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 and studied further by, who introduced the name v-topology, where v stands for valuation.

Definition
A universally subtrusive map is a map f: X &rarr; Y of quasi-compact, quasi-separated schemes such that for any map v: Spec (V) &rarr; Y, where V is a valuation ring, there is an extension (of valuation rings) $$V \subset W$$ and a map Spec W &rarr; 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
have introduced the arc-topology, which is similar in its definition, except that only valuation rings of rank &le; 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).

show that the Amitsur complex of an arc covering of perfect rings is an exact complex.