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In mathematics, especially in algebraic geometry, the v-topology (also known as the universally subtrusive topology) is a Grothendieck topology whose covers are characterized by lifting maps from valuation rings. This topology was introduced by Rydh (2010) and studied further by Bhatt & Scholze (2017), who introduced the name v-topology, where v stands for valuation.

Definition

A universally subtrusive map is a map f: X → Y of quasi-compact, quasi-separated schemes such that for any map v: Spec (V) → Y, where V is a valuation ring, there is an extension (of valuation rings) \( {\displaystyle V\subset W} \) and a map Spec W → X lifting v.
Examples

Examples of v-covers include faithfully flat maps, proper surjective maps.

Voevodsky (1996) introduced the h-topology. It is based on submersive maps, i.e., maps whose underlying map of topological spaces is a quotient map (i.e., surjective and a subset of Y is open if and only if its preimage in X is open). Any such submersive map is a v-cover. The converse holds if Y is Noetherian, but not in general.
Arc topology

Bhatt & Mathew (2018) have introduced the arc-topology, which is similar in its definition, except that only valuation rings of rank ≤ 1 are considered in the definition.

Bhatt & Scholze (2019, §8) show that the Amitsur complex of an arc covering of perfect rings is an exact complex.
See also

List of topologies on the category of schemes

References

Bhatt, Bhargav; Mathew, Akhil (2018), The arc-topology, arXiv:1807.04725v2
Bhatt, Bhargav; Scholze, Peter (2017), "Projectivity of the Witt vector affine Grassmannian", Inventiones Mathematicae, 209 (2): 329–423, arXiv:1507.06490, doi:10.1007/s00222-016-0710-4, MR 3674218
Bhatt, Bhargav; Scholze, Peter (2019), Prisms and Prismatic Cohomology, arXiv:1905.08229
Rydh, David (2010), "Submersions and effective descent of étale morphisms", Bull. Soc. Math. France, 138 (2): 181–230, arXiv:0710.2488, MR 2679038

Voevodsky, Vladimir (1996), "Homology of schemes", Selecta Mathematica. New Series, 2 (1): 111–153, doi:10.1007/BF01587941, MR 1403354

Mathematics Encyclopedia

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