The Stacks project

83.22 Hypercovering by a simplicial object of the site: modules

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 83.21. In this section we study the augmentation

\[ a : (\mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/U)_{total}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X) \]

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 83.16.6. Thus all the results in this section are immediate consequences of the corresponding results in Section 83.20.

Lemma 83.22.1. 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

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

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

Proof. This is a special case of Lemma 83.20.1. $\square$

Lemma 83.22.2. 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

\[ E \longrightarrow Ra_*La^*E \]

is an isomorphism.

Proof. This is a special case of Lemma 83.20.2. $\square$

Lemma 83.22.3. 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

\[ R\Gamma (X, E) = R\Gamma ((\mathcal{C}/U)_{total}, La^*E) \]

for $E \in D(\mathcal{O}_\mathcal {C})$.

Proof. This is a special case of Lemma 83.20.3. $\square$

Lemma 83.22.4. 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

\[ D^+(\mathcal{O}_ X) \longrightarrow D_\mathcal {A}^+(\mathcal{O}) \]

with quasi-inverse $Ra_*$.

Proof. This is a special case of Lemma 83.20.4. $\square$


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