## Local Cohomology and Maximal-Cohen-Macaulay modules

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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?

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## 1 Answer

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[This is from an unpublished manuscript by P. Smith, the first author of the paper]: If $A$ is CM, let $\omega_A = H^d_\mathfrak m(A)^*$ be its dualizing module.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 $i \leq d$)

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