Regularity defect stabilization of powers of an ideal

TL;DR

This paper investigates the behavior of the regularity defect of powers of an ideal in a standard graded algebra, providing bounds and stabilization results for the sequence of regularity defects.

Contribution

It introduces new constraints and bounds on the regularity defect sequence, including stabilization points for monomial ideals and bounds on its differences.

Findings

Regularity defect sequence eventually stabilizes for monomial ideals.

Bounds on the first differences of the regularity defect sequence are established.

The sequence e_i cannot increase beyond a certain point when I is ext{m}-primary.

Abstract

When I is an ideal of a standard graded algebra S with homogeneous maximal ideal \mm, it is known by the work of several authors that the Castelnuovo-Mumford regularity of I^m ultimately becomes a linear function dm + e for m \gg 0. We give several constraints on the behavior of what may be termed the \emph{regularity defect} (the sequence e_m = \reg I^m - dm). When I is \mm-primary we give a family of bounds on the first differences of the e_m, including an upper bound on the increasing part of the sequence; for example, we show that the e_i cannot increase for i \geq \dim(S). When I is a monomial ideal, we show that the e_i become constant for i \geq n(n-1)(d-1), where n = \dim(S).