My question is related to this one. I thought mine could be very elementary but I'm not sure how to look into it.
Let $J$ be an ideal of $R=k[\mathbf{x}]$ where $\mathbf{x}=\{x_1,\dots,x_n\}$. Let $\{f_1,\dots,f_r\}$ be a minimal generating set of $J$. Let $S\subseteq \mathbf{x}$ and $I$ be the ideal generated by all the $f_i$ such only the variables in $S$ divide $f_i$, so that $I\subseteq J$. Is it true that:
- $\mathrm{pd}(I)\leq \mathrm{pd}(J)$?
- Let's further assume that $J$ is the edge ideal of a graph, i.e. is generated by squarefree monomials consisting on two variables each (so that $I$ is the edge ideal of an induced subgraph of $J$). Then by Corollary 4.1.3, $\mathrm{pd}(I)\leq \mathrm{pd}(J)$. Assume that $\beta_{\mathrm{pd}(J),n}(J)>0$. Then does $\mathrm{pd}(I)<\mathrm{pd}(J)$ hold? What if $J$ is the edge ideal of a forest?
As an example for 2, in the case of the complete graph with $n$ vertices, its projective dimension is $n-2$ ($\mathrm{pd}(R/J)=n-1$), $\beta_{n-1,n}(J)>0$ (see Theorem 5.1.1) and every induced subgraph is itself a complete graph, so the property holds.
As another example, if $J$ is the edge ideal of the path with $4$ vertices, say, $J=(x_1x_2,x_2x_3,x_3x_4)\subseteq k[x_1,x_2,x_3,x_4]$, then it has projective dimension $1$, as does the edge ideal of the path with $3$ vertices $I=(x_1x_2,x_2x_3)$, but $\beta_{1,4}(J)=0$.
Any ideas about solving the questions, in particular the second one?
