The Stacks project

Proof. Assume $\mathcal{F}$ is quasi-coherent. Then $\mathcal{F}$ is a sheaf for the fppf topology, hence a sheaf for the étale topology. Moreover, any pullback of $\mathcal{F}$ to a ringed topos is quasi-coherent, hence the restrictions $x^*\mathcal{F}|_{U_{\acute{e}tale}}$ are quasi-coherent. This proves $\mathcal{F}$ is locally quasi-coherent. Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. We have seen that $\mathcal{X}/y = (\mathit{Sch}/V)_{fppf}$. By Descent, Proposition 35.8.9 it follows that $y^*\mathcal{F}$ is the quasi-coherent module associated to a (usual) quasi-coherent module $\mathcal{F}_ V$ on the scheme $V$. Hence certainly the comparison maps (96.9.4.1) are isomorphisms.

Conversely, suppose that $\mathcal{F}$ satisfies (1) and (2). Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. Denote $\mathcal{F}_ V$ the quasi-coherent module on the scheme $V$ corresponding to the restriction $y^*\mathcal{F}|_{V_{\acute{e}tale}}$ which is quasi-coherent by assumption (1), see Descent, Proposition 35.8.9. Condition (2) now signifies that the restrictions $x^*\mathcal{F}|_{U_{\acute{e}tale}}$ for $x$ over $y$ are each isomorphic to the (étale sheaf associated to the) pullback of $\mathcal{F}_ V$ via the corresponding morphism of schemes $U \to V$. Hence $y^*\mathcal{F}$ is the sheaf on $(\mathit{Sch}/V)_{fppf}$ associated to $\mathcal{F}_ V$. Hence it is quasi-coherent (by Descent, Proposition 35.8.9 again) and we see that $\mathcal{F}$ is quasi-coherent on $\mathcal{X}$ by Lemma 96.11.3. $\square$

Proof. Let $\mathcal{G}$ be locally quasi-coherent on $\mathcal{Y}$. Choose an object $x$ of $\mathcal{X}$ lying over the scheme $U$. The restriction $x^*f^*\mathcal{G}|_{U_{\acute{e}tale}}$ equals $(f \circ x)^*\mathcal{G}|_{U_{\acute{e}tale}}$ hence is a quasi-coherent sheaf by assumption on $\mathcal{G}$. $\square$

Proof. In the arguments below $x$ denotes an arbitrary object of $\mathcal{X}$ lying over the scheme $U$. To show that an object $\mathcal{H}$ of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ is in $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ we will show that the restriction $x^*\mathcal{H}|_{U_{\acute{e}tale}} = \mathcal{H}|_{U_{\acute{e}tale}}$ is a quasi-coherent object of $\textit{Mod}(U_{\acute{e}tale}, \mathcal{O}_ U)$.

Proof of (1). Let $\mathcal{I} \to \textit{LQCoh}(\mathcal{O}_\mathcal {X})$, $i \mapsto \mathcal{F}_ i$ be a diagram. Consider the object $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$. The pullback functor $x^*$ commutes with all colimits as it is a left adjoint. Hence $x^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i x^*\mathcal{F}_ i$. Similarly we have $x^*\mathcal{F}|_{U_{\acute{e}tale}} = \mathop{\mathrm{colim}}\nolimits _ i x^*\mathcal{F}_ i|_{U_{\acute{e}tale}}$. Now by assumption each $x^*\mathcal{F}_ i|_{U_{\acute{e}tale}}$ is quasi-coherent. Hence $\mathop{\mathrm{colim}}\nolimits _ i x^*\mathcal{F}_ i|_{U_{\acute{e}tale}}$ is quasi-coherent by Descent, Lemma 35.10.3. Thus $x^*\mathcal{F}|_{U_{\acute{e}tale}}$ is quasi-coherent as desired.

