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:

• Every point x of X is isolated in its fiber f⁻¹(f(x)). In other words, every fiber is a discrete (hence finite) set.
• For every point x of X, the scheme f⁻¹(f(x)) = X ×_YSpec κ(f(x)) is a finite κ(f(x)) scheme. (Here κ(p) is the residue field at a point p.)
• For every point x of X, ${\displaystyle {\mathcal {O}}_{X,x}\otimes \kappa (f(x))}$ is finitely generated over ${\displaystyle \kappa (f(x))}$.

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 f : X → Y 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(x) such that f(U) is contained in V and such that the restriction f : U → V 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.