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$ .