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Notes on regularity stabilization


Authors: David Eisenbud and Bernd Ulrich
Journal: Proc. Amer. Math. Soc. 140 (2012), 1221-1232
MSC (2010): Primary 13D02, 13C99, 13P20, 14N05
DOI: https://doi.org/10.1090/S0002-9939-2011-11270-X
Published electronically: October 18, 2011
MathSciNet review: 2869107
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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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Additional Information

David Eisenbud
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
MR Author ID: 62330
ORCID: 0000-0002-5418-5579
Email: eisenbud@math.berkeley.edu

Bernd Ulrich
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
MR Author ID: 175910
Email: ulrich@math.purdue.edu

Received by editor(s): January 3, 2011
Published electronically: October 18, 2011
Communicated by: Harm Derksen
Article copyright: © Copyright 2011 American Mathematical Society