Depth of factors of square free monomial ideals

Abstract:

Let $I$ be an ideal of a polynomial algebra over a field generated by $r$ square free monomials of degree $d$. If $r$ is bigger than (or equal to, if $I$ is not principal) the number of square free monomials of $I$ of degree $d+1$, then $\mathrm {depth}_SI= d$. Let $J\subsetneq I$, $J\not =0$ be generated by square free monomials of degree $\geq d+1$. If $r$ is bigger than the number of square free monomials of $I\setminus J$ of degree $d+1$ or, more generally, the Stanley depth of $I/J$ is $d$, then $\mathrm {depth}_SI/J= d$. In particular, Stanley’s Conjecture holds in these cases.