Let $ (R,m) $ be a commutative unital noetherian local ring (with $m$ as its maximal ideal), $ I $ an ideal of $ R $, and $ M $ a finite $R$-module with $\dim M\gt 0$. $f_I(M) = \inf\ \{i : H_I^i(M)\ \text{is not finitely generated}\}$ is defined in chapter 9 of Brodmann-Sharp's Local Cohomology book.
I know that $$\operatorname{depth} M\le f_m(M)\le \dim M$$
the question is that how far $f_m(M)$ can be from edges? I specially interested in the case that $M$ is $S/I$ where $S$ is polynomial ring (which one can use software). So is there an example that $\dim S/I\gt \operatorname{depth} S/I+2$ and:
1) $$f_m(S/I)\gt \operatorname{depth} S/I+1$$
and that
2)$$f_m(S/I)\lt \dim S/I+1$$
