The Stacks project

Lemma 103.17.3. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. The module $\mathcal{O}_\mathcal {X}$ is coherent, any invertible $\mathcal{O}_\mathcal {X}$-module is coherent, and more generally any finite locally free $\mathcal{O}_\mathcal {X}$-module is coherent.

Proof. Follows from the definition and Cohomology of Spaces, Lemma 69.12.2. $\square$

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