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 Gin(I) of I and also by those of the Lex-segment Lex(I) of I. We show that \beta_{ijp}(R/I) = \beta_{ijp}(R/Gin(I)) iff I is componentwise linear and that and \beta_{ijp}(R/I) = \beta_{ijp}(R/Lex(I)) iff I is Gotzmann. We also investigate the set Gins(I) of all the gin of I and show that the Koszul-Betti numbers of any ideal in 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 Gins(I) such that the Koszul-Betti numbers of any ideal in Gins(I) are bounded above by those of J.

Koszul homology and extremal properties of Gin and Lex

CONCA, ALDO
2004

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 Gin(I) of I and also by those of the Lex-segment Lex(I) of I. We show that \beta_{ijp}(R/I) = \beta_{ijp}(R/Gin(I)) iff I is componentwise linear and that and \beta_{ijp}(R/I) = \beta_{ijp}(R/Lex(I)) iff I is Gotzmann. We also investigate the set Gins(I) of all the gin of I and show that the Koszul-Betti numbers of any ideal in 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 Gins(I) such that the Koszul-Betti numbers of any ideal in Gins(I) are bounded above by those of J.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11567/208229
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