Finite morphism

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

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 : XY denotes a finite morphism.

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.

See also