such that for each ,
is an open affine subscheme , and the restriction of f to , which induces a map of rings
makes a finitely generated module over .
Properties of finite morphisms
In the following, f : X → Y denotes a finite morphism.
- The composition of two finite maps is finite.
- Any base change of a finite morphism is finite, i.e. if is another (arbitrary) morphism, then the canonical morphism is finite. This corresponds to the following algebraic statement: if A is a finitely generated B-module, then the tensor product is a finitely generated C-module, where is any map. The generators are , where are the generators of A as a B-module.
- Closed immersions are finite, as they are locally given by , where I is the ideal corresponding to the closed subscheme.
- Finite morphisms are closed, hence (because of their stability under base change) proper. Indeed, replacing Y by the closure of f(X), one can assume that f is dominant. Further, one can assume that Y=Spec B is affine, hence so is X=Spec A. Then the morphism corresponds to an integral extension of rings B ⊂ A. Then the statement is a reformulation of the going up theorem of Cohen-Seidenberg.
- Finite morphisms have finite fibres (i.e. they are quasi-finite). This follows from the fact that any finite k-algebra, for any field k is an Artinian ring. Slightly more generally, for a finite surjective morphism f, one has dim X=dim Y.
- Conversely, proper, quasi-finite locally finite-presentation maps are finite. (EGA IV, 8.11.1.)
- Finite morphisms are both projective and affine.
Morphisms of finite type
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.
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.