v-topology#Arc 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 {{harvtxt|Rydh|2010}} and studied further by {{harvtxt|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

{{harvtxt|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).{{Cite journal|last1=Elmanto|first1=Elden|last2=Hoyois|first2=Marc|last3=Iwasa|first3=Ryomei|last4=Kelly|first4=Shane|date=2020-09-23|title=Cdh descent, cdarc descent, and Milnor excision|url=https://doi.org/10.1007/s00208-020-02083-5|journal=Mathematische Annalen|language=en|doi=10.1007/s00208-020-02083-5|issn=1432-1807|arxiv=2002.11647|s2cid=216553105}}

{{harvtxt|Bhatt|Scholze|2019|loc=§8}} show that the Amitsur complex of an arc covering of perfect rings is an exact complex.

References

{{citation|title=The arc-topology|year=2018|first1=Bhargav|last1=Bhatt|first2=Akhil|last2=Mathew|arxiv=1807.04725v2}}
{{citation|first1=Bhargav|last1=Bhatt|first2=Peter|last2=Scholze|title=Projectivity of the Witt vector affine Grassmannian|journal=Inventiones Mathematicae|volume=209|year=2017|issue=2|pages=329–423|mr=3674218|doi=10.1007/s00222-016-0710-4|arxiv=1507.06490|bibcode=2017InMat.209..329B|s2cid=119123398}}
{{Citation|last1=Bhatt|first1=Bhargav|author1-link=Bhargav Bhatt (mathematician)|author2-link=Peter Scholze|last2=Scholze|first2=Peter|title=Prisms and Prismatic Cohomology|year=2019|arxiv=1905.08229}}
{{citation|first1=David|last1=Rydh|title=Submersions and effective descent of étale morphisms|journal=Bull. Soc. Math. France|volume=138|year=2010|issue=2|pages=181–230|doi=10.24033/bsmf.2588|mr=2679038|arxiv=0710.2488|s2cid=17484591}}
{{citation|title=Homology of schemes|author=Voevodsky|first=Vladimir|journal=Selecta Mathematica |series=New Series|volume=2|year=1996|issue=1|pages=111–153|mr=1403354|doi=10.1007/BF01587941|s2cid=9620683}}

Category:Algebraic geometry