  We first provide here a very short proof of a refinement of a theorem of
Kodiyalam and Cutkosky, Herzog and Trung on the regularity of powers of ideals.
This result implies a conjecture of H\`a and generalizes a result of Eisenbud
and Harris concerning the case of ideals primary for the graded maximal ideal
in a standard graded algebra over a field. It also implies a new result on the
regularities of powers of ideal sheaves. We then compare the cohomology of the
stalks and the cohomology of the fibers of a projective morphism to the effect
of comparing the maximum over fibers and over stalks of the Castelnuovo-Mumford
regularities of a family of projective schemes.
