Castelnuovo-Mumford regularity of products of ideals

TL;DR

This paper investigates how the Castelnuovo-Mumford regularity behaves under operations like products and powers of ideals, revealing cases where regularity can increase and conditions for linear resolutions.

Contribution

It provides new insights into the regularity of products of ideals, especially for ideals of linear forms and polymatroidal ideals, and establishes when products have linear resolutions.

Findings

reg(IM) can exceed reg(M)+reg(I) even for linear form ideals

Products of ideals of linear forms can have linear resolutions

Determinantal ideals of generic Hankel matrices have linear resolutions

Abstract

We discuss the behavior of the Castelnuovo-Mumford regularity under certain operations on ideals and modules, like products or powers. In particular, we show that reg(IM) can be larger than reg(M)+reg(I) even when I is an ideal of linear forms and M is a module with a linear resolution. On the other hand, we show that any product of ideals of linear forms has a linear resolution. We also discuss the case of polymatroidal ideals and show that any product of determinantal ideals of a generic Hankel matrix has a linear resolution.