Lemma 103.17.8 (0GRC)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 103: Cohomology of Algebraic Stacks
Section 103.17: Coherent sheaves on locally Noetherian stacks
Lemma 103.17.8 (cite)

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Lemma 103.17.8. In the situation discussed above, the equivalence $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \cong \mathit{QCoh}(U, R, s, t, c)$ sends coherent sheaves to coherent sheaves and vice versa, i.e., induces an equivalence $\textit{Coh}(\mathcal{O}_\mathcal {X}) \cong \textit{Coh}(U, R, s, t, c)$ .

Proof. This is immediate from the definition of coherent $\mathcal{O}_\mathcal {X}$ -modules. For bookkeeping purposes: the material above uses Morphisms of Stacks, Lemma 101.17.5, Algebraic Stacks, Lemma 94.16.1 and Remark 94.16.3, Sheaves on Stacks, Section 96.15, Sheaves on Stacks, Proposition 96.14.3, and Groupoids in Spaces, Section 78.13. $\square$

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