The Stacks project

Lemma 100.12.5. Let $\mathcal{X}$ be an algebraic stack.

  1. $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$.

  2. $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$.

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 100.11.3. By Lemma 100.12.2 we see that $g_!\mathcal{F}$ and $g_!\mathcal{G}$ are quasi-coherent $\mathcal{O}_\mathcal {X}$-modules. Hence we see that $\mathop{\mathrm{Ker}}(g_!\varphi )$ and $\mathop{\mathrm{Coker}}(g_!\varphi )$ are quasi-coherent modules on $\mathcal{X}$. Since $g^*$ is exact (see Lemma 100.11.2) we see that $g^*\mathop{\mathrm{Ker}}(g_!\varphi ) = \mathop{\mathrm{Ker}}(g^*g_!\varphi ) = \mathop{\mathrm{Ker}}(\varphi )$ and $g^*\mathop{\mathrm{Coker}}(g_!\varphi ) = \mathop{\mathrm{Coker}}(g^*g_!\varphi ) = \mathop{\mathrm{Coker}}(\varphi )$ are quasi-coherent too (see Lemma 100.12.3). This proves (3).

Finally, suppose that

\[ 0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0 \]

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. We have to show that $\mathcal{E}$ is quasi-coherent on $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$). We strongly urge the reader to write out what this means on a napkin and prove it him/herself rather than reading the somewhat labyrinthine proof that follows. By Lemma 100.12.3 this is true if and only if $g_!\mathcal{E}$ is quasi-coherent. By Lemmas 100.12.1 and Lemma 100.11.6 we may check this after replacing $\mathcal{X}$ by a smooth (resp. fppf) cover. Choose a scheme $U$ and a smooth surjective morphism $U \to \mathcal{X}$. By definition there exists an ├ętale (resp. fppf) covering $\{ U_ i \to U\} _ i$ such that $\mathcal{G}$ has a global presentation over each $U_ i$. Replacing $\mathcal{X}$ by $U_ i$ (which is permissible by the discussion above) we may assume that the site $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$) has a final object $U$ (in other words $\mathcal{X}$ is representable by the scheme $U$) and that $\mathcal{G}$ has a global presentation

\[ \bigoplus \nolimits _{j \in J} \mathcal{O} \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O} \longrightarrow \mathcal{G} \longrightarrow 0 \]

of $\mathcal{O}$-modules where $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ (resp. $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$). Let $\mathcal{E}'$ be the pullback of $\mathcal{E}$ via the map $\bigoplus \nolimits _{i \in I} \mathcal{O} \to \mathcal{G}$. Then we see that $\mathcal{E}$ is the cokernel of a map $\bigoplus \nolimits _{j \in J} \mathcal{O} \to \mathcal{E}'$ hence by property (3) which we proved above, it suffices to prove that $\mathcal{E}'$ is quasi-coherent. Consider the exact sequence

\[ L_1g_!\left(\bigoplus \nolimits _{i \in I}\mathcal{O}\right) \to g_!\mathcal{F} \to g_!\mathcal{E}' \to g_!\left(\bigoplus \nolimits _{i \in I}\mathcal{O}\right) \to 0 \]

where $L_1g_!$ is the first left derived functor of $g_! : \textit{Mod}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ (resp. $g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \to \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}})$). This derived functor exists by Cohomology on Sites, Lemma 21.36.2. Moreover, since $\mathcal{O} = j_{U!}\mathcal{O}_ U$ we have $Lg_!\mathcal{O} = g_!\mathcal{O} = \mathcal{O}_\mathcal {X}$ also by Cohomology on Sites, Lemma 21.36.2. Thus the left hand term vanishes and we obtain a short exact sequence

\[ 0 \to g_!\mathcal{F} \to g_!\mathcal{E}' \to \bigoplus \nolimits _{i \in I}\mathcal{O}_\mathcal {X} \to 0 \]

By Proposition 100.7.4 it follows that $g_!\mathcal{E}'$ is locally quasi-coherent with the flat base change property. Finally, Lemma 100.12.3 implies that $\mathcal{E}' = g^{-1}g_!\mathcal{E}'$ is quasi-coherent as desired. $\square$


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