The arc-topology

TL;DR

This paper introduces the arc-topology, a refinement of the v-topology on schemes, and demonstrates its utility in establishing descent properties and proving key theorems in étale cohomology and rigid analytic geometry.

Contribution

It defines the arc-topology, shows étale cohomology is an arc-sheaf, and applies arc-descent to reprove and extend fundamental theorems in algebraic and rigid analytic geometry.

Findings

Étale cohomology is an arc-sheaf.

Reproves Gabber-Huber affine proper base change.

Proves a rigid analytic Artin-Grothendieck vanishing theorem.

Abstract

We study a Grothendieck topology on schemes which we call the $\mathrm{arc}$-topology. This topology is a refinement of the $v$-topology (the pro-version of Voevodsky's $h$-topology) where covers are tested via rank $\leq 1$ valuation rings. Functors which are $\mathrm{arc}$-sheaves are forced to satisfy a variety of glueing conditions such as excision in the sense of algebraic $K$-theory. We show that \'etale cohomology is an $\mathrm{arc}$-sheaf and deduce various pullback squares in \'etale cohomology. Using $\mathrm{arc}$-descent, we reprove the Gabber-Huber affine analog of proper base change (in a large class of examples), as well as the Fujiwara-Gabber base change theorem on the \'etale cohomology of the complement of a henselian pair. As a final application we prove a rigid analytic version of the Artin-Grothendieck vanishing theorem from SGA4, extending results of Hansen.