finite morphism

Affine schemes

Let X and Y be affine schemes, so that X=SpecA and Y=SpecB. Let f:XY be a morphismMathworldPlanetmath, so that it induces a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of rings g:BA.

The homomorphism g makes A into a B-algebraMathworldPlanetmathPlanetmath. If A is finitely-generated as a B-algebra, then f is said to be a morphism of finite type.

If A is in fact finitely generatedMathworldPlanetmathPlanetmathPlanetmath as a B-module, then f is said to be a finite morphism.

For example, if k is a field, the scheme 𝔸n(k) has a natural morphism to Speck induced by the ring homomorphism kk[X1,,Xn]. This is a morphism of finite type, but if n>0 then it is not a finite morphism.

On the other hand, if we take the affine scheme Speck[X,Y]/Y2-X3-X, it has a natural morphism to 𝔸1 given by the ring homomorphism k[X]k[X,Y]/Y2-X3-X. Then this morphism is a finite morphism. As a morphism of schemes, we see that every fiber is finite.

General schemes

Now, let X and Y be arbitrary schemes, and let f:XY be a morphism. We say that f is of finite type if there exist an open cover of Y by affine schemes {Ui} and a finite open cover of each Ui by affine schemes {Vij} such that f|Vij is a morphism of finite type for every i and j. We say that f is finite if there exists an open cover of Y by affine schemes {Ui} such that each inverse image, Vi=f-1(Ui) is itself affine, and such that f|Vi is a finite morphism of affine schemes.


Let X=1(k) and Y=Speck. We cover X by two copies of 𝔸1 and consider the natural morphisms from each of these copies to Speck. Both of these affine morphisms are of finite type, but are not finite. The covering morphisms patch together to give a morphism from 1 to Speck. The overall morphism is of finite type, but again is not finite.


D. Eisenbud and J. Harris, The Geometry of Schemes, Springer.

Title finite morphism
Canonical name FiniteMorphism
Date of creation 2013-03-22 12:51:47
Last modified on 2013-03-22 12:51:47
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 9
Author rmilson (146)
Entry type Definition
Classification msc 14A10
Classification msc 14-00
Classification msc 14A15
Defines affine morphism
Defines finite type