# Quasi-finite morphism

In algebraic geometry, a branch of mathematics, a morphism *f* : *X* → *Y* of schemes is **quasi-finite** if it is of finite type and satisfies any of the following equivalent conditions:^{[1]}

- Every point
*x*of*X*is isolated in its fiber*f*^{−1}(*f*(*x*)). In other words, every fiber is a discrete (hence finite) set. - For every point
*x*of*X*, the scheme*f*^{−1}(*f*(*x*)) =*X*×_{Y}Spec κ(*f*(*x*)) is a finite κ(*f*(*x*)) scheme. (Here κ(*p*) is the residue field at a point*p*.) - For every point
*x*of*X*, is finitely generated over .

Quasi-finite morphisms were originally defined by

For a general morphism *f* : *X* → *Y* and a point *x* in *X*, *f* is said to be **quasi-finite** at *x* if there exist open affine neighborhoods *U* of *x* and *V* of *f*(*x*) such that *f*(*U*) is contained in *V* and such that the restriction *f* : *U* → *V* is quasi-finite. *f* is **locally quasi-finite** if it is quasi-finite at every point in *X*.^{}[2] A quasi-compact locally quasi-finite morphism is quasi-finite.

## Properties

For a morphism *f*, the following properties are true.^{[3]}

- If
*f*is quasi-finite, then the induced map*f*_{red}betweenreduced schemesis quasi-finite. - If
*f*is a closed immersion, then*f*is quasi-finite. - If
*X*is noetherian and*f*is an immersion, then*f*is quasi-finite. - If g :
*Y*→*Z*, and if*g*∘*f*is quasi-finite, then*f*is quasi-finite if any of the following are true:*g*is separated,*X*is noetherian,*X*×_{Z}*Y*is locally noetherian.

Quasi-finiteness is preserved by base change. The composite and fiber product of quasi-finite morphisms is quasi-finite.^{[3]}

If *f* is

*x*, then

*f*is quasi-finite at

*x*. Conversely, if

*f*is quasi-finite at

*x*, and if also , the local ring of

*x*in the fiber

*f*

^{−1}(

*f*(

*x*)), is a field and a finite separable extension of κ(

*f*(

*x*)), then

*f*is unramified at

*x*.

^{[4]}

Finite morphisms are quasi-finite.^{[5]} A quasi-finite proper morphism locally of finite presentation is finite.^{[6]} Indeed, a morphism is finite if and only if it is proper and locally quasi-finite.^{[7]} Since proper morphisms are of finite type and finite type morphisms are quasi-compact^{[8]} one may omit the qualification *locally*, i.e., a morphism is finite if and only if it is proper and quasi-finite.

A generalized form of

and quasi-separated. Let*f*be quasi-finite, separated and of finite presentation. Then

*f*factors as where the first morphism is an open immersion and the second is finite. (

*X*is open in a finite scheme over

*Y*.)

## See also

## Notes

**^**EGA II, Définition 6.2.3**^**EGA III, Err_{III}, 20.- ^
^{a}^{b}EGA II, Proposition 6.2.4. **^**EGA IV_{4}, Théorème 17.4.1.**^**EGA II, Corollaire 6.1.7.**^**EGA IV_{3}, Théorème 8.11.1.**^**"Lemma 02LS".*The Stacks Project*. Retrieved 31 January 2022.**^**"Definition 29.15.1".*The Stacks Project*. Retrieved 15 August 2023.**^**EGA IV_{3}, Théorème 8.12.6.

## References

- ISBN 2-85629-141-4.
- .
- Grothendieck, Alexandre; Jean Dieudonné (1966). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie".
*Publications Mathématiques de l'IHÉS*.**28**: 5–255.