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83.19 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 83.16.4. 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 25.3.7. This means that every single result proved for hypercoverings in Section 83.17 has an immediate analogue in the situation in this section.

Lemma 83.19.1. 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

  1. $a^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X) \to \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of sets,

  2. $a^{-1} : \textit{Ab}(\mathcal{C}/X) \to \textit{Ab}((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups.

In both cases $a_*$ provides the quasi-inverse functor.

Proof. Via Remarks 83.15.5 and 83.16.4 and the discussion in the introduction to this section this follows from Lemma 83.17.1. $\square$

Lemma 83.19.2. 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 83.15.5 and 83.16.4 and the discussion in the introduction to this section this follows from Lemma 83.17.3. $\square$

Lemma 83.19.3. 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 $E \in D(\mathcal{C}/X)$.

Lemma 83.19.4. 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 $Ra_*$.

[1] The converse may not be the case, i.e., if $K$ is a simplicial object of $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$ which defines a hypercovering for the site $\mathcal{C}/X$ as in Hypercoverings, Definition 25.6.1, then it may not be true that $K$ defines a hypercovering of $X$. For example, if $K_0 = \{ U_{0, i}\} _{i \in I_0}$ then the latter condition guarantees $\{ U_{0, i} \to X\} $ is a covering of $\mathcal{C}$ whereas the former condition only requires $\coprod h_{U_{0, i}}^\# \to h_ X^\# $ to be a surjective map of sheaves.

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