In this section we assume $\mathcal{C}$ has fibre products and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. We let $K$ be a hypercovering of $X$ as defined in Hypercoverings, Definition 25.3.3. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings on $\mathcal{C}$. Set $\mathcal{O}_ X = \mathcal{O}_\mathcal {C}|_{\mathcal{C}/X}$. We will study the augmentation
of Remark 85.16.6. Observe that $\mathcal{C}/X$ is a site which has equalizers and fibre products and that $K$ is a hypercovering for the site $\mathcal{C}/X$. Therefore the results in this section are immediate consequences of the corresponding results in Section 85.18.
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 is fully faithful with essential image the cartesian $\mathcal{O}$-modules. The functor $a_*$ provides the quasi-inverse.
\[ a^* : \textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}) \]
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. $\square$
Lemma 85.20.2. 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 $X$. For $E \in D(\mathcal{O}_ X)$ the map is an isomorphism.
\[ E \longrightarrow Ra_*La^*E \]
Proof. Via Remarks 85.15.7 and 85.16.6 and the discussion in the introduction to this section this follows from Lemma 85.18.2. $\square$
Lemma 85.20.3. 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 $X$. Then we have a canonical isomorphism for $E \in D(\mathcal{O}_\mathcal {C})$.
\[ R\Gamma (X, E) = R\Gamma ((\mathcal{C}/K)_{total}, La^*E) \]
Proof. Via Remarks 85.15.7 and 85.16.6 and the discussion in the introduction to this section this follows from Lemma 85.18.3. $\square$
Lemma 85.20.4. 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 $X$. Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$ denote the weak Serre subcategory of cartesian $\mathcal{O}$-modules. Then the functor $La^*$ defines an equivalence with quasi-inverse $Ra_*$.
\[ D^+(\mathcal{O}_ X) \longrightarrow D_\mathcal {A}^+(\mathcal{O}) \]
Proof. Via Remarks 85.15.7 and 85.16.6 and the discussion in the introduction to this section this follows from Lemma 85.18.4. $\square$
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