Quasi-finite morphism

From Wikipedia, the free encyclopedia

In mathematics, more specifically in the scheme theory of algebraic geometry, a morphism

f : X → Y

of schemes) is quasi-finite if, for every point y ∈ Y, the fibre

X ×_Y k(y),

where k(y) is the residue field of y and k(y) → Y is the canonical morphism, has only a finite number of points.

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.

[edit] Relationship to other types of morphisms

• Finite morphisms are quasifinite.