projective dimension and localization

Question

Let $R$ be a Noetherian ring and $M$ a non-zero finite $R$-module with finite projective dimension equal to $n$. For any $P$ inside the support of $M$ we have that $\operatorname{projdim} M_P \le \operatorname{projdim} M$.

Question 1: Is it true that there exists some $P \in \operatorname{Supp} M$ such that $\operatorname{projdim} M_P = \operatorname{projdim} M$? If yes, please provide either proof or reference.

Question 2: If the answer to Question 1 is no, then can we "nicely" characterize modules $M$ having the required property described in Question 1?

Answer 1

Hint. Try to prove that $\operatorname{pd} M=\sup\{\operatorname{pd}M_P:P\in\operatorname{Supp}M\}$. (Since the property of a module to be $0$ localizes, the equality holds without any finiteness hypothesis on $R$ and $M$, only $\operatorname{pd} M<\infty$ is necessary.)

Answer 2

In this case (i.e. $R$ Noetherian, $M$ finite), $\text{projdim}_R M = \sup \{\text{projdim}_{R_m} M_m \mid m \in \text{mSpec}(R) \}$, so if $\text{projdim} M < \infty$, then this value is achieved locally.