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

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A universally subtrusive map is a map f: XY 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)   and a map Spec WX lifting v.

Examples

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Examples of v-covers include faithfully flat maps, proper surjective maps. In particular, any Zariski covering is a v-covering. Moreover, universal homeomorphisms, such as  , the normalisation of the cusp, and the Frobenius in positive characteristic are v-coverings. In fact, the perfection   of a scheme is a v-covering.

Voevodsky's h topology

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See h-topology, relation to the v-topology

Arc topology

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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. A variant of this topology, with an analogous relationship that the h-topology has with the cdh topology, called the cdarc-topology was later introduced by Elmanto, Hoyois, Iwasa and Kelly (2020).[1]

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

See also

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References

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  1. ^ Elmanto, Elden; Hoyois, Marc; Iwasa, Ryomei; Kelly, Shane (2020-09-23). "Cdh descent, cdarc descent, and Milnor excision". Mathematische Annalen. arXiv:2002.11647. doi:10.1007/s00208-020-02083-5. ISSN 1432-1807. S2CID 216553105.