Regularity defect stabilization of powers of an ideal

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).