In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is of finite type and satisfies any of the following equivalent conditions:[1] 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.
, 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.
[6] Indeed, a morphism is finite if and only if it is proper and locally quasi-finite.
A generalized form of Zariski Main Theorem is the following:[9] Suppose Y is quasi-compact and quasi-separated.