---
tag: 9537RKBZPP
kind: concept
name: Proper morphism
msc: [14A15]
specializes: [W0B0BBAH35, X30G4V7VAJ]
sources:
  - url: https://en.wikipedia.org/wiki/Proper_morphism
same_as:
  wikidata: Q7250176
  wikipedia: Proper morphism
---

## Summary

A proper morphism of schemes is a morphism that is separated, of finite type and universally closed.
It is the algebraic analogue of a proper map of topological spaces, and of a compact space when the target is a point.
Many finiteness results, such as the finiteness of cohomology of coherent sheaves and the existence of the Stein factorization, hold for proper morphisms.

## Explanations

### L1 human

A proper morphism is a family of spaces in which no fibre has points "missing at infinity".
Projective space over a field is proper over that field, while the affine line is not.
The affine line misses the point at infinity: a point moving off to infinity has no limit in it.
Properness rules out this kind of escape, and so it plays the role of compactness in algebraic geometry.