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

Abstract

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 ⁢ ( 𝒪 X ) {{\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.