Lemma 102.16.3. Let $\mathcal{X}$ be an algebraic stack.

With $g$ as in Lemma 102.14.2 for the lisse-étale site we have

the functors $g^{-1}$ and $g_!$ define mutually inverse functors

\[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^{-1}} & \mathit{QCoh}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \ar@<1ex>[l]^-{g_!} } \]if $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then $g^{-1}\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ and

$Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma 102.10.1.

With $g$ as in Lemma 102.14.2 for the flat-fppf site we have

the functors $g^{-1}$ and $g_!$ define mutually inverse functors

\[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^{-1}} & \mathit{QCoh}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{g_!} } \]if $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then $g^{-1}\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$ and

$Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma 102.10.1.

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