## Why is \$pd_R R_p=gl.dim R_p\$?

Let $$R$$ be a Noetherian regular local ring, and let $$P\subset R$$ be a prime ideal. It should be true that $$pd_RR_P=gl.dim R_P,$$ where $$pd_RR_P$$ is the projective dimension of $$R_P$$ as an $$R$$-module, and $$gl.dim R_P$$ is the global dimension of the ring $$R_P$$.