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 on $\mathcal{C}$. Let $U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined in Section 85.21. In this section we study the augmentation
we obtain by thinking of $U$ as a simiplical semi-representable object of $\mathcal{C}/X$ whose degree $n$ part is the singleton element $\{ U_ n/X\} $ and applying the constructions in Remark 85.16.6. Thus all the results in this section are immediate consequences of the corresponding results in Section 85.20.
Lemma 85.22.1. 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 $U$ be a hypercovering of $X$ in $\mathcal{C}$. 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. This is a special case of Lemma 85.20.1. $\square$
Lemma 85.22.2. 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 $U$ be a hypercovering of $X$ in $\mathcal{C}$. For $E \in D(\mathcal{O}_ X)$ the map is an isomorphism.
\[ E \longrightarrow Ra_*La^*E \]
Proof. This is a special case of Lemma 85.20.2. $\square$
Lemma 85.22.3. 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 $U$ be a hypercovering of $X$ in $\mathcal{C}$. Then we have a canonical isomorphism for $E \in D(\mathcal{O}_\mathcal {C})$.
\[ R\Gamma (X, E) = R\Gamma ((\mathcal{C}/U)_{total}, La^*E) \]
Proof. This is a special case of Lemma 85.20.3. $\square$
Lemma 85.22.4. 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 $U$ be a hypercovering of $X$ in $\mathcal{C}$. 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. This is a special case of Lemma 85.20.4. $\square$
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