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.
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.
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.
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.
D. Eisenbud and J. Harris, The Geometry of Schemes, Springer.
|Date of creation||2013-03-22 12:51:47|
|Last modified on||2013-03-22 12:51:47|
|Last modified by||rmilson (146)|