TITLE: Projective dimension of the residue field of a noetherian local ring.
QUESTION [5 upvotes]: Let $R$ be a commutative Noetherian local ring with maximal ideal $\mathfrak m$. 

Is it true that the projective dimension of $R/\mathfrak m$ is finite knowing that its injective dimension is finite? If yes, why?!?

I would need this to prove something else but I'm not sure I can use it.
Can you help me please?
Thanks.

REPLY [4 votes]: If $\operatorname {injdim} (k)\lt \infty$, then $R$ is regular : this is stated as exercise 3.1.26 of Bruns-Herzog's Cohen-Macaulay rings.
From this follows from Serre's theorem that $R$ has finite global dimension and then of course this implies that $k$, like all $R$-modules, has finite projective dimension.