A note on the regularity of products
Seyed Hamid Hassanzadeh, Siamak Yassemi
TL;DR
The paper investigates conditions under which the Castelnuovo-Mumford regularity satisfies subadditivity for products IM of a monomial ideal I and a multigraded S-module M, namely $\mathrm{reg}(IM)\le \mathrm{reg}(I)+\mathrm{reg}(M)$. It employs Herzog's multigraded $*$-product to construct a free resolution of $M/IM$ from minimal resolutions of $S/I$ and $M$, assuming $\mathrm{Gens}(I)\cap\mathrm{Gens}(M)=\emptyset$. The main contributions include proving $\mathrm{reg}(IM)\le \mathrm{reg}(I)+\mathrm{reg}(M)$, establishing $\mathrm{reg}(IJ)\le \mathrm{reg}(I)+\mathrm{reg}(J)$ when $|\mathrm{Gens}(I)\cap \mathrm{Gens}(J)|\le 1$, and providing counterexamples when the intersection has size at least $2$. These results clarify how the interaction between generator sets governs regularity in multigraded settings and extend known inequalities for products of ideals.
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.