Universal homeomorphism

In algebraic geometry, a universal homeomorphism is a morphism of schemes [math]\displaystyle{ f: X \to Y }[/math] such that, for each morphism [math]\displaystyle{ Y' \to Y }[/math], the base change [math]\displaystyle{ X \times_Y Y' \to Y' }[/math] is a homeomorphism of topological spaces. A morphism of schemes is a universal homeomorphism if and only if it is integral, radicial and surjective. In particular, a morphism of locally of finite type is a universal homeomorphism if and only if it is finite, radicial and surjective.

For example, an absolute Frobenius morphism is a universal homeomorphism.

References

• Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS 32. doi:10.1007/bf02732123. http://www.numdam.org/articles/PMIHES_1967__32__5_0. 

External links

• Universal homeomorphisms and the étale topology
• Do pushouts along universal homeomorphisms exist?