is smooth: each fibre is a line.
The map from the curve $y^2 = x^2(x + 1)$ to a point is not smooth, because the curve has a node at the origin.