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Joachim Cuntz, David Masser, Tobias Colding, Daniel Huybrechts, Jun-Muk Hwang, Olivier Schiffmann, Jakob Stix, Gábor Székelyhidi, Franz Stückle, Druck Und Verlag

Abstract

Let X/s{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{p>0}. In this article we prove that the cohomology of 𝒲⁢(𝒪X){{\mathcal{W}}({\mathcal{O}}_{X})} is a finitely generated 𝒲⁢(κ){{\mathcal{W}}(\kappa)}-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{Y/\kappa} 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{X/s} if X/s{X/s} satisfies natural conditions.

Locations

• Journal für die reine und angewandte Mathematik (Crelles Journal) - View - PDF