The Stacks project

Proof. Proof of (1). We may replace each of the algebraic stacks $\mathcal{X}_ j$ by a scheme $U_ j$ (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma 101.33.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_ j)_{\acute{e}tale}$ is still quasi-coherent, see Modules on Sites, Lemma 18.23.4. Then $f = \coprod f_ j : U = \coprod U_ j \to \mathcal{X}$ is a smooth surjective morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 96.19.10 there exists an étale covering $\{ x_ i \to x\} _{i \in I}$ such that each $x_ i$ lifts to an object $u_ i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. This just means that $x_ i$ lives over a scheme $V_ i$, that $\{ V_ i \to V\} $ is an étale covering, and that $x_ i$ comes from a morphism $u_ i : V_ i \to U$. Then $x_ i^*\mathcal{F} = u_ i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 96.11.4 we see that $x^*\mathcal{F}$ is an fppf sheaf and since $x$ was arbitrary we see that $\mathcal{F}$ is a sheaf in the fppf topology. Applying Sheaves on Stacks, Lemma 96.11.3 we see that $\mathcal{F}$ is quasi-coherent.

Proof of (2). This is proved using exactly the same argument, which we fully write out here. We may replace each of the algebraic stacks $\mathcal{X}_ j$ by a scheme $U_ j$ (using that any algebraic stack has a smooth covering by a scheme and that flat and locally finite presented morphisms are preserved by composition, see Morphisms of Stacks, Lemmas 101.25.2 and 101.27.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_ j)_{\acute{e}tale}$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma 96.11.2. Then $f = \coprod f_ j : U = \coprod U_ j \to \mathcal{X}$ is a surjective, flat, and locally finitely presented morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 96.19.10 there exists an fppf covering $\{ x_ i \to x\} _{i \in I}$ such that each $x_ i$ lifts to an object $u_ i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. This just means that $x_ i$ lives over a scheme $V_ i$, that $\{ V_ i \to V\} $ is an fppf covering, and that $x_ i$ comes from a morphism $u_ i : V_ i \to U$. Then $x_ i^*\mathcal{F} = u_ i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 96.11.3 we see that $\mathcal{F}$ is quasi-coherent. $\square$

Proof. Pick a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By Modules on Sites, Definition 18.23.1 there exists an étale (resp. fppf) covering $\{ U_ i \to U\} _{i \in I}$ such that each pullback $f_ i^{-1}\mathcal{H}$ has a global presentation (see Modules on Sites, Definition 18.17.1). Here $f_ i : U_ i \to \mathcal{X}$ is the composition $U_ i \to U \to \mathcal{X}$ which is a morphism of algebraic stacks. (Recall that the pullback “is” the restriction to $\mathcal{X}/f_ i$, see Sheaves on Stacks, Definition 96.9.2 and the discussion following.) Since each $f_ i$ is smooth (resp. flat) by Lemma 103.15.1 we see that $f_ i^{-1}g_!\mathcal{H} = g_{i, !}(f'_ i)^{-1}\mathcal{H}$. Using Lemma 103.16.1 we reduce the statement of the lemma to the case where $\mathcal{H}$ has a global presentation. Say we have

of $\mathcal{O}$-modules where $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ (resp. $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$). Since $g_!$ commutes with arbitrary colimits (as a left adjoint functor, see Lemma 103.14.4 and Categories, Lemma 4.24.5) we conclude that there exists an exact sequence

Lemma 103.14.5 shows that $g_!\mathcal{O} = \mathcal{O}_\mathcal {X}$. In case (2) we are done. In case (1) we apply Sheaves on Stacks, Lemma 96.11.4 to conclude. $\square$

Proof. Pullback by any morphism of ringed topoi preserves categories of quasi-coherent modules, see Modules on Sites, Lemma 18.23.4. Hence $g^{-1}$ preserves the categories of quasi-coherent modules; here we use that $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) = \mathit{QCoh}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ by Sheaves on Stacks, Lemma 96.11.4. The same is true for $g_!$ by Lemma 103.16.2. We know that $\mathcal{H} \to g^{-1}g_!\mathcal{H}$ is an isomorphism by Lemma 103.14.2. Conversely, if $\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ then the map $g_!g^{-1}\mathcal{F} \to \mathcal{F}$ is a map of quasi-coherent modules on $\mathcal{X}$ whose restriction to any scheme smooth over $\mathcal{X}$ is an isomorphism. Then the discussion in Sheaves on Stacks, Sections 96.14 and 96.15 (comparing with quasi-coherent modules on presentations) shows it is an isomorphism. This proves (1)(a) and (2)(a).

