A pro-cdh topology on formal schemes

TL;DR

This paper introduces a pro-cdh topology on formal schemes, establishing bounds on homotopy dimension and providing new interpretations for negative K-theory and motivic cohomology.

Contribution

It defines a pro-cdh topology on formal schemes and proves homotopy dimension bounds, improving previous topology definitions on schemes.

Findings

Pro-cdh sheaves of spaces have bounded homotopy dimension.

Provides a topos-theoretic interpretation of negative K-theory vanishing.

Remedies a defect in the pro-cdh topology on schemes.

Abstract

We introduce a pro-cdh topology on formal schemes and prove that the $\infty$-topos of pro-cdh sheaves of spaces has an optimal bound of homotopy dimension. This remedies a defect for a pro-cdh topology on schemes introduced in [KS23]. As an application, we give a topos-theoretic interpretation of Weibel's vanishing of negative K-theory and motivic cohomology of Elmanto and Morrow.