Let $I \subset J $ be two monomial ideals in $S=k[x_1,..,x_n]$ minimally generated by $(a_1,...,a_s)$ and $(a_1,...,a_s,b_1,...,b_r)$. I want to show that depth$_S S/I \geq$ depth$_S S/J$.
Let $c = c_1,...,c_t$ be a maximal $S/J$ regular sequence. If $c$ is a $S/I$ regular sequence it is done. First it is clear that $c_1$ is not a zero divisor in $S/I$.
My question is :
Is $c_2$ a non zero divisor for $S/(I,c_1)$ ? I think the answer is yes, since $c_2$ is a non zero divisor for $S/(J,c_1)$. If this is true, then step by step we get depth$_S S/I \geq$ depth$_S S/J$?
