Let $R$ be a Noetherian standard graded algebra with $R_0 = k$, a field, then it is finitely generated over $R_0$ by $R_1$ and is the homomorphic image of some $k[x_1, \ldots, x_n]$, hence isomorphic to $k[x_1, \ldots, x_n]/I$ where $I = (a_1, \ldots, a_s)$ is a homogeneous ideal of height $g$.
When is a Noetherian standard graded algebra over a field Cohen-Macaulay? Any easy counter-example(s) when it is not the case?
Edit: A standard graded algebra not CM is $k[x,y]/(x^2, xy)$, which has dim = 1 but depth 0.