Quasi-finite Morphism - Properties

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 g : Y → Z, and if g ∘ f is quasi-finite, then f is quasi-finite if any of the following are true:
    1. g is separated,
    2. X is noetherian,
    3. X ×Z Y is locally noetherian.

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(f(x)), is a field and a finite separable extension of κ(f(x)), 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. (X is open in a finite scheme over Y.)

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