Regularity bounds for Koszul cycles

TL;DR

This paper investigates bounds on the regularity of Koszul cycles in polynomial rings, establishing subadditivity for 0-dimensional ideals and bounds for Borel-fixed ideals, advancing understanding of their algebraic properties.

Contribution

It introduces new bounds on the regularity of Koszul cycles, including subadditivity results and bounds for Borel-fixed ideals, under mild assumptions.

Findings

Regularity of Koszul cycles is subadditive for 0-dimensional ideals.

Bounded the regularity of Koszul cycles for Borel-fixed ideals.

Provided explicit upper bounds involving ideal regularity.

Abstract

We study the module of Koszul cycles $Z_t(I,M)$ of a homogeneous ideal $I$ in a polynomial ring $S$ with respect to a graded module $M$. Under mild assumptions on the base field we prove that the regularity of $Z_t(I,S)$ is a subadditive function of the homological position t when I is 0-dimensional. For Borel-fixed ideals $I$ and $J$ we prove that the regularity of $Z_t(I,S/J)$ is bounded above by $t(1+\reg I)+\reg S/J$.