Quasi-finite morphism

Quasi-finite morphism

In algebraic geometry, a branch of mathematics, a morphism "f" : "X" → "Y" of schemes is quasi-finite if it satisfies the following two conditions:

* "f" is locally of finite type.
* For every point "y" ∈ "Y", the scheme-theoretic fiber "X" ×"Y" "k"("y") has only a finite number of points. Here "k"("y") is the residue field of "y" and "k"("y") → "Y" is the inclusion morphism.

Note that the underlying topological space of the fibre is homeomorphic to the preimage of "f" −1("y") when "f" is regarded as a map of topological spaces.

Quasi-finite morphisms were originally defined by Alexander Grothendieck in SGA 1 and did not include the locally of 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.

Relationship to other types of morphisms

*Finite morphisms are quasi-finite. Conversely, by Zariski's main theorem, a quasi-finite proper morphism is finite. See EGA IV3, 8.12.6.
*Unramified and in particular étale morphisms are quasi-finite.

References

*cite book
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Michèle Raynaud
title = Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques 3)
origyear = 1971
edition = Updated edition
year = 2003
publisher = Société Mathématique de France
language = French
pages = xviii+327
id = ISBN 2-85629-141-4
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1961
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes | journal = Publications Mathématiques de l'IHÉS
volume = 8
pages = 5–222
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1961__8_
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1966
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie
journal = Publications Mathématiques de l'IHÉS
volume = 28
pages = 5–255
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1966__28_


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