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

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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 \(X{\setminus }D\).

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Acknowledgments

This paper is based on my master thesis in the University of Tokyo under the guidance of my supervisor Atsushi Shiho. I would like to express my sincere gratitude to him for the helpful discussions, reading the draft several times and providing valuable suggestions for improvement. This work would not have been possible without his advice. I would also like to thank Yukiyoshi Nakkajima for sending me his preprint [29].

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  1. Planning Dept., Treasury Unit, Sumitomo Mitsui Banking Corporation, 1-1-2, Marunouchi, Chiyoda-ku, Tokyo, Japan

    Hironori Matsuue

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  1. Hironori Matsuue
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Correspondence to Hironori Matsuue.

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Matsuue, H. On relative and overconvergent de Rham–Witt cohomology for log schemes. Math. Z. 286, 19–87 (2017). https://doi.org/10.1007/s00209-016-1755-1

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