The Stacks project

Proof. We will use the results of Lemma 47.16.5 without further mention. Fix a finite module $M$. If $M$ is Cohen-Macaulay, then only $\mathop{\mathrm{Ext}}\nolimits ^{-d}_ A(M, \omega _ A^\bullet )$ can be nonzero, hence (1) $\Rightarrow$ (3). The implication (3) $\Rightarrow$ (2) is immediate. Assume (2) and let $N = \mathop{\mathrm{Ext}}\nolimits ^{-\delta }_ A(M, \omega _ A^\bullet )$ be the nonzero $\mathop{\mathrm{Ext}}\nolimits$ where $\delta = \text{depth}(M)$. Then, since

(Lemma 47.15.3) we conclude that $M = \mathop{\mathrm{Ext}}\nolimits _ A^{-\delta }(N, \omega _ A^\bullet )$. Thus $\delta \geq \dim (\text{Supp}(M))$. However, since we also know that $\delta \leq \dim (\text{Supp}(M))$ (Algebra, Lemma 10.72.3) we conclude that $M$ is Cohen-Macaulay.

To prove the final statement, it suffices to show that $N = \mathop{\mathrm{Ext}}\nolimits ^{-d}_ A(M, \omega _ A^\bullet )$ is in $CM_ d$ for $M$ in $CM_ d$. Above we have seen that $M[0] = R\mathop{\mathrm{Hom}}\nolimits _ A(N[d], \omega _ A^\bullet )$ and this proves the desired result by the equivalence of (1) and (3). $\square$