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), 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) [math]\displaystyle{ V \subset W }[/math] 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 [math]\displaystyle{ X_{red} \to X }[/math], the normalisation of the cusp, and the Frobenius in positive characteristic are v-coverings. In fact, the perfection [math]\displaystyle{ X_{perf} \to X }[/math] of a scheme is a v-covering.
Voevodsky's h topology
See h-topology, relation to the v-topology
Arc topology
(Bhatt Mathew) 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) 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
- ↑ Elmanto, Elden; Hoyois, Marc; Iwasa, Ryomei; Kelly, Shane (2020-09-23). "Cdh descent, cdarc descent, and Milnor excision" (in en). Mathematische Annalen. doi:10.1007/s00208-020-02083-5. ISSN 1432-1807. https://doi.org/10.1007/s00208-020-02083-5.
- Bhatt, Bhargav; Mathew, Akhil (2018), The arc-topology
- Bhatt, Bhargav; Scholze, Peter (2017), "Projectivity of the Witt vector affine Grassmannian", Inventiones Mathematicae 209 (2): 329–423, doi:10.1007/s00222-016-0710-4, Bibcode: 2017InMat.209..329B
- Bhatt, Bhargav; Scholze, Peter (2019), Prisms and Prismatic Cohomology
- Rydh, David (2010), "Submersions and effective descent of étale morphisms", Bull. Soc. Math. France 138 (2): 181–230, doi:10.24033/bsmf.2588
- Voevodsky, Vladimir (1996), "Homology of schemes", Selecta Mathematica, New Series 2 (1): 111–153, doi:10.1007/BF01587941
 |  |
