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:[1]
- 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 .
Quasi-finite morphisms were originally defined by
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.[2] A quasi-compact locally quasi-finite morphism is quasi-finite.
Properties
For a morphism f, the following properties are true.[3]
- If f is quasi-finite, then the induced map fred between reduced schemesis 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.[3]
If f is
Finite morphisms are quasi-finite.[5] A quasi-finite proper morphism locally of finite presentation is finite.[6] Indeed, a morphism is finite if and only if it is proper and locally quasi-finite.[7] Since proper morphisms are of finite type and finite type morphisms are quasi-compact[8] 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
See also
Notes
- ^ EGA II, Définition 6.2.3
- ^ EGA III, ErrIII, 20.
- ^ a b EGA II, Proposition 6.2.4.
- ^ EGA IV4, Théorème 17.4.1.
- ^ EGA II, Corollaire 6.1.7.
- ^ EGA IV3, Théorème 8.11.1.
- ^ "Lemma 02LS". The Stacks Project. Retrieved 31 January 2022.
- ^ "Definition 29.15.1". The Stacks Project. Retrieved 15 August 2023.
- ^ EGA IV3, Théorème 8.12.6.
References
- ISBN 2-85629-141-4.
- .
- . Publications Mathématiques de l'IHÉS. 28: 5–255.