The Stacks project

83.20 Cohomological descent for hypercoverings of an object: modules

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

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

of Remark 83.16.6. 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 83.18.

Lemma 83.20.1. 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 83.15.7 and 83.16.6 and the discussion in the introduction to this section this follows from Lemma 83.18.1. $\square$

Lemma 83.20.2. 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

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

is an isomorphism.

Proof. Via Remarks 83.15.7 and 83.16.6 and the discussion in the introduction to this section this follows from Lemma 83.18.2. $\square$

Lemma 83.20.3. 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

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

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

Proof. Via Remarks 83.15.7 and 83.16.6 and the discussion in the introduction to this section this follows from Lemma 83.18.3. $\square$

Lemma 83.20.4. 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

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

with quasi-inverse $Ra_*$.

Proof. Via Remarks 83.15.7 and 83.16.6 and the discussion in the introduction to this section this follows from Lemma 83.18.4. $\square$


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