Lemma 104.5.3. Let $\mathcal{X}$ be an algebraic stack. Set $\mathcal{P}_\mathcal {X} = \textit{Parasitic}(\mathcal{O}_\mathcal {X}) \cap \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.
Let $\mathcal{F}^\bullet $ be an object of $D_{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})} (\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$. With $g$ as in Cohomology of Stacks, Lemma 103.14.2 for the lisse-étale site we have
$g^*\mathcal{F}^\bullet $ is in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$,
$g^*\mathcal{F}^\bullet = 0$ if and only if $\mathcal{F}^\bullet $ is in $D_{\mathcal{P}_\mathcal {X}}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$,
$Lg_!\mathcal{H}^\bullet $ is in $D_{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ for $\mathcal{H}^\bullet $ in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$, and
the functors $g^*$ and $Lg_!$ define mutually inverse functors
\[ \xymatrix{ D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^*} & D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \ar@<1ex>[l]^-{Lg_!} } \]
Let $\mathcal{F}^\bullet $ be an object of $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. With $g$ as in Cohomology of Stacks, Lemma 103.14.2 for the flat-fppf site we have
$g^*\mathcal{F}^\bullet $ is in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$,
$g^*\mathcal{F}^\bullet = 0$ if and only if $\mathcal{F}^\bullet $ is in $D_{\mathcal{P}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$,
$Lg_!\mathcal{H}^\bullet $ is in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ for $\mathcal{H}^\bullet $ in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$, and
the functors $g^*$ and $Lg_!$ define mutually inverse functors
\[ \xymatrix{ D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^*} & D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{Lg_!} } \]
Comments (2)
Comment #1148 by Olaf Schnürer on
Comment #1169 by Johan on