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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, is finitely generated over .

Properties
For a morphism f, the following properties are true. • If f is quasi-finite, then the induced map fred between reduced schemes is quasi-finite. • If f is a closed immersion, then f is quasi-finite. • If X is noetherian and f is an immersion, then f is quasi-finite. • If , and if is quasi-finite, then f is quasi-finite if any of the following are true: • g is separated, • X is noetherian, • is locally noetherian. Quasi-finiteness is preserved by base change. The composite and fiber product of quasi-finite morphisms is quasi-finite. Finite morphisms are quasi-finite. A quasi-finite proper morphism locally of finite presentation is finite. Indeed, a morphism is finite if and only if it is proper and locally quasi-finite. Since proper morphisms are of finite type and finite type morphisms are quasi-compact one may omit the qualification locally, i.e., a morphism is finite if and only if it is proper and quasi-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 X \hookrightarrow X' \to Y where the first morphism is an open immersion and the second is finite. (X is open in a finite scheme over Y.) ==See also==
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