  In this paper we show how, given a complex of graded modules and knowing some
partial Castelnuovo-Mumford regularities for all the modules in the complex and
for all the positive homologies, it is possible to get a bound on the
regularity of the zero homology. We use this to prove that if $\dim
\tor_1^R(M,N)\leq1$ then $\reg(M\otimes N)\leq \reg(M)+\reg(N)$, generalizing
results of Chandler, Conca and Herzog, and Sidman. Finally we give a
description of the regularity of a module in terms of the postulation numbers
of filter regular hyperplane restrictions.
