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Hilbert functions of Gorenstein monomial curves


Authors: Feza Arslan and Pinar Mete
Journal: Proc. Amer. Math. Soc. 135 (2007), 1993-2002
MSC (2000): Primary 13H10, 14H20; Secondary 13P10
DOI: https://doi.org/10.1090/S0002-9939-07-08793-X
Published electronically: March 2, 2007
MathSciNet review: 2299471
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Abstract: It is a conjecture due to M. E. Rossi that the Hilbert function of a one-dimensional Gorenstein local ring is non-decreasing. In this article, we show that the Hilbert function is non-decreasing for local Gorenstein rings with embedding dimension four associated to monomial curves, under some arithmetic assumptions on the generators of their defining ideals in the non-complete intersection case. In order to obtain this result, we determine the generators of their tangent cones explicitly by using standard basis computations under these arithmetic assumptions and show that the tangent cones are Cohen-Macaulay. In the complete intersection case, by characterizing certain families of complete intersection numerical semigroups, we give an inductive method to obtain large families of complete intersection local rings with arbitrary embedding dimension having non-decreasing Hilbert functions.


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

  • 1. F. Arslan, Cohen-Macaulayness of tangent cones, Proc. Amer. Math. Soc. 128 (2000), 2243-2251. MR 1653409 (2000k:13021)
  • 2. D. Bayer and M. Stillman, Macaulay, A system for computation in algebraic geometry and commutative algebra, 1992.
  • 3. H. Bresinsky, Symmetric semigroups of integers generated by four elements, Manuscripta Math. 17, (1975), 205-219. MR 0414559 (54:2660)
  • 4. C. Delorme, Sous-monoïdes d'intersection complète de $ N$, Ann. Sci. École Norm. Sup. (4) 9 No.1 (1976), 145-154. MR 0407038 (53:10821)
  • 5. J. Elias, The Conjecture of Sally on the Hilbert Function for Curve Singularities, Journal of Algebra 160 No.1 (1993), 42-49. MR 1237076 (94j:13018)
  • 6. J. Elias, Private communication.
  • 7. J. Eakin, A. Sathaye, Prestable ideals, Journal of Algebra 41 (1976), 439-454. MR 0419428 (54:7449)
  • 8. G.-M. Greuel, G. Pfister, A Singular Introduction to Commutative Algebra, Springer-Verlag, 2002. MR 1930604 (2003k:13001)
  • 9. G.-M. Greuel, G. Pfister, and H. Schönemann. SINGULAR 2.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001). http://www.singular.uni-kl.de.
  • 10. S.K. Gupta, L.G. Roberts, Cartesian squares and ordinary singularities of curves, Comm. in Algebra 11 No.2 (1983), 127-182. MR 0688046 (84h:13036)
  • 11. J. Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 (1970), 175-193. MR 0269762 (42:4657)
  • 12. J. Herzog, R. Waldi, A note on the Hilbert function of a one-dimensional Cohen-Macaulay ring, Manuscripta Math. 16, (1975), 251-260. MR 0384785 (52:5658)
  • 13. E. Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc. 25 (1970), 748-751. MR 0265353 (42:263)
  • 14. E. Matlis, One-dimensional Cohen-Macaulay Rings, Lecture Notes in Mathematics 327, Springer-Verlag, 1977. MR 0357391 (50:9859)
  • 15. M. Morales, Syzygies of a monomial curve and a linear Diophantine problem of Frobenius, Max-Planck-Institut für Mathematik, preprint, 1987.
  • 16. O. Orecchia, One-dimensional local rings with reduced associated graded ring and their Hilbert functions, Manuscripta Math. 32 (1980), 391-405. MR 0595429 (83c:13011)
  • 17. L.G. Roberts, Ordinary singularities with decreasing Hilbert function, Canad. J. Math. 34 (1982), 169-180. MR 0650856 (83f:13007)
  • 18. J. Sally, Number of generators of ideals in local rings, Lecture Notes in Pure and Appl. Math. 35, Marcel Dekker, 1978. MR 0485852 (58:5654)
  • 19. G. Valla, Problems and results on Hilbert functions of graded algebras, Six Lectures on Commutative Algebra, Progr. Math. vol. 66, Birkhäuser, Basel, 1998, pp. 293-344. MR 1648668 (99k:13022)

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Additional Information

Feza Arslan
Affiliation: Department of Mathematics, Middle East Technical University, Ankara, 06531 Turkey
Email: sarslan@metu.edu.tr

Pinar Mete
Affiliation: Department of Mathematics, Middle East Technical University, Ankara, 06531 Turkey
Address at time of publication: Department of Mathematics, Balıkesir University, Balıkesir, 10145 Turkey
Email: pinarm@metu.edu.tr, pinarm@balikesir.edu.tr

DOI: https://doi.org/10.1090/S0002-9939-07-08793-X
Keywords: Gorenstein local ring, Hilbert function of a local ring, tangent cone, monomial curve, numerical semigroup, standard basis.
Received by editor(s): December 17, 2005
Received by editor(s) in revised form: April 1, 2006
Published electronically: March 2, 2007
Additional Notes: The second author was supported by TÜBİTAK with grant no. TBAG-HD/108(105T543).
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2007 American Mathematical Society

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