Quasi-finite morphism

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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 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.

[edit] Relationship to other types of morphisms

• Finite morphisms are quasi-finite. By Zariski's main theorem, a finite morphism is a quasi-finite proper morphism. See EGA IV₃, 8.12.6.

[edit] References

• Grothendieck, Alexandre; Michèle Raynaud [1971] (2003). Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques 3), Updated edition (in French), Société Mathématique de France, xviii+327. ISBN 2-85629-141-4. 
• Grothendieck, Alexandre; Jean Dieudonné (1961). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS 8: 5-222. 
• 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.