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Artin–Mazur heights and Yobuko heights of proper log smooth schemes of Cartier type, and Hodge–Witt decompositions and Chow groups of quasi-F-split threefolds

  • Yukiyoshi Nakkajima EMAIL logo
Published/Copyright: April 20, 2022

Abstract

Let X/s be a proper log smooth scheme of Cartier type over a fine log scheme whose underlying scheme is the spectrum of a perfect field κ of characteristic p>0. In this article we prove that the cohomology of 𝒲(𝒪X) is a finitely generated 𝒲(κ)-module if the Yobuko height of X is finite. As an application of this result, we prove that, if the Yobuko height of a proper smooth threefold Y over κ is finite, then the crystalline cohomology of Y/κ has the Hodge–Witt decomposition and the p-primary torsion part of the Chow group of codimension 2 of Y is of finite cotype. These are nontrivial generalizations of results in [K. Joshi and C. S. Rajan, Frobenius splitting and ordinarity, Int. Math. Res. Not. IMRN 2003 2003, 2, 109–121] and [K. Joshi, Exotic torsion, Frobenius splitting and the slope spectral sequence, Canad. Math. Bull. 50 2007, 4, 567–578]. We also prove a fundamental inequality between the Artin–Mazur heights and the Yobuko height of X/s if X/s satisfies natural conditions.

Funding statement: The author is supported by two JSPS Grant-in-Aid’s for Scientific Research (C) (Grant No. 80287440, 18K03224).

Acknowledgements

I would like to express my sincere gratitude to F. Yobuko for sending me very attractive articles [48] and [49]. Without his articles, I could not write this article. I would also like to express my gratitude to the referee for recommending me to give more examples of (log) varieties such that the strict inequality hq<hF (q1) holds.

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Received: 2019-07-11
Revised: 2022-01-27
Published Online: 2022-04-20
Published in Print: 2022-06-01

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