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83.18 Cohomological descent for hypercoverings: modules

Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Assume $\mathcal{C}$ has equalizers and fibre products and let $K$ be a hypercovering as defined in Hypercoverings, Definition 25.6.1. We will study cohomological descent for the augmentation

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

of Remark 83.16.5.

Lemma 83.18.1. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $K$ be a hypercovering. With notation as above

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

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

Proof. Since $a^{-1}\mathcal{O}_\mathcal {C} = \mathcal{O}$ we have $a^* = a^{-1}$. Hence the lemma follows immediately from Lemma 83.17.1. $\square$

Lemma 83.18.2. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $K$ be a hypercovering. For $E \in D(\mathcal{O}_\mathcal {C})$ the map

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

is an isomorphism.

Proof. Since $a^{-1}\mathcal{O}_\mathcal {C} = \mathcal{O}$ we have $La^* = a^* = a^{-1}$. Moreover $Ra_*$ agrees with $Ra_*$ on abelian sheaves, see Cohomology on Sites, Lemma 21.20.7. Hence the lemma follows immediately from Lemma 83.17.3. $\square$

Lemma 83.18.3. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $K$ be a hypercovering. Then we have a canonical isomorphism

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

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

Proof. This follows from Lemma 83.18.2 because $R\Gamma ((\mathcal{C}/K)_{total}, -) = R\Gamma (\mathcal{C}, -) \circ Ra_*$ by Cohomology on Sites, Remark 21.14.4 or by Cohomology on Sites, Lemma 21.20.5. $\square$

Lemma 83.18.4. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $K$ be a hypercovering. 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

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

with quasi-inverse $Ra_*$.

Proof. Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma 83.12.6 (the required hypotheses hold by the discussion in Remark 83.16.5). The equivalence is a formal consequence of the results obtained so far. Use Lemmas 83.18.1 and 83.18.2 and Cohomology on Sites, Lemma 21.27.5. $\square$


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