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. We introduce a category of cochain complexes equipped with an endomorphism F (of underlying graded abelian groups) satisfying dF = pF d, 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 Lηp on the p-complete derived category. We give various applications of this approach, including a simplification of the crystalline comparison in AΩ-cohomology theory.
@article{bhattRevisitingRhamWittComplex2021,
title = {Revisiting the de {Rham}-{Witt} complex},
volume = {424},
issn = {03031179, 24925926},
url = {https://smf.emath.fr/publications/revisiter-le-complexe-de-de-rham-witt},
doi = {10.24033/ast.1146},
language = {en},
journal = {Astérisque},
author = {Bhatt, Bhargav and Lurie, Jacob and Mathew, Akhil},
year = {2021},
}