Revisiting the de Rham-Witt complex

TL;DR

This paper introduces a new homological algebra framework for constructing the de Rham-Witt complex, simplifying crystalline comparison and broadening understanding of cohomology theories in characteristic p.

Contribution

It develops a novel categorical and homological approach to the de Rham-Witt complex, generalizing its construction and applications in p-adic cohomology.

Findings

Simplified crystalline comparison for AΩ-cohomology.

Established a new categorical framework for de Rham-Witt complexes.

Connected fixed points of Berthelot-Ogus operator to homological algebra.

Abstract

The goal of this paper is to offer a new construction of the de Rham-Witt complex of smooth varieties over perfect fields of characteristic $p>0$. We introduce a category of cochain complexes equipped with an endomorphism $F$ of underlying graded abelian groups satisfying $dF = pFd$, whose homological algebra we study in detail. To any such object satisfying an abstract analog of the Cartier isomorphism, an elementary homological process associates a generalization of the de Rham-Witt construction. Abstractly, the homological algebra can be viewed as a calculation of the fixed points of the Berthelot-Ogus operator $L \eta_p$ on the $p$-complete derived category. We give various applications of this approach, including a simplification of the crystalline comparison for the $A \Omega$-cohomology theory introduced in [BMS18].