  In this article we prove a fundamental inequality between Artin-Mazur heights
and Yobuko heights of certain proper log smooth schemes of Cartier type over a
fine log scheme whose underlying scheme is the spectrum of a perfect field
$\kappa$ of characteristic $p>0$. We also prove that the cohomologies of
Witt-sheaves of them are finitely generated ${\cal W}(\kappa)$-modules if the
Yobuko heights of them are finite. As an application, we prove that the
$p$-primary torsion parts of the Chow groups of codimension $2$ of proper
smooth threefolds over $\kappa$ is of finite cotype if the Yobuko heights of
them are finite. These are nontrivial generalizations of results in [JR] and
[J].
