DOI QR코드

DOI QR Code

ON THE FAILURE OF GORENSTEINESS FOR THE SEQUENCE (1, 125, 95, 77, 70, 77, 95, 125, 1)

  • Ahn, Jeaman (Department of Mathematics Education Kongju National University)
  • Received : 2015.05.04
  • Accepted : 2015.10.26
  • Published : 2015.11.15

Abstract

In [9], the authors determine an infinite class of non-unimodal Gorenstein sequence, which includes the example $$\bar{h}_1\text{ = (1, 125, 95, 77, 71, 77, 95, 125, 1)}$$. They raise a question whether there is a Gorenstein algebra with Hilbert function $$\bar{h}_2\text{= (1, 125, 95, 77, 70, 77, 95, 125, 1)}$$, which has remained an open question. In this paper, we prove that there is no Gorenstein algebra with Hilbert function $\bar{h}_2$.

Keywords

Acknowledgement

Supported by : Kongju National University

References

  1. D. Bernstein and A. Iarrobino, A non-unimodal graded Gorenstein Artin algebra in codimension five, Comm. Algebra 20 (1992), no. 8, 2323-2336. https://doi.org/10.1080/00927879208824466
  2. A. M. Bigatti and A. V. Geramita, Level Algebras, Lex Segments and Minimal Hilbert Functions, Comm. Algebra 31 (2003), 1427-1451. https://doi.org/10.1081/AGB-120017774
  3. M. Boij, Graded Gorenstein Artin algebras whose Hilbert functions have a large number of valleys, Comm. Algebra 23 (1995), no. 1, 97-103. https://doi.org/10.1080/00927879508825208
  4. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge studies in advanced Mathematics, 39, Revised edition (1998), Cambridge, U.K.
  5. M. Boij and D. Laksov, Nonunimodality of graded Gorenstein Artin algebras, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1083-1092. https://doi.org/10.1090/S0002-9939-1994-1227512-2
  6. M. Green. Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann. In Algebraic curves and projective geometry (Trento, 1988), volume 1389 of Lecture Notes in Math., pages 76-86. Springer, Berlin, 1989.
  7. M. Kreuzer and L. Robbiano. Computational commutative algebra. 2. Springer-Verlag, Berlin, 2005.
  8. F. S. Macaulay, The algebraic theory of modular systems, Revised reprint of the 1916 original. With an introduction by Paul Roberts. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1994.
  9. J. Migliore, U. Nagel, and F. Zanello, Bounds and asymptotic minimal growth for Gorenstein Hilbert functions, J. Algebra 321 (2009), no. 5, 1510-1521. https://doi.org/10.1016/j.jalgebra.2008.11.026
  10. J. Migliore, U. Nagel, and F. Zanello, On the degree two entry of a Gorenstein h-vector and a conjecture of Stanley, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2755-2762. https://doi.org/10.1090/S0002-9939-08-09456-2
  11. R. P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57-83. https://doi.org/10.1016/0001-8708(78)90045-2
  12. F. Zanello, Stanley's theorem on codimension 3 Gorenstein h-vectors, Proc. Amer. Math. Soc. 134 (2006), no. 1, 5-8 (electronic) https://doi.org/10.1090/S0002-9939-05-08276-6
  13. J. Migliore(1-NDM), U. Nagel(1-KY), and F. Zanello(1-NDM), On the degree two entry of a Gorenstein h-vector and a conjecture of Stanley. (English sum- mary) Proc. Amer. Math. Soc. 136 (2008), no. 8, 2755-2762. https://doi.org/10.1090/S0002-9939-08-09456-2