Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Reduction numbers and initial ideals


Author: Aldo Conca
Journal: Proc. Amer. Math. Soc. 131 (2003), 1015-1020
MSC (2000): Primary 13P10, 13A30; Secondary 13F20
DOI: https://doi.org/10.1090/S0002-9939-02-06607-8
Published electronically: June 12, 2002
MathSciNet review: 1948090
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The reduction number $r(A)$ of a standard graded algebra $A$ is the least integer $k$ such that there exists a minimal reduction $J$ of the homogeneous maximal ideal $\mathbf m$ of $A$such that $J\mathbf m^k=\mathbf m^{k+1}$. Vasconcelos conjectured that $r(R/I)\leq r(R/\mathrm{in}(I))$ where $\mathrm{in}(I)$ is the initial ideal of an ideal $I$ in a polynomial ring $R$ with respect to a term order. The goal of this note is to prove the conjecture.


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

  • 1. A. Bigatti, Upper bounds for the Betti numbers of a given Hilbert function, Comm. Algebra 21 (1993), no. 7, 2317-2334. MR 94c:13014
  • 2. H. Bresinsky, L. Hoa, On the reduction number of some graded algebras, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1257-1263. MR 99h:13027
  • 3. A. Capani, G. Niesi, L. Robbiano, CoCoA, a system for doing Computations in Commutative Algebra, Available ftp from cocoa.dima.unige.it.
  • 4. H. Hulett, Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm. Algebra 21 (1993), no. 7, 2335-2350. MR 94c:13015
  • 5. S. Iyengar, K. Pardue, Maximal minimal resolutions, J. Reine Angew. Math. 512 (1999), 27-48. MR 2000d:13023
  • 6. K. Pardue, Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), no. 4, 564-585. MR 97g:13029
  • 7. E. Sbarra, Upper bounds for local cohomology for rings with given Hilbert function, Comm. Algebra 29 (2001), 5383-5409.
  • 8. N. Trung, Gröbner bases, local cohomology and reduction number, Proc. Amer. Math. Soc. 129 (2001), no. 1, 9-18. MR 2001c:13042
  • 9. N. Trung, Constructive characterization of the reduction numbers, preprint 2001.
  • 10. W. Vasconcelos, Cohomological degrees of graded modules, in Six lectures on commutative algebra (Bellaterra, 1996), 345-392, Progr. Math., 166, Birkhäuser, Basel, 1998. MR 99j:13012

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13P10, 13A30, 13F20

Retrieve articles in all journals with MSC (2000): 13P10, 13A30, 13F20


Additional Information

Aldo Conca
Affiliation: Dipartimento di Matematica, Universitá di Genova, Via Dodecaneso 35, I-16146 Genova, Italia
Email: conca@dima.unige.it

DOI: https://doi.org/10.1090/S0002-9939-02-06607-8
Keywords: Gr\"obner bases, initial ideal, reduction number, Lex-segment ideal
Received by editor(s): September 24, 2001
Received by editor(s) in revised form: October 29, 2001
Published electronically: June 12, 2002
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society