nLab h-topology

Idea

Let $S$ be a noetherian scheme. The h-topology is a Grothendieck topology on the category of schemes, used by Vladimir Voevodsky to construct a triangulated category $DM$ of mixed motives.

Definition

A morphism $p : X \to Y$ of schemes is called a topological epimorphism? if the underlying topological space of $Y$ is a quotient space of the underlying topological space of $X$, i.e. $p$ is surjective and a subset $A \subset Y$ is open iff its preimage $p^{-1}(A) \subset X$ is open. Such a morphism is further called a universal topological epimorphism if this property is preserved under any base change.

Consider the pretopology on the category of schemes whose covering families are finite families $(p_i : U_i \to X)_i$ such that $p_i$ are of finite type and $\sqcup p_i : \sqcup U_i \to X$ is a universal topological epimorphism?. The Grothendieck topology generated by this pretopology is called the h-topology on the category of schemes.

The h-topology is stronger than the etale and proper topologies.

See also

• motives
• qfh-topology?
• cdh-topology?
• etale topology

References

• Vladimir Voevodsky, Homology of schemes and covariant motives, thesis, ProQuest LLC, Ann Arbor, MI, pp. 64, 1992, pdf.

• Vladimir Voevodsky, Homology of schemes, I. 1994, K-theory Preprint Archives, url

• David Rydh, Submersions and effective descent of étale morphisms,

  Bull. Soc. Math. France 138(2) (2010), 181–230, publ, pdf.