I am reading Cohen-Macaulay Rings by Bruns & Herzog, and trying to prove the lemma below:
Lemma. Let $R$ be a Noetherian ring, and let $M$ be a finitely generated $R$-module. Then for all $\mathfrak{p}\in\mathop{\mathrm{Supp}}M$, $$ \mathop{\mathrm{grade}}M\leq \mathop{\mathrm{grade}}M_\mathfrak{p}\leq \mathop{\mathrm{proj\,dim}}M_\mathfrak{p}\leq \mathop{\mathrm{proj\,dim}}M. $$ What I have already thought
- The first inequality immediately follows from the definition of grade $$ \mathop{\mathrm{grade}} M = \inf\{i\geq 0 \mid\mathop{\mathrm{Ext}}\nolimits_{R}^{i}(R/\mathop{\mathrm{Ann}} M, R)\neq 0\},\\ \mathop{\mathrm{grade}} M_\mathfrak{p} = \inf\{i\geq 0 \mid\mathop{\mathrm{Ext}}\nolimits_{R_\mathfrak{p}}^{i}(R_\mathfrak{p}/\mathop{\mathrm{Ann}} M_\mathfrak{p}, R_\mathfrak{p})\neq 0\} $$ and the fact that under this circumstance Ext commutes with the localization $-\otimes R_\mathfrak{p}$.
- The third inequality immediately follows from the fact that, by the localization $-\otimes R_\mathfrak{p}$, all projective resolutions of $M$ give projective resolutions of $M_\mathfrak{p}$.
Question. How to prove the second inequality $\mathop{\mathrm{grade}}M_\mathfrak{p}\leq \mathop{\mathrm{proj\,dim}}M_\mathfrak{p}$?