Universal homeomorphism

In algebraic geometry, a universal homeomorphism is a morphism of schemes ${\displaystyle f:X\to Y}$ such that, for each morphism ${\displaystyle Y'\to Y}$, the base change ${\displaystyle X\times _{Y}Y'\to Y'}$ is a homeomorphism of topological spaces.

A morphism of schemes is a universal homeomorphism if and only if it is integral, radicial and surjective.^[1] 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. MR 0238860.

External links[]

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

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