Witt differentials in the h-topology

TL;DR

This paper explores the h-topology framework for de Rham-Witt complexes, aiming to unify and extend recent approaches to differential forms in algebraic geometry, with a focus on cohomological descent and sheafification.

Contribution

It demonstrates how to incorporate Illusie's de Rham-Witt complex into the h-topology framework, establishing h-descent results and analyzing sheafification of rational de Rham-Witt differentials.

Findings

Complete cohomological h-descent for de Rham-Witt complexes under resolution of singularities.

Unconditional h-descent for global sections of the complex.

Identification of a right Kan extension as a qfh-sheaf, with new results on qfh-sheaves.

Abstract

Recent important and powerful frameworks for the study of differential forms by Huber-Joerder and Huber-Kebekus-Kelly based on Voevodsky's h-topology have greatly simplified and unified many approaches. This article builds towards the goal of putting Illusie's de Rham-Witt complex in the same framework by exploring the h-sheafification of the rational de Rham-Witt differentials. Assuming resolution of singularities in positive characteristic one recovers a complete cohomological h-descent for all terms of the complex. We also provide unconditional h-descent for the global sections and draw the expected conclusions. The approach is to realize that a certain right Kan extension introduced by Huber-Kebekus-Kelly takes the sheaf of rational de Rham-Witt forms to a qfh-sheaf. As such, we state and prove many results about qfh-sheaves which are of independent interest.