Regularity of deficiency modules through spectral sequences

TL;DR

This paper develops spectral sequence techniques to establish upper bounds on the regularity of graded deficiency modules, extending previous results from monomial ideals to broader classes of rings like toric face rings and binomial edge rings.

Contribution

It introduces a spectral sequence approach to bound regularity, generalizing Kumini--Murai's results from monomial ideals to more complex ring types.

Findings

Recovered Kumini--Murai's upper bound for monomial ideals

Extended upper bounds to toric face rings

Provided new bounds for deficiency modules of binomial edge rings

Abstract

The main goal of this paper is to obtain upper bounds for the regularity of graded deficiency modules in the spirit of the one obtained by Kumini--Murai in the monomial case building upon the spectral sequence formalism developed by \`Alvarez Montaner, Boix and Zarzuela. This spectral sequence formalism allows us not only to recover Kumini--Murai's upper bound for monomial ideals, but also to extend it for other types of rings, which include toric face rings and some binomial edge rings, producing to the best of our knowledge new upper bounds for the regularity of graded deficiency modules of this type of rings.