Proposition 103.11.1. Let f : \mathcal{X} \to \mathcal{Y} be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor f^* : \mathit{QCoh}(\mathcal{O}_\mathcal {Y}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {X}) has a right adjoint
f_{\mathit{QCoh}, *} : \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \longrightarrow \mathit{QCoh}(\mathcal{O}_\mathcal {Y})
which can be defined as the composition
\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \xrightarrow {f_*} \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) \xrightarrow {Q} \mathit{QCoh}(\mathcal{O}_\mathcal {Y})
where the functors f_* and Q are as in Proposition 103.8.1 and Lemma 103.10.1. Moreover, if we define R^ if_{\mathit{QCoh}, *} as the composition
\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \xrightarrow {R^ if_*} \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) \xrightarrow {Q} \mathit{QCoh}(\mathcal{O}_\mathcal {Y})
then the sequence of functors \{ R^ if_{\mathit{QCoh}, *}\} _{i \geq 0} forms a cohomological \delta -functor.
Proof.
This is a combination of the results mentioned in the statement. The adjointness can be shown as follows: Let \mathcal{F} be a quasi-coherent \mathcal{O}_\mathcal {X}-module and let \mathcal{G} be a quasi-coherent \mathcal{O}_\mathcal {Y}-module. Then we have
\begin{align*} \mathop{\mathrm{Mor}}\nolimits _{\mathit{QCoh}(\mathcal{O}_\mathcal {X})}(f^*\mathcal{G}, \mathcal{F}) & = \mathop{\mathrm{Mor}}\nolimits _{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y})} (\mathcal{G}, f_*\mathcal{F}) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathit{QCoh}(\mathcal{O}_\mathcal {Y})}(\mathcal{G}, Q(f_*\mathcal{F})) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathit{QCoh}(\mathcal{O}_\mathcal {Y})}(\mathcal{G}, f_{\mathit{QCoh}, *}\mathcal{F}) \end{align*}
the first equality by adjointness of f_* and f^* (for arbitrary sheaves of modules). By Proposition 103.8.1 we see that f_*\mathcal{F} is an object of \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) (and can be computed in either the fppf or étale topology) and we obtain the second equality by Lemma 103.10.1. The third equality is the definition of f_{\mathit{QCoh}, *}.
To see that \{ R^ if_{\mathit{QCoh}, *}\} _{i \geq 0} is a cohomological \delta -functor as defined in Homology, Definition 12.12.1 let
0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0
be a short exact sequence of \mathit{QCoh}(\mathcal{O}_\mathcal {X}). This sequence may not be an exact sequence in \textit{Mod}(\mathcal{O}_\mathcal {X}) but we know that it is up to parasitic modules, see Lemma 103.9.4. Thus we may break up the sequence into short exact sequences
\begin{matrix} 0 \to \mathcal{P}_1 \to \mathcal{F}_1 \to \mathcal{I}_2 \to 0
\\ 0 \to \mathcal{I}_2 \to \mathcal{F}_2 \to \mathcal{Q}_2 \to 0
\\ 0 \to \mathcal{P}_2 \to \mathcal{Q}_2 \to \mathcal{I}_3 \to 0
\\ 0 \to \mathcal{I}_3 \to \mathcal{F}_3 \to \mathcal{P}_3 \to 0
\end{matrix}
of \textit{Mod}(\mathcal{O}_\mathcal {X}) with \mathcal{P}_ i parasitic. Note that each of the sheaves \mathcal{P}_ j, \mathcal{I}_ j, \mathcal{Q}_ j is an object of \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}), see Proposition 103.8.1. Applying R^ if_* we obtain long exact sequences
\begin{matrix} 0 \to f_*\mathcal{P}_1 \to f_*\mathcal{F}_1 \to f_*\mathcal{I}_2 \to R^1f_*\mathcal{P}_1 \to \ldots
\\ 0 \to f_*\mathcal{I}_2 \to f_*\mathcal{F}_2 \to f_*\mathcal{Q}_2 \to R^1f_*\mathcal{I}_2 \to \ldots
\\ 0 \to f_*\mathcal{P}_2 \to f_*\mathcal{Q}_2 \to f_*\mathcal{I}_3 \to R^1f_*\mathcal{P}_2 \to \ldots
\\ 0 \to f_*\mathcal{I}_3 \to f_*\mathcal{F}_3 \to f_*\mathcal{P}_3 \to R^1f_*\mathcal{I}_3 \to \ldots
\end{matrix}
where are the terms are objects of \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) by Proposition 103.8.1. By Lemma 103.9.3 the sheaves R^ if_*\mathcal{P}_ j are parasitic, hence vanish on applying the functor Q, see Lemma 103.10.2. Since Q is exact the maps
Q(R^ if_*\mathcal{F}_3) \cong Q(R^ if_*\mathcal{I}_3) \cong Q(R^ if_*\mathcal{Q}_2) \rightarrow Q(R^{i + 1}f_*\mathcal{I}_2) \cong Q(R^{i + 1}f_*\mathcal{F}_1)
can serve as the connecting map which turns the family of functors \{ R^ if_{\mathit{QCoh}, *}\} _{i \geq 0} into a cohomological \delta -functor.
\square
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