Let $A$ be a Noetherian local ring and $M$ a finitely generated $A$-module such that $\operatorname{depth}M= \dim M=\dim A.$ I can prove that $\operatorname{depth}M_{\mathfrak{p}}= \dim M_{\mathfrak{p}}\leq\dim A_{\mathfrak{p}}\quad\forall\mathfrak{p}\in\operatorname{Supp}M.$ I guess the inequality is in fact an equality, but can't seem to be able to prove it. Does anyone have any idea?
P.S. If $A$ is Cohen-Macaulay and pd$(M)<\infty$ (i.e., if $M$ is perfect), then it can be shown using $\operatorname{pd}M=\operatorname{depth}A-\operatorname{depth}M.$