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LetX  /  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 characteristicp  &gt;  0{p&gt;0}   . In this article we prove that the cohomology of  ⁢   (     X   ){{\mathcal{W}}({\mathcal{O}}_{X})}    is a finitely generated  ⁢   (  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 ofY  /  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 ofX  /  s{X/s}    ifX  /  s{X/s}    satisfies natural conditions.

journal für die reine und angewandte mathematik (crelles journal)

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