The Stacks project

Lemma 85.20.1. Let $\mathcal{C}$ be a site with fibre products and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $K$ be a hypercovering of $X$. With notation as above

\[ a^* : \textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}) \]

is fully faithful with essential image the cartesian $\mathcal{O}$-modules. The functor $a_*$ provides the quasi-inverse.

Proof. Via Remarks 85.15.7 and 85.16.6 and the discussion in the introduction to this section this follows from Lemma 85.18.1. $\square$

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View Lemma 85.20.1 as pdf