85.19 Cohomological descent for hypercoverings of an object
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. We will study the augmentation
\[ a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X) \]
of Remark 85.16.4. 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$1 by Hypercoverings, Lemma 25.3.9. This means that every single result proved for hypercoverings in Section 85.17 has an immediate analogue in the situation in this section.
Lemma 85.19.1. Let $\mathcal{C}$ be a site with fibre products and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $K$ be a hypercovering of $X$. Then
$a^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X) \to \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of sets,
$a^{-1} : \textit{Ab}(\mathcal{C}/X) \to \textit{Ab}((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups.
In both cases $a_*$ provides the quasi-inverse functor.
Proof.
Via Remarks 85.15.5 and 85.16.4 and the discussion in the introduction to this section this follows from Lemma 85.17.1.
$\square$
Lemma 85.19.2. Let $\mathcal{C}$ be a site with fibre product and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $K$ be a hypercovering of $X$. For $E \in D(\mathcal{C}/X)$ the map
\[ E \longrightarrow Ra_*a^{-1}E \]
is an isomorphism.
Proof.
Via Remarks 85.15.5 and 85.16.4 and the discussion in the introduction to this section this follows from Lemma 85.17.3.
$\square$
Lemma 85.19.3. Let $\mathcal{C}$ be a site with fibre products and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $K$ be a hypercovering of $X$. Then we have a canonical isomorphism
\[ R\Gamma (X, E) = R\Gamma ((\mathcal{C}/K)_{total}, a^{-1}E) \]
for $E \in D(\mathcal{C}/X)$.
Proof.
Via Remarks 85.15.5 and 85.16.4 this follows from Lemma 85.17.4.
$\square$
Lemma 85.19.4. Let $\mathcal{C}$ be a site with fibre products and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $K$ be a hypercovering of $X$. Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})$ denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor $a^{-1}$ defines an equivalence
\[ D^+(\mathcal{C}/X) \longrightarrow D_\mathcal {A}^+((\mathcal{C}/K)_{total}) \]
with quasi-inverse $Ra_*$.
Proof.
Via Remarks 85.15.5 and 85.16.4 this follows from Lemma 85.17.5.
$\square$
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