When a graded ring is Cohen-Macaulay? I am trying to solve exercise 19.10 from Eisenbud's Commutative Algebra. 

I want to show that if $R=k[x_0,...,x_n]/I$ is a graded ring, then $R$ is Cohen-Macaulay iff $R_{\mathfrak p}$ is Cohen-Macaulay, where $\mathfrak p=(x_0,...,x_n)$. 

The hint is to use the graded Auslander-Buchsbaum formula. 
I believe currently I am able to prove if $R_P$ is Cohen-Macaulay ring, then codim($p$)=depth($p$), but I do not know how to proceed. Is it enough to prove this homogeneous maximal ideal satisfies Cohen-Macaulay condition? If it is not, could someone give some hint? Thanks!

This is consequence of

Bruns-Herzog, Exercise 2.1.27 (c): Let $R$ be a Noetherian graded ring and $M$ a finite graded R-module. Suppose in addition that $(R,m)$  is $^*$local. Then $M$ is Cohen-Macaulay if and only if $M_m$ is.

Assuming that we know proofs of former parts, this part can be easily proved:
Let $p$ be a graded prime ideal. Then $p\subseteq m.$ Hence $M_p=(M_m)_{pR_m}$ is Cohen-Macaulay. So $M$ is Cohen-Macaulay, by part (b):(ii)$\to$ (i).
