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:
- g is separated,
- X is noetherian,
- 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.)
Read more about this topic: Quasi-finite Morphism
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)