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On the upper bound of the multiplicity conjecture


Author: Tony J. Puthenpurakal
Journal: Proc. Amer. Math. Soc. 136 (2008), 3429-3434
MSC (2000): Primary 13H15, 13D02; Secondary 13D40, 13A30
DOI: https://doi.org/10.1090/S0002-9939-08-09426-4
Published electronically: May 15, 2008
MathSciNet review: 2415026
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Abstract: Let $ A = K[X_1,\ldots,X_n]$ and let $ I$ be a graded ideal in $ A$. We show that the upper bound of the multiplicity conjecture of Herzog, Huneke and Srinivasan holds asymptotically (i.e., for $ I^k$ and all $ k \gg 0$) if $ I$ belongs to any of the following large classes of ideals:

(1)
radical ideals,
(2)
monomial ideals with generators in different degrees,
(3)
zero-dimensional ideals with generators in different degrees.
Surprisingly, our proof uses local techniques like analyticity, reductions, equimultiplicity and local results like Rees's theorem on multiplicities.


References [Enhancements On Off] (What's this?)

  • 1. M. Boij and J. Söderberg, Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture. Preprint, 2006.
  • 2. W. Bruns and J. Herzog, Cohen-Macaulay rings, Revised edition, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998). MR 1251956 (95h:13020)
  • 3. R.C. Cowsik and M.V. Nori, On the fibres of blowing up. J. Indian Math. Soc. 40, no. 1-4, 217-222 (1976). MR 0572990 (58:28011)
  • 4. S.D. Cutkosky, J. Herzog and N.V. Trung, Asymptotic behavior of the Castelnuovo-Mumford regularity, Compositio Math. 118, no. 3, 243-261 (1999). MR 1711319 (2000f:13037)
  • 5. C.A. Francisco, New approaches to bounding the multiplicity of an ideal. J. Algebra 299, no. 1, 309-328 (2006). MR 2225778 (2007b:13043)
  • 6. L.H. Gold, A degree bound for codimension two lattice ideals. J. Pure Appl. Algebra 182, no. 2-3, 201-207 (2003). MR 1903053 (2004i:13023)
  • 7. L. Gold, H. Schenck and H. Srinivasan, Betti numbers and degree bounds for some linked zero-schemes. J. Pure Appl. Algebra 210, 481-491 (2007). MR 2320011
  • 8. U. Grothe, M. Herrmann and U. Orbanz, Graded Cohen-Macaulay rings associated to equimultiple ideals. Math. Z. 186, no. 4, 531-556 (1984). MR 744964 (86c:13019)
  • 9. E. Guardo and A. Van Tuyl, Powers of complete intersections: Graded Betti numbers and applications. Ill. J. Math. 49, no. 1, 265-279 (2005). MR 2157379 (2006k:13035)
  • 10. J. Herzog and H. Srinivasan, Bounds for multiplicities. Trans. Am. Math. Soc. 350, no. 7, 2879-2902 (1998). MR 1458304 (99g:13033)
  • 11. J. Herzog and X. Zheng, Notes on the multiplicity conjecture. Collect. Math. 57, no. 2, 211-226 (2006). MR 2223853 (2007a:13029)
  • 12. V. Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc. 128, no. 2, 407-411 (2000). MR 1621961 (2000c:13027)
  • 13. H. Matsumura, Comutative ring theory. Cambridge Studies in Advanced Math. 8, Cambridge University Press, Cambridge (1986). MR 879273 (88h:13001)
  • 14. J. Migliore, U. Nagel and T. Römer, The multiplicity conjecture in low codimensions. Math. Res. Lett. 12, no. 5-6, 731-747 (2005). MR 2189234 (2006i:13042)
  • 15. J. Migliore, U. Nagel and T. Römer, Extensions of the multiplicity conjecture. Trans. Am. Math. Soc. 360, 2965-2985 (2008). MR 2379783
  • 16. J. Migliore, U. Nagel and F. Zanello, An improved multiplicity conjecture for codimension three Gorenstein algebras. Preprint, 2006.
  • 17. R.M. Miró-Roig, A note on the multiplicity of determinantal ideals. J. Algebra 299, no. 2, 714-724 (2006). MR 2228336 (2007a:13030)
  • 18. R.M. Miró-Roig, Betti numbers of determinantal ideals. Preprint, 2007.
  • 19. D.G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge. Philos. Soc. 50, 145-158 (1954) MR 0059889 (15:596a)
  • 20. I. Novik and E. Swartz, Face ring multiplicity via CM-connectivity sequences. Preprint, 2006.
  • 21. D. Rees, $ \mathfrak{a}$-transforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Philos. Soc. (2) 57, 8-17 (1961). MR 0118750 (22:9521)
  • 22. T. Römer, Note on bounds for multiplicities. J. Pure Appl. Algebra 195, no. 1, 113-123 (2005). MR 2100313 (2005g:13041)
  • 23. T. Römer, Betti numbers and shifts in minimal graded free resolutions. Preprint, 2007.
  • 24. H. Srinivasan, A note on the multiplicities of Gorenstein algebras. J. Algebra 208, no. 2, 425-443 (1998). MR 1655460 (99m:13046)
  • 25. F. Zanello, Improving the bounds in the multiplicity conjecture: the codimension 3 level case. J. Pure Appl. Algebra 209, no. 1, 79-89 (2007). MR 2292118 (2008a:13034)

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Additional Information

Tony J. Puthenpurakal
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, India 400 076
Email: tputhen@math.iitb.ac.in

DOI: https://doi.org/10.1090/S0002-9939-08-09426-4
Keywords: Multiplicity conjecture, regularity, reduction, analyticity
Received by editor(s): January 29, 2007
Received by editor(s) in revised form: September 3, 2007
Published electronically: May 15, 2008
Additional Notes: The author thanks the Universtät Duisburg-Essen for its hospitality during November and December of 2006. The author also thanks DFG for financial support, which made this visit possible
Dedicated: Dedicated to Juergen Herzog on the occasion of his 65th birthday
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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