Suppose that $R\to S$ is a ring morphism, $M$ an $S$-module and $I\subseteq S$ an ideal.
When is it true that $\mathrm{depth}_I(M)=\mathrm{depth}_{f^{-1}(I)}(M)$?
Particularly I'm interested in the case where $D\subseteq X$ is an effective Cartier divisor on some (sufficiently nice) scheme, $R=\mathcal{O}_{X,x}, S=\mathcal{O}_{D,x}$ and $M=\mathcal{F}_x$ for some sheaf of modules on $D$.
I think this should be true but I'm missing some idea.