Notes on regularity stabilization
Authors:
David Eisenbud and Bernd Ulrich
Journal:
Proc. Amer. Math. Soc. 140 (2012), 12211232
MSC (2010):
Primary 13D02, 13C99, 13P20, 14N05
Published electronically:
October 18, 2011
MathSciNet review:
2869107
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Abstract: When is a finitely generated graded module over a standard graded algebra and is an ideal of , it is known from work of Cutkosky, Herzog, Kodiyalam, Römer, Trung and Wang that the CastelnuovoMumford regularity of has the form when . We give an explicit bound on the for which this is true, under the hypotheses that is generated in a single degree and 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 EisenbudGoto conjecture, under the additional hypotheses that the scheme lies on a quadric and has nice singularities.
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 S. Cutkosky, J. Herzog, and N.V. Trung, Asymptotic behavior of the CastelnuovoMumford regularity, Compositio Math. 118 (1999), 243261. MR 1711319 (2000f:13037)
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 H. Derksen and J. Sidman, A sharp bound for the CastelnuovoMumford regularity of subspace arrangements, Adv. Math. 172 (2002), 151157. MR 1942401 (2003m:13013)
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 D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89133. MR 741934 (85f:13023)
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 D. Eisenbud and J. Harris, Powers of ideals and fibers of morphisms, Math. Res. Lett. 17 (2010), 267273. MR 2644374 (2011e:14003)
 [G]
 D. Giaimo, On the CastelnuovoMumford regularity of connected curves, Trans. Amer. Math. Soc. 358 (2006), 267284. MR 2171233 (2006f:13013)
 [Kod]
 V. Kodiyalam, Asymptotic behavior of CastelnuovoMumford regularity, Proc. Amer. Math. Soc. 128 (2000), 407411. MR 1621961 (2000c:13027)
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 M. Johnson and B. Ulrich, ArtinNagata properties and CohenMacaulay associated graded rings, Compositio Math 103 (1996), 729. MR 1404996 (97f:13006)
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 J. Kollár, Higher direct images of dualizing sheaves I, Ann. of Math. (2) 123 (1986), 1142. MR 825838 (87c:14038)
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 E. Mayr and A. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. Math. 46 (1982), 305329. MR 683204 (84g:20099)
 [O]
 T. Ohsawa, Vanishing theorems on complete Kähler manifolds, Publ. Res. Inst. Math. Sci. 20 (1984), 2138. MR 736089 (85g:32046)
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 T. Römer, Homological properties of bigraded algebras, Illinois J. Math. 45 (2001),
13611376. MR 1895463 (2003d:13015)
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 N.V. Trung and H.J. Wang, On the asymptotic linearity of CastelnuovoMumford
regularity, J. Pure and Appl. Alg. 201 (2005), 4248. MR 2158746 (2006k:13039)
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Additional Information
David Eisenbud
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
Email:
eisenbud@math.berkeley.edu
Bernd Ulrich
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email:
ulrich@math.purdue.edu
DOI:
http://dx.doi.org/10.1090/S00029939201111270X
Received by editor(s):
January 3, 2011
Published electronically:
October 18, 2011
Communicated by:
Harm Derksen
Article copyright:
© Copyright 2011
American Mathematical Society
