Depth in a pathological case

Abstract

Let $I$ be a squarefree monomial ideal of a polynomial algebra over a field minimally generated by $f_1,...,f_r$ of degree $ d\geq 1$, and a set $E$ of monomials of degree $\geq d+1$. Let $J\subsetneq I$ be a squarefree monomial ideal generated in degree $\geq d+1$. Suppose that all squarefree monomials of $I\setminus (J\cup E)$ of degree $d+1$ are some least common multiples of $f_i$. If $J$ contains all least common multiples of two of $(f_i)$ of degree $d+2$ then $\depth_SI/J\leq d+1$ and Stanley's Conjecture holds for $I/J$.