The arc-topology

Abstract

We study a Grothendieck topology on schemes which we call the 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 ≤1 valuation rings. Functors which are arc-sheaves are forced to satisfy a variety of gluing conditions such as excision in the sense of algebraic K-theory. We show that etale cohomology is an arc-sheaf, and we deduce various pullback squares in etale cohomology. Using 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 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, extending results of Hansen.