Bounds on the Castelnuovo-Mumford regularity of tensor products

TL;DR

This paper establishes bounds on the Castelnuovo-Mumford regularity of tensor products of modules, extending previous results and providing new insights into regularity behavior via homological and hyperplane restriction techniques.

Contribution

It introduces a method to bound the regularity of the zero homology using partial regularities and generalizes existing bounds for tensor products under specific conditions.

Findings

If $ ext{dim} or_1^R(M,N) extless=1$, then $ eg(Migotimes N) extless= eg(M)+ eg(N)$

Provides a new bound on regularity based on partial regularities of modules and homologies

Describes regularity in terms of postulation numbers of filter regular hyperplane restrictions

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 $\dim \tor_1^R(M,N)\leq1$ 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.