Quasi-finite morphism

Quasi-finite morphism

In algebraic geometry, a branch of mathematics, a morphism "f" : "X" → "Y" of schemes is quasi-finite if it satisfies the following two conditions:

* "f" is locally of finite type.
* For every point "y" ∈ "Y", the scheme-theoretic fiber "X" ×"Y" "k"("y") has only a finite number of points. Here "k"("y") is the residue field of "y" and "k"("y") → "Y" is the inclusion morphism.

Note that the underlying topological space of the fibre is homeomorphic to the preimage of "f" −1("y") when "f" is regarded as a map of topological spaces.

Quasi-finite morphisms were originally defined by Alexander Grothendieck in SGA 1 and did not include the locally of 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.

Relationship to other types of morphisms

*Finite morphisms are quasi-finite. Conversely, by Zariski's main theorem, a quasi-finite proper morphism is finite. See EGA IV3, 8.12.6.
*Unramified and in particular étale morphisms are quasi-finite.

References

*cite book
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Michèle Raynaud
title = Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques 3)
origyear = 1971
edition = Updated edition
year = 2003
publisher = Société Mathématique de France
language = French
pages = xviii+327
id = ISBN 2-85629-141-4
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = 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_
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = 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
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1966__28_


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Finite morphism — In algebraic geometry, a branch of mathematics, a morphism of schemes is a finite morphism, if Y has an open cover by affine schemes Vi = SpecBi such that for each i, f − 1(Vi) = Ui is an open affine subscheme SpecAi, and the restriction of …   Wikipedia

  • Étale morphism — In algebraic geometry, a field of mathematics, an étale morphism (pronunciation IPA|) is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem,… …   Wikipedia

  • Proper morphism — In algebraic geometry, a proper morphism between schemes is an analogue of a proper map between topological spaces. Contents 1 Definition 2 Examples 3 Properties and characterizations of proper morphisms …   Wikipedia

  • Zariski's main theorem — In algebraic geometry, a field in mathematics, Zariski s main theorem, or Zariski s connectedness theorem, is a theorem proved by harvs|txt=yes|last=Zariski|year1=1943|year2=1949 which implies that fibers over normal points of birational… …   Wikipedia

  • List of mathematics articles (Q) — NOTOC Q Q analog Q analysis Q derivative Q difference polynomial Q exponential Q factor Q Pochhammer symbol Q Q plot Q statistic Q systems Q test Q theta function Q Vandermonde identity Q.E.D. QED project QR algorithm QR decomposition Quadratic… …   Wikipedia

  • List of algebraic geometry topics — This is a list of algebraic geometry topics, by Wikipedia page. Contents 1 Classical topics in projective geometry 2 Algebraic curves 3 Algebraic surfaces 4 …   Wikipedia

  • Exceptional inverse image functor — In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier… …   Wikipedia

  • Glossary of scheme theory — This is a glossary of scheme theory. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of… …   Wikipedia

  • Sheaf (mathematics) — This article is about sheaves on topological spaces. For sheaves on a site see Grothendieck topology and Topos. In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.… …   Wikipedia

  • Flat topology — In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also has played a fundamental role in the theory of descent (faithfully flat descent). [… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”