Year: 2018
Type: preprint
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 étale cohomology is an $\mathrm{arc}$-sheaf and deduce various pullback squares in étale 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 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 from SGA4, extending results of Hansen.
Source: arXiv (Cornell University)
Authors: Bhatt, Bhargav, Mathew, Akhil
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4394660770