Lemma 85.12.6. In Situation 85.3.3.
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.
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.
Comments (0)