A note on the regularity of products

TL;DR

This paper investigates the regularity of products of monomial ideals and modules over polynomial rings, establishing bounds on their Castelnuovo-Mumford regularity using multigraded free resolutions.

Contribution

It introduces a construction of multigraded free resolutions for modules over polynomial rings to prove regularity inequalities for products of ideals and modules.

Findings

Proves $ ext{Reg}(IM) \\leq ext{Reg}(I) + ext{Reg}(M)$ for a large class of ideals and modules.

Provides a specific condition under which the inequality holds for ideals.

Uses Herzog's method to construct resolutions and derive regularity bounds.

Abstract

Let $S={\Bbb K}[x_1,\dots,x_n]$ denote a polynomial ring over a field $\Bbb K$. Given a monomial ideal $I$ and a finitely generated multigraded $M$ over $S$, we follow Herzog's method to construct a multigraded free $S$-resolution of $M/IM$ by using multigraded $S$-free resolutions of $S/I$ and $M$. The complex constructed in this paper is used to prove the inequality $\Reg(IM)\leq \Reg(I)+\Reg(M)$ for a large class of ideals and modules. In the case where $M$ is an ideal, under one relative condition on the generators which specially does not involve the dimensions, the inequality $\Reg(IM)\leq \Reg(I)+\Reg(M)$ is proven.