Let $R$ be a commutative ring and $M$ a simple $R$-module. Then $\mathfrak{m}=Ann(M)$ is a maximal ideal of $R$. Then it is known that $ \mathrm{pdim}_{R}(M)=\mathrm{pdim}_{R_{\mathfrak{m}}}(M), $ where $R_{\mathfrak{m}}$ is the localization of $R$ with respect to $\mathfrak{m}$.
Does anyone know the proof of this simple fact?