Lemma 30.11.5. Let $X$ be a regular scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. The following are equivalent

$\mathcal{F}$ is Cohen-Macaulay and $\text{Supp}(\mathcal{F}) = X$,

$\mathcal{F}$ is finite locally free of rank $> 0$.

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