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'' → ''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 $X_ \to X$, the normalisation of the cusp, and the Frobenius in positive characteristic are ''v''-coverings. In fact, the perfection $X_ \to X$ 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 ≤ 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.

See also

* List of topologies on the category of schemes

References

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* {{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

Algebraic geometry