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 Tor(1)(R) (M, N) <= 1, then reg (M circle times N) = 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.