nLab
h-topology

Contents

Idea

Let SS 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 DMDM of mixed motives.

Definition

A morphism p:XYp : X \to Y of schemes is called a topological epimorphism? if the underlying topological space of YY is a quotient space of the underlying topological space of XX, i.e. pp is surjective and a subset AYA \subset Y is open iff its preimage p 1(A)Xp^{-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 iX) i(p_i : U_i \to X)_i such that p ip_i are of finite type and p i:U iX\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

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.

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