On relative and overconvergent de Rham-Witt cohomology for log schemes

TL;DR

This paper develops a generalized framework for log de Rham-Witt complexes, establishing comparison theorems with crystalline cohomology, and extends overconvergent complexes to log schemes, linking them to rigid cohomology.

Contribution

It constructs the relative log de Rham-Witt complex for log schemes, proves comparison theorems, and extends overconvergent complexes to new settings.

Findings

Comparison theorem between hypercohomology and crystalline cohomology

Degeneration of the weight spectral sequence at E2

Extension of overconvergent de Rham-Witt complex to log schemes

Abstract

We construct the relative log de Rham-Witt complex. This is a generalization of the relative de Rham-Witt complex of Langer-Zink to log schemes. We prove the comparison theorem between the hypercohomology of the log de Rham-Witt complex and the relative log crystalline cohomology in certain cases. We construct the $p$-adic weight spectral sequence for relative proper strict semistable log schemes. When the base log scheme is a log point, We show it degenerates at $E_2$ after tensoring with the fraction field of the Witt ring. We also extend the definition of the overconvergent de Rham-Witt complex of Davis-Langer-Zink to log schemes $(X,D)$ associated with smooth schemes with simple normal crossing divisor over a perfect field. Finally, we compare its hypercohomology with the rigid cohomology of $X \setminus D$.