  In this paper we give bounds on the Castelnuovo-Mumford regularity of
products of ideals and ideal sheaves. In particular, we show that the
regularity of a product of ideals is bounded by the sum of the regularities of
its factors if the corresponding schemes intersect in a finite set of points.
We also show how approximations of sheaves can be used to bound the regularity
of an arrangement of two-planes in projective space.
