The Stacks project

Lemma 102.10.1. Let $\mathcal{X}$ be an algebraic stack. Let $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ be the category of locally quasi-coherent modules with the flat base change property, see Section 102.8. The inclusion functor $i : \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ has a right adjoint

\[ Q : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \]

such that $Q \circ i$ is the identity functor.

Proof. Choose a scheme $U$ and a surjective smooth morphism $f : U \to \mathcal{X}$. Set $R = U \times _\mathcal {X} U$ so that we obtain a smooth groupoid $(U, R, s, t, c)$ in algebraic spaces with the property that $\mathcal{X} = [U/R]$, see Algebraic Stacks, Lemma 93.16.2. We may and do replace $\mathcal{X}$ by $[U/R]$. By Sheaves on Stacks, Proposition 95.14.3 there is an equivalence

\[ q_1 : \mathit{QCoh}(U, R, s, t, c) \longrightarrow \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \]

Let us construct a functor

\[ q_2 : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \longrightarrow \mathit{QCoh}(U, R, s, t, c) \]

by the following rule: if $\mathcal{F}$ is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then we set

\[ q_2(\mathcal{F}) = (f^*\mathcal{F}|_{U_{\acute{e}tale}}, \alpha ) \]

where $\alpha $ is the isomorphism

\[ t_{small}^*(f^*\mathcal{F}|_{U_{\acute{e}tale}}) \to t^*f^*\mathcal{F}|_{R_{\acute{e}tale}} \to s^*f^*\mathcal{F}|_{R_{\acute{e}tale}} \to s_{small}^*(f^*\mathcal{F}|_{U_{\acute{e}tale}}) \]

where the outer two morphisms are the comparison maps. Note that $q_2(\mathcal{F})$ is quasi-coherent precisely because $\mathcal{F}$ is locally quasi-coherent and that we used (and needed) the flat base change property in the construction of the descent datum $\alpha $. We omit the verification that the cocycle condition (see Groupoids in Spaces, Definition 77.12.1) holds. Looking at the proof of Sheaves on Stacks, Proposition 95.14.3 we see that $q_2 \circ i$ is the quasi-inverse to $q_1$. We define $Q = q_1 \circ q_2$. Let $\mathcal{F}$ be an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ and let $\mathcal{G}$ be an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. We have

\begin{align*} \mathop{\mathrm{Mor}}\nolimits _{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})} (i(\mathcal{G}), \mathcal{F}) & = \mathop{\mathrm{Mor}}\nolimits _{\mathit{QCoh}(U, R, s, t, c)}(q_2(i(\mathcal{G})), q_2(\mathcal{F})) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathit{QCoh}(\mathcal{O}_\mathcal {X})}(\mathcal{G}, Q(\mathcal{F})) \end{align*}

where the first equality is Sheaves on Stacks, Lemma 95.14.4 and the second equality holds because $q_1 \circ i$ and $q_2$ are quasi-inverse equivalences of categories. The assertion $Q \circ i \cong \text{id}$ is a formal consequence of the fact that $i$ is fully faithful. $\square$


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