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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.

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