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:
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 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 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. A quasi-compact locally quasi-finite morphism is quasi-finite.
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.
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism.
En géométrie algébrique, un morphisme de type fini peut être pensé comme une famille de variétés algébriques paramétrée par un schéma de base. C'est un des types de morphismes les plus couramment étudiés. Soit un morphisme de schémas. On dit que est de type fini si pour tout ouvert affine de , est quasi-compact (i.e. réunion finie d'ouverts affines) et que pour tout ouvert affine contenu dans , le morphisme canonique est de type fini.
En mathématiques, les schémas sont les objets de base de la géométrie algébrique, généralisant la notion de variété algébrique de plusieurs façons, telles que la prise en compte des multiplicités, l'unicité des points génériques et le fait d'autoriser des équations à coefficients dans un anneau commutatif quelconque.
This course is an introduction to the non-perturbative bootstrap approach to Conformal Field Theory and to the Gauge/Gravity duality, emphasizing the fruitful interplay between these two ideas.
This is a course about group schemes, with an emphasis on structural theorems for algebraic groups (e.g. Barsotti--Chevalley's theorem). All the basics will be covered towards the proof of such theore
Explore la décomposition primaire et les schémas en géométrie algébrique, soulignant l'importance de travailler sur les champs non-algébriques fermés et le concept de fibres de morphismes.
In nature, one observes that a K-theory of an object is defined in two steps. First a “structured” category is associated to the object. Second, a K-theory machine is applied to the latter category that produces an infinite loop space. We develop a general ...