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Transactions of the American Mathematical Society

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Koszul homology and extremal properties of Gin and Lex


Author: Aldo Conca
Journal: Trans. Amer. Math. Soc. 356 (2004), 2945-2961
MSC (2000): Primary 13D02; Secondary 13P10, 13Fxx
DOI: https://doi.org/10.1090/S0002-9947-03-03393-2
Published electronically: November 18, 2003
MathSciNet review: 2052603
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Abstract: For every homogeneous ideal $I$ in a polynomial ring $R$ and for every $p\leq\dim R$ we consider the Koszul homology $H_i(p,R/I)$ with respect to a sequence of $p$ of generic linear forms. The Koszul-Betti number $\beta_{ijp}(R/I)$ is, by definition, the dimension of the degree $j$ part of $H_i(p,R/I)$. In characteristic $0$, we show that the Koszul-Betti numbers of any ideal $I$ are bounded above by those of the gin-revlex $\mathrm{Gin}(I)$ of $I$ and also by those of the Lex-segment $\mathrm{Lex}(I)$ of $I$. We show that $\beta_{ijp}(R/I)=\beta_{ijp}(R/\mathrm{Gin}(I))$ iff $I$ is componentwise linear and that and $\beta_{ijp}(R/I)=\beta_{ijp}(R/\mathrm{Lex}(I))$ iff $I$is Gotzmann. We also investigate the set $\mathrm{Gins}(I)$ of all the gin of $I$ and show that the Koszul-Betti numbers of any ideal in $\mathrm{Gins}(I)$are bounded below by those of the gin-revlex of $I$. On the other hand, we present examples showing that in general there is no $J$ is $\mathrm{Gins}(I)$such that the Koszul-Betti numbers of any ideal in $\mathrm{Gins}(I)$ are bounded above by those of $J$.


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Additional Information

Aldo Conca
Affiliation: Dipartimento di Matematica, Universita’ di Genova, Genova, I-16146 Italy
Email: conca@dima.unige.it

DOI: https://doi.org/10.1090/S0002-9947-03-03393-2
Keywords: Koszul homology, Betti numbers, generic initial ideal, Lex-segment
Received by editor(s): December 3, 2002
Received by editor(s) in revised form: April 30, 2003
Published electronically: November 18, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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