Koszul homology and extremal properties of Gin and Lex

Conca, Aldo

doi:10.1090/S0002-9947-03-03393-2

Koszul homology and extremal properties of Gin and Lex
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Trans. Amer. Math. Soc. 356 (2004), 2945-2961 Request permission

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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