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 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.

[edit] Relationship to other types of morphisms

Finite morphisms are quasi-finite. Conversely, by Zariski's main theorem, a quasi-finite proper morphism of finite presentation is finite. See EGA IV₃, 8.11.1.
Unramified and in particular étale morphisms are quasi-finite.