algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a universal homeomorphism is a

morphism of schemes In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generalizes a ...

f: X \to Y

such that, for each morphism

Y' \to Y

, the base change

X \times_Y Y' \to Y'

is a

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

of topological spaces. A morphism of schemes is a universal homeomorphism if and only if it is

integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

, radicial and surjective.EGA IV₄, 18.12.11. In particular, a morphism of locally of finite type is a universal homeomorphism if and only if it is

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

, radicial and surjective. For example, an

absolute Frobenius morphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism ma ...

is a universal homeomorphism.

References

External links

Universal homeomorphisms and the étale topologyDo pushouts along universal homeomorphisms exist?
{{algebraic-geometry-stub Homeomorphisms Morphisms of schemes