---
tag: 41009J4M5D
kind: concept
name: Quasi-compact morphism
msc: [14A15]
specializes: [JXRKH6P1Z4]
sources:
  - url: https://en.wikipedia.org/wiki/Quasi-compact_morphism
same_as:
  wikidata: Q7269441
  wikipedia: Quasi-compact morphism
---

## Summary

A quasi-compact morphism of schemes is one for which the preimage of every open affine subscheme is quasi-compact.
Quasi-compact means that every open cover has a finite subcover; no Hausdorff condition is asked.
It is a finiteness condition on a [[morphism of schemes|JXRKH6P1Z4]].
It is part of the definition of morphisms of finite type and of proper morphisms.

## Explanations

### L1 human

Spaces in algebraic geometry are rarely Hausdorff, so "compact" is replaced by "quasi-compact": every open cover has a finite subcover.
Every affine scheme is quasi-compact.
A morphism is quasi-compact when, over each affine piece of the target, the source can be covered by finitely many affine pieces.
Mapping infinitely many separate points onto a single point fails this test.