Smooth morphism

Builds on Morphism of schemes Flat morphism Fiber of a morphism Regular scheme Finitely presented algebra

Definition

Let $f \colon X \to Y$ be a morphism of schemes. The morphism $f$ is smooth if it satisfies the following three conditions.

$f$ is locally of finite presentation. This means that, for all affine opens $\operatorname{Spec} B \subseteq X$ and $\operatorname{Spec} A \subseteq Y$ with $f(\operatorname{Spec} B) \subseteq \operatorname{Spec} A$ , the $A$ -algebra $B$ is finitely presented.
$f$ is flat.
For every $y \in Y$ with residue field $\kappa(y)$ , the fiber $X_y$ is geometrically regular over $\kappa(y)$ . This means that $X_y \times_{\kappa(y)} \operatorname{Spec} \overline{\kappa(y)}$ is a regular scheme, where $\overline{\kappa(y)}$ is an algebraic closure of $\kappa(y)$ .

Remarks

For a ring $A$ , the projection $\mathbb{A}^n_A \to \operatorname{Spec} A$ is smooth. It is of finite presentation and flat, and each fibre is affine space over a field, which stays regular over the algebraic closure. For a field $k$ , the morphism $X = \operatorname{Spec} k[x, y]/(y^2 - x^3) \to \operatorname{Spec} k$ is not smooth. Clause 3 fails, since the local ring at the origin has maximal ideal $(x, y)$ , which needs two generators while the curve has dimension one. Equivalently, $f$ is smooth of relative dimension $n$ if and only if three conditions hold. It is flat and locally of finite presentation, its nonempty fibres have pure dimension $n$ , and $\Omega_{X/Y}$ is locally free of rank $n$ . The ring version of this notion is a smooth morphism of rings.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`R123NEWYZR`
URI	`m2://R123NEWYZR`
Kind	definition
Writing	W0 stub
MSC	14A15, 14B25
Defines	Smooth morphism
Uses	Morphism of schemes, Flat morphism, Fiber of a morphism, Regular scheme, Finitely presented algebra
Changed	2 November 2026

Smooth morphism

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here