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Proceedings of the American Mathematical Society
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
Published electronically: June 12, 2002
MathSciNet review: 1948090
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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.


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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: http://dx.doi.org/10.1090/S0002-9939-02-06607-8
PII: S 0002-9939(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