The Stacks project

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. $\square$