On relative and overconvergent de Rham–Witt cohomology for 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 $X \setminus D$.