---
tag: C9NE52R27S
kind: definition
name: Universally closed morphism
msc: [14A15]
defines: W0B0BBAH35
uses: [JXRKH6P1Z4, 1DZMT4S9FB, 00FSJTXPJM]
sources:
  - url: https://en.wikipedia.org/wiki/Proper_morphism
---

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].
The morphism $f$ is *universally closed* if the following holds for every morphism of schemes $Y' \to Y$.
The base change
$$f' \colon X \times_Y Y' \to Y'$$
is a [[closed map|00FSJTXPJM]] of the underlying topological spaces.
Here $X \times_Y Y'$ is the [[fiber product|1DZMT4S9FB]] and $f'$ is the second projection.

## Remarks

A closed immersion $Z \to Y$ is universally closed.
Its base change along any $Y' \to Y$ is again a closed immersion, and a closed immersion is a homeomorphism onto a closed subset, so it is a closed map.
For a field $k$, the morphism $\mathbb{A}^1_k \to \operatorname{Spec} k$ is closed but not universally closed.
Its base change along $\mathbb{A}^1_k \to \operatorname{Spec} k$ is the projection $\mathbb{A}^2_k \to \mathbb{A}^1_k$, which sends the closed set $V(xy - 1)$ onto $\mathbb{A}^1_k \setminus \set{0}$, a set that is not closed.
Universally closed morphisms are stable under composition and base change.