  For a characteristic $p > 0$ variety $X$ with controlled $F$-singularities,
we state conditions which imply that a divisorial sheaf is Cohen-Macaulay or at
least has depth $\geq 3$ at certain points. This mirrors results of Koll\'ar
for varieties in characteristic zero. As an application, we show that that
relative canonical sheaves are compatible with arbitrary base change for
certain families with sharply $F$-pure fibers.
