On the maximal graded shifts of ideals and modules

TL;DR

This paper extends known bounds on the maximal graded shifts of modules and establishes a linear regularity bound for ideals in polynomial rings based on early steps of their resolutions.

Contribution

It generalizes a result on maximal graded shifts and introduces a linear regularity bound depending only on initial resolution steps.

Findings

Generalized Eisenbud-Huneke-Ulrich's result on graded shifts

Proved a linear regularity bound for ideals in polynomial rings

Bound depends only on first p - c steps of the resolution

Abstract

We generalize a result of Eisenbud-Huneke-Ulrich on the maximal graded shifts of a module with prescribed annihilator and prove a linear regularity bound for ideals in a polynomial ring depending only on the first $p - c$ steps in the resolution, where $p = \mathrm{pd}(S/I)$ and $c = \mathrm{codim}(I)$.