On comparison between relative log de Rham-Witt cohomology and relative log crystalline cohomology

TL;DR

This paper establishes a comparison theorem between relative log de Rham-Witt cohomology and relative log crystalline cohomology for certain log schemes, extending previous results and ensuring compatibility with Hyodo-Kato theory.

Contribution

It proves a new comparison theorem for relative log cohomologies in broader settings, generalizing Matsuue's results and confirming compatibility with Hyodo-Kato structures.

Findings

Established the comparison theorem for a broader class of log schemes.

Extended previous results to include cases with trivial or hollow log structures.

Demonstrated compatibility with Hyodo-Kato cohomology maps.

Abstract

In this article, we prove the comparison theorem between the relative log de Rham-Witt cohomology and the relative log crystalline cohomology for a log smooth saturated morphism of fs log schemes satisfying certain condition. Our result covers the case where the base fs log scheme is etale locally log smooth over a scheme with trivial log structure or the case where the base fs log scheme is hollow, and so it generalizes the previously known results of Matsuue. In Appendix, we prove that our relative log de Rham-Witt complex and our comparison map are compatible with those of Hyodo-Kato.