The Stacks project

Lemma 83.6.3. With assumptions as in Lemma 83.6.1 the functor $g_{n!} : \textit{Mod}(\mathcal{O}_ n) \to \textit{Mod}(\mathcal{O})$ is exact if the maps $f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ are flat for all $\varphi : [n] \to [m]$.

Proof. Recall that $g_{n!}\mathcal{G}$ is the $\mathcal{O}$-module whose degree $m$ part is the $\mathcal{O}_ m$-module

\[ \bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^*\mathcal{G} \]

Here the morphism of ringed topoi $f_\varphi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ m), \mathcal{O}_ m) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n)$ uses the map $f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ of the statement of the lemma. If these maps are flat, then $f_\varphi ^*$ is exact (Modules on Sites, Lemma 18.31.2). By definition of the site $\mathcal{C}_{total}$ we see that these functors have the desired exactness properties and we conclude. $\square$


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