The Stacks project

Lemma 85.18.1. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $K$ be a hypercovering. With notation as above

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

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

Proof. Since $a^{-1}\mathcal{O}_\mathcal {C} = \mathcal{O}$ we have $a^* = a^{-1}$. Hence the lemma follows immediately from Lemma 85.17.1. $\square$

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