**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Proper morphism

Summary

In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
Some authors call a proper variety over a field k a complete variety. For example, every projective variety over a field k is proper over k. A scheme X of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space X(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff.
A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite.
A morphism f: X → Y of schemes is called universally closed if for every scheme Z with a morphism Z → Y, the projection from the fiber product
is a closed map of the underlying topological spaces. A morphism of schemes is called proper if it is separated, of finite type, and universally closed ([EGA] II, 5.4.1 ). One also says that X is proper over Y. In particular, a variety X over a field k is said to be proper over k if the morphism X → Spec(k) is proper.
For any natural number n, projective space Pn over a commutative ring R is proper over R. Projective morphisms are proper, but not all proper morphisms are projective. For example, there is a smooth proper complex variety of dimension 3 which is not projective over C. Affine varieties of positive dimension over a field k are never proper over k. More generally, a proper affine morphism of schemes must be finite. For example, it is not hard to see that the affine line A1 over a field k is not proper over k, because the morphism A1 → Spec(k) is not universally closed. Indeed, the pulled-back morphism
(given by (x,y) ↦ y) is not closed, because the image of the closed subset xy = 1 in A1 × A1 = A2 is A1 − 0, which is not closed in A1.
In the following, let f: X → Y be a morphism of schemes.
The composition of two proper morphisms is proper.
Any base change of a proper morphism f: X → Y is proper. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×Y Z → Z is proper.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (7)

Related concepts (35)

Related lectures (46)

MATH-101(e): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

PHYS-739: Conformal Field theory and Gravity

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.

MATH-510: Algebraic geometry II - schemes and sheaves

The aim of this course is to learn the basics of the modern scheme theoretic language of algebraic geometry.

Proper morphism

In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field k a complete variety. For example, every projective variety over a field k is proper over k. A scheme X of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space X(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff. A closed immersion is proper.

Glossary of algebraic geometry

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.

Scheme (mathematics)

In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne).

Explores hints of holography in classical black hole thermodynamics.

Explains the concept of Noetherian schemes and their local properties using open sets and coverings.