In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasifinite if it is 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, \mathcal{O}_{X,x}\otimes \kappa(f(x)) is finitely generated over \kappa(f(x)).
Quasifinite morphisms were originally defined by Alexander Grothendieck in SGA 1 and did not include the finite type hypothesis. This hypothesis was added to the definition in EGA II 6.2 because it makes it possible to give an algebraic characterization of quasifiniteness in terms of stalks.
For a general morphism f : X → Y and a point x in X, f is said to be quasifinite 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 quasifinite. f is locally quasifinite if it is quasifinite at every point in X.^{[2]} A quasicompact locally quasifinite morphism is quasifinite.
Properties
For a morphism f, the following properties are true.^{[3]}

If f is quasifinite, then the induced map f_{red} between reduced schemes is quasifinite.

If f is a closed immersion, then f is quasifinite.

If X is noetherian and f is an immersion, then f is quasifinite.

If g : Y → Z, and if g ∘ f is quasifinite, then f is quasifinite if any of the following are true:

g is separated,

X is noetherian,

X ×_{Z} Y is locally noetherian.
Quasifiniteness is preserved by base change. The composite and fiber product of quasifinite morphisms is quasifinite.^{[3]}
If f is unramified at a point x, then f is quasifinite at x. Conversely, if f is quasifinite at x, and if also \mathcal{O}_{f^{1}(f(x)),x}, 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 quasifinite.^{[5]} A quasifinite proper morphism locally of finite presentation is finite.^{[6]}
A generalized form of Zariski Main Theorem is the following:^{[7]} Suppose Y is quasicompact and quasiseparated. Let f be quasifinite, separated and of finite presentation. Then f factors as X \hookrightarrow X' \to Y where the first morphism is an open immersion and the second is finite. (X is open in a finite scheme over Y.)
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.

^ EGA IV_{3}, Théorème 8.12.6.
References
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.