Let $\mathcal{F}$ be an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. By Lemma 103.10.2 the kernel and cokernel of the map $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic. Hence by Lemma 103.14.6 and since $g^* = g^{-1}$ is exact, we conclude $g^*Q(\mathcal{F}) \to g^*\mathcal{F}$ is an isomorphism. Thus $g^*\mathcal{F}$ is quasi-coherent. This proves (1)(b) and (2)(b). Finally, (1)(c) and (2)(c) follow because $g_!g^*Q(\mathcal{F}) \to Q(\mathcal{F})$ is an isomorphism by our arguments above. $\square$

Proof. We will verify conditions (1), (2), (3), (4) of Homology, Lemma 12.10.3.

Since $0$ is a quasi-coherent module on any ringed site we see that (1) holds.

By definition $\mathit{QCoh}(\mathcal{O})$ is a strictly full subcategory $\textit{Mod}(\mathcal{O})$, so (2) holds.

Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a morphism of quasi-coherent modules on $\mathcal{X}_{lisse,{\acute{e}tale}}$ or $\mathcal{X}_{flat,fppf}$. We have $g^*g_!\mathcal{F} = \mathcal{F}$ and similarly for $\mathcal{G}$ and $\varphi $, see Lemma 103.14.4. By Lemma 103.16.2 we see that $g_!\mathcal{F}$ and $g_!\mathcal{G}$ are quasi-coherent $\mathcal{O}_\mathcal {X}$-modules. By Sheaves on Stacks, Lemma 96.15.1 we have that $\mathop{\mathrm{Coker}}(g_!\varphi )$ is a quasi-coherent module on $\mathcal{X}$ (and the cokernel in the category of quasi-coherent modules on $\mathcal{X}$). Since $g^*$ is exact (see Lemma 103.14.2) $g^*\mathop{\mathrm{Coker}}(g_!\varphi ) = \mathop{\mathrm{Coker}}(g^*g_!\varphi ) = \mathop{\mathrm{Coker}}(\varphi )$ is quasi-coherent too (see Lemma 103.16.3). By Proposition 103.8.1 the kernel $\mathop{\mathrm{Ker}}(g_!\varphi )$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. Since $g^*$ is exact, we have $g^*\mathop{\mathrm{Ker}}(g_!\varphi ) = \mathop{\mathrm{Ker}}(g^*g_!\varphi ) = \mathop{\mathrm{Ker}}(\varphi )$. Since $g^*$ maps objects of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ to quasi-coherent modules by Lemma 103.16.3 we conclude that $\mathop{\mathrm{Ker}}(\varphi )$ is quasi-coherent as well. This proves (3).

Finally, suppose that

is an extension of $\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$-modules (resp. $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-modules) with $\mathcal{F}$ and $\mathcal{G}$ quasi-coherent. To prove (4) and finish the proof we have to show that $\mathcal{E}$ is quasi-coherent on $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$). Let $U$ be an object of $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$; we think of $U$ as a scheme smooth (resp. flat) over $\mathcal{X}$. We have to show that the restriction of $\mathcal{E}$ to $U_{lisse,{\acute{e}tale}}$ (resp. $=U_{flat,fppf}$) is quasi-coherent. Thus we may assume that $\mathcal{X} = U$ is a scheme. Because $\mathcal{G}$ is quasi-coherent on $U_{lisse,{\acute{e}tale}}$ (resp. $U_{flat,fppf}$), we may assume, after replacing $U$ by the members of an étale (resp. fppf) covering, that $\mathcal{G}$ has a presentation

on $U_{lisse,{\acute{e}tale}}$ (resp. $U_{flat,fppf}$) where $\mathcal{O}$ is the structure sheaf on the site. We may also assume $U$ is affine. Since $\mathcal{F}$ is quasi-coherent, we have

Namely, $\mathcal{F}$ is the pullback of a quasi-coherent module $\mathcal{F}'$ on the big site of $U$ (by Lemma 103.16.3), cohomology of $\mathcal{F}$ and $\mathcal{F}'$ agree (by Lemma 103.14.3), and we know that the cohomology of $\mathcal{F}'$ on the big site of the affine scheme $U$ is zero (to get this in the current situation you have to combine Descent, Propositions 35.8.9 and 35.9.3 with Cohomology of Schemes, Lemma 30.2.2). Thus we can lift the map $\bigoplus _{i \in I} \mathcal{O} \to \mathcal{G}$ to $\mathcal{E}$. A diagram chase shows that we obtain an exact sequence

By (3) proved above, we conclude that $\mathcal{E}$ is quasi-coherent as desired. $\square$