Let $\mathbf{k}$ be a field, and let $S=\mathbf{k}[x_1,x_2,\ldots,x_n]$. Let $I\subset J$ be finitely generated monomial ideals in $S$. Is it true that the projective dimension of $I$ is either smaller than or equal to the projective dimension of $J$?
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2$\begingroup$ No. $(x_1)$ has projective dimension $0$, while $(x_1x_2,x_1x_3)$ has projective dimension $>0$. $\endgroup$– abxCommented Oct 22, 2014 at 8:40
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Interestingly, the equality you seek holds in one important special case. If $I$ is any monomial ideal and $J$ is the radical of $I$, then $pd_S(I)\leq pd_S(J)$. See the proof of Theorem 2.6 in this paper by Herzog-Takayama-Terai.
answered Feb 28, 2021 at 21:29
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$\begingroup$ Mr. Long_ math.stackexchange.com/q/4031531/822157, I need to the help. $\endgroup$– user164469Commented Mar 13, 2021 at 11:55