The arc-topology

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 $\le 1$ valuation rings. Functors which are $\mathrm{arc}$-sheaves are forced to satisfy a variety of gluing conditions such as excision in the sense of algebraic K-theory. We show that étale cohomology is an $\mathrm{arc}$-sheaf, and we deduce various pullback squares in étale cohomology. Using $\mathrm{arc}$-descent, we re-prove the Gabber–Huber affine analogue of proper base change (in a large class of examples), as well as the Fujiwara–Gabber base change theorem on the étale cohomology of the complement of a Henselian pair. As a final application, we prove a rigid analytic version of the Artin–Grothendieck vanishing theorem, extending results of Hansen.