Checking a recent article [this one, specifically section 3.1] I found the following claim (I'm paraphrasing, of course):

Let $A$ be a graded connected noetherian algebra (not necessarily commutative), and suppose it is AS-Cohen-Macaulay of depth $d$. If $M$ is a finitely generated graded module over $A$, and it is Maximal Cohen Macaulay (MCM, ie, its only non-zero local cohomology module is precisely the $d$-th), then its first syzygy is also MCM.

I have a proof for this in the commutative ungraded case, but it deppends on the fact that $\lbrace i|H^i_{\mathfrak m}(M) \neq 0 \rbrace$ is non-empty and contained in the interval $[0,d]$ (consider the short exact sequence involving $M$ and its first syzygy and look at the long exact sequence of local cohomology). I found results regarding the non-vanishing of this groups in the non-commutative case, but they demand much more strict conditions than in the paper (finite GK-dimension, enough normal elements, etc.). Any idea on how to prove this in this more general context?