Bounds on the Castelnuovo-Mumford regularity of tensor products

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 $\dim \operatorname{Tor} _1^R(M,N)\leq1$, then $\operatorname{reg}(M\otimes N)\leq \operatorname{reg}( M)+\operatorname{reg}(N)$, 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.