Bounds on the Castelnuovo-Mumford regularity of tensor products

Abstract

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 \dim \operatorname {Tor} _1^R(M,N)\leq 1 , then reg ⁡ ( M ⊗ N ) ≤ reg ⁡ ( M ) + reg ⁡ ( N ) \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.