The Stacks project

Lemma 85.12.6. In Situation 85.3.3.

  1. The full subcategory of cartesian abelian sheaves forms a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{total})$. Colimits of systems of cartesian abelian sheaves are cartesian.

  2. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ such that the morphisms

    \[ f_{\delta ^ n_ j} : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{n - 1}), \mathcal{O}_{n - 1}) \]

    are flat. The full subcategory of cartesian $\mathcal{O}$-modules forms a weak Serre subcategory of $\textit{Mod}(\mathcal{O})$. Colimits of systems of cartesian $\mathcal{O}$-modules are cartesian.

Proof. To see we obtain a weak Serre subcategory in (1) we check the conditions listed in Homology, Lemma 12.10.3. First, if $\varphi : \mathcal{F} \to \mathcal{G}$ is a map between cartesian abelian sheaves, then $\mathop{\mathrm{Ker}}(\varphi )$ and $\mathop{\mathrm{Coker}}(\varphi )$ are cartesian too because the restriction functors $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n)$ and the functors $f_\varphi ^{-1}$ are exact. Similarly, if

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

is a short exact sequence of abelian sheaves on $\mathcal{C}_{total}$ with $\mathcal{F}$ and $\mathcal{G}$ cartesian, then it follows that $\mathcal{H}$ is cartesian from the 5-lemma. To see the property of colimits, use that colimits commute with pullback as pullback is a left adjoint. In the case of modules we argue in the same manner, using the exactness of flat pullback (Modules on Sites, Lemma 18.31.2) and the fact that it suffices to check the condition for $f_{\delta ^ n_ j}$, see Lemma 85.12.2. $\square$

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