Castelnuovo-Mumford regularity of products of ideals

TLDR: In this article, it was shown that the Castelnuovo-Mumford regularity of a module can be doubly exponential in the degrees of the minimal generators and in the number of variables.

Abstract: The Castelnuovo-Mumford regularity reg$(M)$ is one of the most important invariants of a finitely generated graded module $M$ over a polynomial ring $R$. For instance, it measures the amount of computational resources that working with $M$ requires. In general one knows that the regularity of a module can be doubly exponential in the degrees of the minimal generators and in the number of the variables. On the other hand, in many situations one has or one conjectures a much better behavior. One may ask, for instance, whether the Castelnuovo-Mumford regularity reg($IM$) of the product of an ideal $I$ with a module $M$ is bounded by the sum reg($I$) + reg($M$). In general this is not the case. But we show that it is indeed the case if either dim $R/I\leq 1$ or $I$ is generic (in a very precise sense). Further we show that products of ideals of linear forms have always a linear resolution and that the same is true for products of determinantal ideals of a generic Hankel matrix.