The Stacks project

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 94.16.2. We may and do replace $\mathcal{X}$ by $[U/R]$. By Sheaves on Stacks, Proposition 96.14.3 there is an equivalence

Let us construct a functor

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

where $\alpha$ is the isomorphism

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 78.12.1) holds. Looking at the proof of Sheaves on Stacks, Proposition 96.14.3 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

where the first equality is Sheaves on Stacks, Lemma 96.14.4 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. $\square$

Proof. Write $\mathcal{X} = [U/R]$ as in the proof of Lemma 103.10.1. Let $\mathcal{F}$ be an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. It is clear from the proof of Lemma 103.10.1 that $\mathcal{F}$ is in the kernel of $Q$ if and only if $\mathcal{F}|_{U_{\acute{e}tale}} = 0$. In particular, if $\mathcal{F}$ is parasitic then $\mathcal{F}$ is in the kernel. Next, let $x : V \to \mathcal{X}$ be a flat morphism, where $V$ is a scheme. Set $W = V \times _\mathcal {X} U$ and consider the diagram

Note that the projection $p : W \to U$ is flat and the projection $q : W \to V$ is smooth and surjective. This implies that $q_{small}^*$ is a faithful functor on quasi-coherent modules. By assumption $\mathcal{F}$ has the flat base change property so that we obtain $p_{small}^*\mathcal{F}|_{U_{\acute{e}tale}} \cong q_{small}^*\mathcal{F}|_{V_{\acute{e}tale}}$. Thus if $\mathcal{F}$ is in the kernel of $Q$, then $\mathcal{F}|_{V_{\acute{e}tale}} = 0$ which completes the proof of (1).

Part (2) follows from the discussion above and the fact that the map $Q(\mathcal{F}) \to \mathcal{F}$ becomes an isomorphism after restricting to $U_{\acute{e}tale}$.

To see part (3) note that $Q$ is left exact as a right adjoint. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. Consider the following commutative diagram

Since the kernels and cokernels of $a$, $b$, and $c$ are parasitic by part (2) and since the bottom row is a short exact sequence, we see that the top row as a complex of $\mathcal{O}_\mathcal {X}$-modules has parasitic cohomology sheaves (details omitted; this uses that the category of parasitic modules is a Serre subcategory of the category of all modules). By left exactness of $Q$ we see that only exactness at $Q(\mathcal{H})$ is at issue. However, the cokernel $\mathcal{Q}$ of $Q(\mathcal{G}) \to Q(\mathcal{H}))$ may be computed either in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ or in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ with the same result because the inclusion functor $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ is a left adjoint and hence right exact. Hence $\mathcal{Q} = Q(\mathcal{Q})$ is both quasi-coherent and parasitic, whence $0$ by part (1) as desired.

As a right adjoint $Q$ commutes with all limits. Since $Q$ is exact, to show that $Q$ commutes with all colimits it suffices to show that $Q$ commutes with direct sums, see Categories, Lemma 4.14.12. Let $\mathcal{F}_ i$, $i \in I$ be a family of objects of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. To see that $Q(\bigoplus \mathcal{F}_ i)$ is equal to $\bigoplus Q(\mathcal{F}_ i)$ we look at the construction of $Q$ in the proof of Lemma 103.10.1. This uses a presentation $\mathcal{X} = [U/R]$ where $U$ is a scheme. Then $Q(\mathcal{F})$ is computed by first taking the pair $(\mathcal{F}|_{U_{\acute{e}tale}}, \alpha )$ in $\mathit{QCoh}(U, R, s, t, c)$ and then using the equivalence $\mathit{QCoh}(U, R, s, t, c) \cong \mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Since the restriction functor $\textit{Mod}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_{U_{\acute{e}tale}})$, $\mathcal{F} \mapsto \mathcal{F}|_{U_{\acute{e}tale}}$ commutes with direct sums, the desired equality is clear. $\square$

Proof. Observe that $f^*$ preserves both $\mathit{QCoh}$ and $\textit{LQCoh}^{fbc}$, see Sheaves on Stacks, Lemma 96.11.2 and Proposition 103.8.1. If $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y})$ then $Q_\mathcal {Y}(\mathcal{F}) \to \mathcal{F}$ has parasitic kernel and cokernel by Lemma 103.10.2. As $f$ is flat we get that $f^*Q_\mathcal {Y}(\mathcal{F}) \to f^*\mathcal{F}$ has parasitic kernel and cokernel by Lemma 103.9.2. Thus the induced map $f^*Q_\mathcal {Y}(\mathcal{F}) \to Q_\mathcal {X}(f^*\mathcal{F})$ has parasitic kernel and cokernel and hence is an isomorphism for example by Lemma 103.9.4. $\square$

Proof. True because the kernel and cokernel of $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic, see Lemma 103.10.2. $\square$

Proof. The construction of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ in Modules on Sites, Section 18.27 depends only on $\mathcal{F}$ and $\mathcal{G}$ as presheaves of modules; the output $\mathop{\mathcal{H}\! \mathit{om}}\nolimits$ is a sheaf for the fppf topology because $\mathcal{F}$ and $\mathcal{G}$ are assumed sheaves in the fppf topology, see Modules on Sites, Lemma 18.27.1. By Sheaves on Stacks, Lemma 96.12.4 we see that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ is locally quasi-coherent. By Lemma 103.7.2 we see that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ has the flat base change property. Hence $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ and it makes sense to apply the functor $Q$ of Lemma 103.10.1 to it. By the universal property of $Q$ we have

for $\mathcal{H}$ quasi-coherent, hence the displayed formula of the lemma follows from Modules on Sites, Lemma 18.27.6. $\square$

Proof. We have $f^*\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {Y}}(\mathcal{F}, \mathcal{G}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(f^*\mathcal{F}, f^*\mathcal{G})$ by Modules on Sites, Lemma 18.31.4. (Observe that this step is not where the flatness of $f$ is used as the morphism of ringed topoi associated to $f$ is always flat, see Sheaves on Stacks, Remark 96.6.3.) Then apply Lemma 103.10.3 (and here we do use flatness of $f$). $\square$