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quasi-finite morphism

{{Short description|Type of morphism in algebraic geometry}}

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:EGA II, Définition 6.2.3

  • 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 {{nowrap|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, \mathcal{O}_{X,x}\otimes \kappa(f(x)) is finitely generated over \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 {{nowrap|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 {{nowrap|f : U → V}} is quasi-finite. f is locally quasi-finite if it is quasi-finite at every point in X.EGA III, ErrIII, 20. A quasi-compact locally quasi-finite morphism is quasi-finite.

Properties

For a morphism f, the following properties are true.EGA II, Proposition 6.2.4.

  • 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 {{nowrap|g : Y → Z}}, and if {{nowrap|g ∘ f}} is quasi-finite, then f is quasi-finite if any of the following are true:
  • #g is separated,
  • #X is noetherian,
  • #{{nowrap|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 \mathcal{O}_{f^{-1}(f(x)),x}, 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.EGA IV4, Théorème 17.4.1.

Finite morphisms are quasi-finite.EGA II, Corollaire 6.1.7. A quasi-finite proper morphism locally of finite presentation is finite.EGA IV3, Théorème 8.11.1. Indeed, a morphism is finite if and only if it is proper and locally quasi-finite.{{cite web |url=https://stacks.math.columbia.edu/tag/02LS |website=The Stacks Project |access-date=31 January 2022 |title=Lemma 02LS}} Since proper morphisms are of finite type and finite type morphisms are quasi-compact{{cite web |url=https://stacks.math.columbia.edu/tag/01T1 |website=The Stacks Project |access-date=15 August 2023 |title=Definition 29.15.1.}} 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:EGA IV3, Théorème 8.12.6. 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

Notes

{{Reflist}}

References

  • {{cite book

| last = Grothendieck

| first = Alexandre

| author-link = Alexandre Grothendieck

|author2=Michèle Raynaud

| title = Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques 3)

| orig-year = 1971

| edition = Updated

| year = 2003

| publisher = Société Mathématique de France

| language = fr

| isbn = 2-85629-141-4

| no-pp = true

| page = xviii+327

}}

  • {{cite journal

| last = Grothendieck

| first = Alexandre

| author-link = Alexandre Grothendieck

|author2=Jean Dieudonné

|author2-link=Jean Dieudonné

| year = 1961

| title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes | journal = Publications Mathématiques de l'IHÉS

| volume = 8

| pages = 5–222

| url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1961__8_

| doi=10.1007/bf02699291

}}

  • {{cite journal

| last = Grothendieck

| first = Alexandre

| author-link = Alexandre Grothendieck

|author2=Jean Dieudonné

|author2-link=Jean Dieudonné

| year = 1966

| title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie

| journal = Publications Mathématiques de l'IHÉS

| volume = 28

| pages = 5–255

| doi = 10.1007/BF02684343

| url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1966__28_

}}

Category:Morphisms of schemes