There is another finiteness condition on morphisms of schemes, morphisms of finite type, which is much weaker than being finite.
Morally, a morphism of finite type corresponds to a set of polynomial equations with finitely many variables. For example, the algebraic equation
corresponds to the map of (affine) schemes or equivalently to the inclusion of rings . This is an example of a morphism of finite type.
The technical definition is as follows: let be an open cover of by affine schemes, and for each let be an open cover of by affine schemes. The restriction of f to induces a morphism of rings . The morphism f is called locally of finite type, if is a finitely generated algebra over (via the above map of rings). If in addition the open cover can be chosen to be finite, then f is called of finite type.
For example, if is a field, the scheme has a natural morphism to induced by the inclusion of rings 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.
Read more about this topic: Finite Morphism
Famous quotes containing the words finite and/or type:
“Any language is necessarily a finite system applied with different degrees of creativity to an infinite variety of situations, and most of the words and phrases we use are “prefabricated” in the sense that we don’t coin new ones every time we speak.”
—David Lodge (b. 1935)
“The ideal American type is perfectly expressed by the Protestant, individualist, anti-conformist, and this is the type that is in the process of disappearing. In reality there are few left.”
—Orson Welles (1915–1984)