The Stacks project

Lemma 83.12.10. In Situation 83.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Then $\mathcal{F}$ is quasi-coherent in the sense of Modules on Sites, Definition 18.23.1 if and only if $\mathcal{F}$ is cartesian and $\mathcal{F}_ n$ is a quasi-coherent $\mathcal{O}_ n$-module for all $n$.

Proof. Assume $\mathcal{F}$ is quasi-coherent. Since pullbacks of quasi-coherent modules are quasi-coherent (Modules on Sites, Lemma 18.23.4) we see that $\mathcal{F}_ n$ is a quasi-coherent $\mathcal{O}_ n$-module for all $n$. To show that $\mathcal{F}$ is cartesian, let $U$ be an object of $\mathcal{C}_ n$ for some $n$. Let us view $U$ as an object of $\mathcal{C}_{total}$. Because $\mathcal{F}$ is quasi-coherent there exists a covering $\{ U_ i \to U\} $ and for each $i$ a presentation

\[ \bigoplus \nolimits _{j \in J_ i} \mathcal{O}_{\mathcal{C}_{total}/U_ i} \to \bigoplus \nolimits _{k \in K_ i} \mathcal{O}_{\mathcal{C}_{total}/U_ i} \to \mathcal{F}|_{\mathcal{C}_{total}/U_ i} \to 0 \]

Observe that $\{ U_ i \to U\} $ is a covering of $\mathcal{C}_ n$ by the construction of the site $\mathcal{C}_{total}$. Next, let $V$ be an object of $\mathcal{C}_ m$ for some $m$ and let $V \to U$ be a morphism of $\mathcal{C}_{total}$ lying over $\varphi : [n] \to [m]$. The fibre products $V_ i = V \times _ U U_ i$ exist and we get an induced covering $\{ V_ i \to V\} $ in $\mathcal{C}_ m$. Restricting the presentation above to the sites $\mathcal{C}_ n/U_ i$ and $\mathcal{C}_ m/V_ i$ we obtain presentations

\[ \bigoplus \nolimits _{j \in J_ i} \mathcal{O}_{\mathcal{C}_ m/U_ i} \to \bigoplus \nolimits _{k \in K_ i} \mathcal{O}_{\mathcal{C}_ m/U_ i} \to \mathcal{F}_ n|_{\mathcal{C}_ n/U_ i} \to 0 \]

and

\[ \bigoplus \nolimits _{j \in J_ i} \mathcal{O}_{\mathcal{C}_ m/V_ i} \to \bigoplus \nolimits _{k \in K_ i} \mathcal{O}_{\mathcal{C}_ m/V_ i} \to \mathcal{F}_ m|_{\mathcal{C}_ m/V_ i} \to 0 \]

These presentations are compatible with the map $\mathcal{F}(\varphi ) : f_\varphi ^*\mathcal{F}_ n \to \mathcal{F}_ m$ (as this map is defined using the restriction maps of $\mathcal{F}$ along morphisms of $\mathcal{C}_{total}$ lying over $\varphi $). We conclude that $\mathcal{F}(\varphi )|_{\mathcal{C}_ m/V_ i}$ is an isomorphism. As $\{ V_ i \to V\} $ is a covering we conclude $\mathcal{F}(\varphi )|_{\mathcal{C}_ m/V}$ is an isomorphism. Since $V$ and $U$ were arbitrary this proves that $\mathcal{F}$ is cartesian. (In case A use Sites, Lemma 7.14.10.)

Conversely, assume $\mathcal{F}_ n$ is quasi-coherent for all $n$ and that $\mathcal{F}$ is cartesian. Then for any $n$ and object $U$ of $\mathcal{C}_ n$ we can choose a covering $\{ U_ i \to U\} $ of $\mathcal{C}_ n$ and for each $i$ a presentation

\[ \bigoplus \nolimits _{j \in J_ i} \mathcal{O}_{\mathcal{C}_ m/U_ i} \to \bigoplus \nolimits _{k \in K_ i} \mathcal{O}_{\mathcal{C}_ m/U_ i} \to \mathcal{F}_ n|_{\mathcal{C}_ n/U_ i} \to 0 \]

Pulling back to $\mathcal{C}_{total}/U_ i$ we obtain complexes

\[ \bigoplus \nolimits _{j \in J_ i} \mathcal{O}_{\mathcal{C}_{total}/U_ i} \to \bigoplus \nolimits _{k \in K_ i} \mathcal{O}_{\mathcal{C}_{total}/U_ i} \to \mathcal{F}|_{\mathcal{C}_{total}/U_ i} \to 0 \]

of modules on $\mathcal{C}_{total}/U_ i$. Then the property that $\mathcal{F}$ is cartesian implies that this is exact. We omit the details. $\square$


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