Koszul homology and extremal properties of Gin and Lex

Abstract

In a polynomial ring $R$ with $n$ variables, for every homogeneous ideal $I$ and for every $p\leq n$ we consider the Koszul homology $H_i(p,R/I)$ with respect to a sequence of $p$ of generic linear forms and define the Koszul-Betti number $β_{ijp}(R/I)$ of $R/I$ to be 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 any gin of $I$ and also by those of the Lex-segment of $I$. 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 $I$ and 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 $J$.