In algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

 f : X → Y of schemes

Scheme (mathematics)

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...

 is quasi-finite if it is finite type

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 scheme theory...

 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 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

Alexander Grothendieck

Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...

 in SGA 1 and did not include the finite type hypothesis. This hypothesis was added to the definition in EGA

Éléments de géométrie algébrique

The Éléments de géométrie algébrique by Alexander Grothendieck , or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published from 1960 through 1967 by the Institut des Hautes Études Scientifiques...

 II 6.2 because it makes it possible to give an algebraic characterization of quasi-finiteness in terms of stalks

Stalk (sheaf)

The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.-Motivation and definition:Sheaves are defined on open sets, but the underlying topological space X consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a...

.

For a general morphism 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 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.

Properties

For a morphism f, the following properties are true.

• If f is quasi-finite, then the induced map f_red 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 , and if is quasi-finite, then f is quasi-finite if any of the following are true:
  1. g is separated,
  2. X is noetherian,
  3. 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.)