finite morphism
Affine schemes
Let and be affine schemes, so that and . Let be a morphism, so that it induces a homomorphism of rings .
The homomorphism makes into a -algebra. If is finitely-generated as a -algebra, then is said to be a morphism of finite type.
If is in fact finitely generated as a -module, then is said to be a finite morphism.
For example, if is a field, the scheme has a natural morphism to induced by the ring homomorphism . This is a morphism of finite type, but if then it is not a finite morphism.
On the other hand, if we take the affine scheme , it has a natural morphism to given by the ring homomorphism . Then this morphism is a finite morphism. As a morphism of schemes, we see that every fiber is finite.
General schemes
Now, let and be arbitrary schemes, and let be a morphism. We say that is of finite type if there exist an open cover of by affine schemes and a finite open cover of each by affine schemes such that is a morphism of finite type for every and . We say that is finite if there exists an open cover of by affine schemes such that each inverse image, is itself affine, and such that is a finite morphism of affine schemes.
Example.
Let and . We cover by two copies of and consider the natural morphisms from each of these copies to . Both of these affine morphisms are of finite type, but are not finite. The covering morphisms patch together to give a morphism from to . The overall morphism is of finite type, but again is not finite.
References.
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 |