We study a Grothendieck topology on schemes which we call the -topology. This topology is a refinement of the v-topology (the pro-version of Voevodsky’s h-topology), where covers are tested via rank valuation rings. Functors which are -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 -sheaf, and we deduce various pullback squares in étale cohomology. Using -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.
"The -topology." Duke Math. J. 170 (9) 1899 - 1988, 15 June 2021. https://doi.org/10.1215/00127094-2020-0088