Proof of (2). It follows from (1) that cokernels exist in $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ and agree with the cokernels computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ and let $\mathcal{K} = \mathop{\mathrm{Ker}}(\varphi )$ computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$. If we can show that $\mathcal{K}$ is a locally quasi-coherent module, then the proof of (2) is complete. To see this, note that kernels are computed in the category of presheaves (no sheafification necessary). Hence $\mathcal{K}|_{U_{\acute{e}tale}}$ is the kernel of the map $\mathcal{F}|_{U_{\acute{e}tale}} \to \mathcal{G}|_{U_{\acute{e}tale}}$, i.e., is the kernel of a map of quasi-coherent sheaves on $U_{\acute{e}tale}$ whence quasi-coherent by Descent, Lemma 35.10.3. This proves (2).

Proof of (3). Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$. Since we are using the étale topology, the restriction $0 \to \mathcal{F}_1|_{U_{\acute{e}tale}} \to \mathcal{F}_2|_{U_{\acute{e}tale}} \to \mathcal{F}_3|_{U_{\acute{e}tale}} \to 0$ is a short exact sequence too. Hence (3) follows from the corresponding statement in Descent, Lemma 35.10.3.

Proof of (4). Let $\mathcal{F}$ and $\mathcal{G}$ be in $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$. Since restriction to $U_{\acute{e}tale}$ is given by pullback along the morphism of ringed topoi $U_{\acute{e}tale}\to (\mathit{Sch}/U)_{\acute{e}tale}\to \mathcal{X}_{\acute{e}tale}$ we see that the restriction of the tensor product $\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}$ to $U_{\acute{e}tale}$ is equal to $\mathcal{F}|_{U_{\acute{e}tale}} \otimes _{\mathcal{O}_ U} \mathcal{G}|_{U_{\acute{e}tale}}$, see Modules on Sites, Lemma 18.26.2. Since $\mathcal{F}|_{U_{\acute{e}tale}}$ and $\mathcal{G}|_{U_{\acute{e}tale}}$ are quasi-coherent, so is their tensor product, see Descent, Lemma 35.10.3.

Proof of (5). Let $\mathcal{F}$ and $\mathcal{G}$ be in $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$ with $\mathcal{F}$ of finite presentation. Since $(\mathit{Sch}/U)_{\acute{e}tale}= \mathcal{X}_{\acute{e}tale}/x$ is a localization of $\mathcal{X}_{\acute{e}tale}$ at an object we see that the restriction of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ to $(\mathit{Sch}/U)_{\acute{e}tale}$ is equal to

by Modules on Sites, Lemma 18.27.2. The morphism of ringed topoi $(U_{\acute{e}tale}, \mathcal{O}_ U) \to ((\mathit{Sch}/U)_{\acute{e}tale}, \mathcal{O})$ is flat as the pullback of $\mathcal{O}$ is $\mathcal{O}_ U$. Hence the pullback of $\mathcal{H}$ by this morphism is equal to $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_{U_{\acute{e}tale}}, \mathcal{G}|_{U_{\acute{e}tale}})$ by Modules on Sites, Lemma 18.31.4. In other words, the restriction of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ to $U_{\acute{e}tale}$ is $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_{U_{\acute{e}tale}}, \mathcal{G}|_{U_{\acute{e}tale}})$. Since $\mathcal{F}|_{U_{\acute{e}tale}}$ and $\mathcal{G}|_{U_{\acute{e}tale}}$ are quasi-coherent, so is $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_{U_{\acute{e}tale}}, \mathcal{G}|_{U_{\acute{e}tale}})$, see Descent, Lemma 35.10.3. We conclude as before. $\square$

Proof. Let $x$ be an arbitrary object of $\mathcal{X}$ lying over the scheme $U$. Let $\tau \in \{ Zariski, {\acute{e}tale}, fppf\} $. To show that an object $\mathcal{H}$ of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$ is in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ it suffices show that the restriction $x^*\mathcal{H}$ (Section 96.9) is a quasi-coherent object of $\textit{Mod}((\mathit{Sch}/U)_\tau , \mathcal{O})$. See Lemmas 96.11.3 and 96.11.4. Similarly for being finite locally free. Recall that $(\mathit{Sch}/U)_\tau = \mathcal{X}_\tau /x$ is a localization of $\mathcal{X}_\tau $ at an object. Hence restriction commutes with colimits, tensor products, and forming internal hom (see Modules on Sites, Lemmas 18.14.3, 18.26.2, and 18.27.2). This reduces the lemma to Descent, Lemma 35.10.6. $\square$