A note on the regularity of products

Abstract

Let $S=\mathbb{K}[x_1,\dots ,x_n]$ denote a polynomial ring over a field $\mathbb{K}$. 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.

7 pages

References in corpus (1)

• Castelnuovo-Mumford regularity of products of ideals