Lemma 103.10.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a flat morphism of algebraic stacks. Then $Q_\mathcal {X} \circ f^* = f^* \circ Q_\mathcal {Y}$ where $Q_\mathcal {X}$ and $Q_\mathcal {Y}$ are as in Lemma 103.10.1.
Proof. Observe that $f^*$ preserves both $\mathit{QCoh}$ and $\textit{LQCoh}^{fbc}$, see Sheaves on Stacks, Lemma 96.11.2 and Proposition 103.8.1. If $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y})$ then $Q_\mathcal {Y}(\mathcal{F}) \to \mathcal{F}$ has parasitic kernel and cokernel by Lemma 103.10.2. As $f$ is flat we get that $f^*Q_\mathcal {Y}(\mathcal{F}) \to f^*\mathcal{F}$ has parasitic kernel and cokernel by Lemma 103.9.2. Thus the induced map $f^*Q_\mathcal {Y}(\mathcal{F}) \to Q_\mathcal {X}(f^*\mathcal{F})$ has parasitic kernel and cokernel and hence is an isomorphism for example by Lemma 103.9.4. $\square$
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