Lemma 103.10.4 (0GQK)—The Stacks project

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Part 7: Algebraic Stacks
Chapter 103: Cohomology of Algebraic Stacks
Section 103.10: Quasi-coherent modules
Lemma 103.10.4 (cite)

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Lemma 103.10.4. Let $\mathcal{X}$ be an algebraic stack. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$ such that $x : U \to \mathcal{X}$ is flat. Then for $\mathcal{F}$ in $\mathit{QCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ we have $Q(\mathcal{F})|_{U_{\acute{e}tale}} = \mathcal{F}|_{U_{\acute{e}tale}}$ .

Proof. True because the kernel and cokernel of $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic, see Lemma 103.10.2. $\square$

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