v-topology

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) $V\subset W$ and a map Spec W → X lifting v.

Examples

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 $X_{red}\to X$ , the normalisation of the cusp, and the Frobenius in positive characteristic are v-coverings. In fact, the perfection $X_{perf}\to X$ of a scheme is a v-covering.

Voevodsky's h topology

See h-topology, relation to the v-topology

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. 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.

References

^ 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.

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, Bibcode:2017InMat.209..329B, doi:10.1007/s00222-016-0710-4, MR 3674218, S2CID 119123398
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, doi:10.24033/bsmf.2588, MR 2679038, S2CID 17484591
Voevodsky, Vladimir (1996), "Homology of schemes", Selecta Mathematica, New Series, 2 (1): 111–153, doi:10.1007/BF01587941, MR 1403354, S2CID 9620683

[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.

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