The Stacks project

Proposition 102.11.1. 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 102.8.1 and Lemma 102.10.1. 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 102.8.1 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 ├ętale topology) and we obtain the second equality by Lemma 102.10.1. 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 102.9.4. 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 102.8.1. 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 102.8.1. By Lemma 102.9.3 the sheaves $R^ if_*\mathcal{P}_ j$ are parasitic, hence vanish on applying the functor $Q$, see Lemma 102.10.2. 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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