---
tag: AHTE5DVX61
kind: concept
name: Morphism of finite type
msc: [14A15]
specializes: [41009J4M5D]
sources:
  - url: https://en.wikipedia.org/wiki/Morphism_of_finite_type
same_as:
  wikidata: Q112108702
  wikipedia: Morphism of finite type
---

## Summary

A morphism of finite type $X \to Y$ is a [[morphism of schemes|JXRKH6P1Z4]] described by finitely many coordinates over $Y$.
Over each affine piece $\operatorname{Spec} B$ of $Y$, the source is covered by finitely many spectra of finitely generated $B$-algebras.
It is the relative version of a [[finitely generated algebra|AC5TFFQT9M]].
Varieties over a field $k$ are schemes of finite type over $\operatorname{Spec} k$.

## Explanations

### L1 human

A scheme of finite type over a field is one that can be described by finitely many coordinates and equations on each of finitely many pieces.
The plane curve $y^2 = x^3 - x$ over $\Q$ is of finite type: it is $\operatorname{Spec} \Q[x, y]/(y^2 - x^3 + x)$.
The spectrum of the field $\Q(x)$ of rational functions is not of finite type over $\Q$, because no finite set of elements generates $\Q(x)$ as an algebra.
Most of classical algebraic geometry deals with schemes of finite type over a field.