World Library  
Flag as Inappropriate
Email this Article

Quasi-finite morphism

Article Id: WHEBN0002143056
Reproduction Date:

Title: Quasi-finite morphism  
Author: World Heritage Encyclopedia
Language: English
Subject: Étale morphism, Locally factorial scheme, Proof of Zariski's main theorem, Morphisms of schemes, WikiProject Mathematics/PlanetMath Exchange/14-XX Algebraic geometry
Publisher: World Heritage Encyclopedia

Quasi-finite morphism

In algebraic geometry, a branch of mathematics, a morphism f : XY of schemes is quasi-finite 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 ×YSpec 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)).

Quasi-finite 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 quasi-finiteness in terms of stalks.

For a general morphism f : XY 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 : UV 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.


For a morphism f, the following properties are true.[3]

  • If f is quasi-finite, then the induced map fred between reduced schemes is 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 : YZ, and if gf 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 unramified at a point x, then f is quasi-finite at x. Conversely, if f is quasi-finite 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 quasi-finite.[5] A quasi-finite proper morphism locally of finite presentation is finite.[6]

A generalized form of Zariski Main Theorem is the following:[7] Suppose Y is quasi-compact and quasi-separated. Let f be quasi-finite, 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.)


  1. ^ EGA II, Définition 6.2.3
  2. ^ EGA III, ErrIII, 20.
  3. ^ a b EGA II, Proposition 6.2.4.
  4. ^ EGA IV4, Théorème 17.4.1.
  5. ^ EGA II, Corollaire 6.1.7.
  6. ^ EGA IV3, Théorème 8.11.1.
  7. ^ EGA IV3, Théorème 8.12.6.


This article was sourced from Creative Commons Attribution-ShareAlike 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, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government 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 non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.