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

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Abstract | References | Similar Articles | Additional Information

Abstract: The reduction number of a standard graded algebra is the least integer such that there exists a minimal reduction of the homogeneous maximal ideal of such that . Vasconcelos conjectured that where is the initial ideal of an ideal in a polynomial ring with respect to a term order. The goal of this note is to prove the conjecture.

**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**

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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