Lemma 103.11.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $y : V \to \mathcal{Y}$ in $\mathop{\mathrm{Ob}}\nolimits (\mathcal{Y})$ with $y$ a flat morphism. Let $\mathcal{F}$ be in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Then $(f_*\mathcal{F})(y) = (f_{\mathit{QCoh}, *}\mathcal{F})(y)$ and $(R^ if_*\mathcal{F})(y) = (R^ if_{\mathit{QCoh}, *}\mathcal{F})(y)$ for all $i \in \mathbf{Z}$.
Proof. This follows from the construction of the functors $R^ if_{\mathit{QCoh}, *}$ in Proposition 103.11.1, the definition of parasitic modules in Definition 103.9.1, and Lemma 103.10.2 part (2). $\square$
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