Checking a recent article [[this one](http://arxiv.org/abs/1108.1552), 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?



Local Cohomology and Maximal-Cohen-Macaulay modules
Well, I don't know if I'm supposed to, but since I found a solution, I'll write the general idea here.


[This is from an unpublished manuscript by P. Smith, the first author of the paper]: If $A$ is CM, let $\omega\_A = H^d\_\mathfrak m(A)^\*$ be its dualizing module. Then there is a spectral sequence 
$$ E^{pq}\_2 = \underline{Ext}^p\_A(\underline{Ext}^q(M, \omega\_A),\omega\_A) \Rightarrow \begin{cases}M&\mbox{ if p = q} \\\ 0 &\mbox{ otherwise}\end{cases}$$


Since $H\_\mathfrak{m}^i(M)^\* \cong \underline{Ext}\_A^i(M,\omega\_A)$, the convergence of this SS to a non-zero result guarantees that there must be a non-zero local cohomology module (and in fact, that there is a non-zero one with $i \leq d$)


