---
tag: N9C4X7XKE4
kind: concept
name: Finite morphism
msc: [14A15]
specializes: [AFC6JMYWEH, AHTE5DVX61]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_morphism
same_as:
  wikidata: Q2370289
  wikipedia: Finite morphism
---

## Summary

A finite morphism $X \to Y$ of schemes is affine over $Y$.
Over each affine piece $\operatorname{Spec} B$ of $Y$, it comes from a $B$-algebra that is a finitely generated $B$-module.
It is the scheme version of a [[finite morphism of rings|1NBACAEC2V]].
Finite morphisms have finite fibers and are closed maps; branched covers of curves are typical examples.

## Explanations

### L1 human

A finite morphism is like a map where each point has finitely many points above it, and nothing escapes to infinity.
The map $t \mapsto t^2$ from the complex line to the complex line is finite: above each nonzero point there are two points, and above $0$ there is one.
The inclusion of the punctured line into the line is not finite, because points near the removed origin escape.
Closed immersions are finite as well.