# Finite morphism

In algebraic geometry, a branch of mathematics, a morphism of schemes is a **finite morphism** if has an open cover by affine schemes

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*.

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.