Quasi-finite morphism


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:
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.
For a general morphism and a point x in X, f is said to be quasi-finite at x if there exist open affine neighborhoods U of x and V of f such that f is contained in V and such that the restriction is quasi-finite. f is locally quasi-finite if it is quasi-finite at every point in X. A quasi-compact locally quasi-finite morphism is quasi-finite.

Properties

For a morphism f, the following properties are true.
Quasi-finiteness is preserved by base change. The composite and fiber product of quasi-finite morphisms is quasi-finite.
If f is unramified at a point x, then f is quasi-finite at x. Conversely, if f is quasi-finite at x, and if also, the local ring of x in the fiber f−1, is a field and a finite separable extension of κ, then f is unramified at x.
Finite morphisms are quasi-finite. A quasi-finite proper morphism locally of finite presentation is finite.
A generalized form of Zariski Main Theorem is the following: Suppose Y is quasi-compact and quasi-separated. Let f be quasi-finite, separated and of finite presentation. Then f factors as where the first morphism is an open immersion and the second is finite.