Notes on regularity stabilization

Eisenbud, David; Ulrich, Bernd

doi:10.1090/S0002-9939-2011-11270-X

Notes on regularity stabilization
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by David Eisenbud and Bernd Ulrich

Proc. Amer. Math. Soc. 140 (2012), 1221-1232

DOI: https://doi.org/10.1090/S0002-9939-2011-11270-X

Published electronically: October 18, 2011

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Abstract:

When $M$ is a finitely generated graded module over a standard graded algebra $S$ and $I$ is an ideal of $S$, it is known from work of Cutkosky, Herzog, Kodiyalam, Römer, Trung and Wang that the Castelnuovo-Mumford regularity of $I^mM$ has the form $dm+e$ when $m\gg 0$. We give an explicit bound on the $m$ for which this is true, under the hypotheses that $I$ is generated in a single degree and $M/IM$ has finite length, and we explore the phenomena that occur when these hypotheses are not satisfied. Finally, we prove a regularity bound for a reduced, equidimensional projective scheme of codimension 2 that is similar to the bound in the Eisenbud-Goto conjecture, under the additional hypotheses that the scheme lies on a quadric and has nice singularities.

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