Mathematical Research Letters

Volume 19 (2012)

Number 1

Regularity defect stabilization of powers of an ideal

Pages: 109 – 119



David Berlekamp (Department of Mathematics, University of California at Berkeley, CA 94720-3840, USA)


When $I$ is an ideal of a standard graded algebra $S$ with homogeneous maximal ideal $\mm$, itis 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 $e_m = \reg I^m - dm$) in various cases. When $I$ is $\mm$-primary we give afamily of bounds on the first differences of the $e_m$, including an upper bound on the increasingpart 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 $n = \dim(S)$.

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