The Stacks project

Lemma 83.12.8. In Situation 83.3.3.

  1. An object $K$ of $D(\mathcal{C}_{total})$ is cartesian if and only if $H^ q(K)$ is a cartesian abelian sheaf for all $q$.

  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. Then an object $K$ of $D(\mathcal{O})$ is cartesian if and only if $H^ q(K)$ is a cartesian $\mathcal{O}$-module for all $q$.

Proof. Part (1) is true because the pullback functors $(f_\varphi )^{-1}$ are exact. Part (2) follows from the characterization in Lemma 83.12.2 and the fact that $L(f_{\delta ^ n_ j})^* = (f_{\delta ^ n_ j})^*$ by flatness. $\square$


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