The Stacks project

Lemma 85.34.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. Assume $a : U \to X$ is an fppf hypercovering of $X$. Then $\mathit{QCoh}(\mathcal{O}_ U)$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_ U)$ and

\[ a^* : D_\mathit{QCoh}(\mathcal{O}_ X) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ U) \]

is an equivalence of categories with quasi-inverse given by $Ra_*$. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 85.32.

Proof. First observe that the maps $a_ n : U_ n \to X$ and $d^ n_ i : U_ n \to U_{n - 1}$ are flat, locally of finite presentation, and surjective by Hypercoverings, Remark 25.8.4.

Recall that an $\mathcal{O}_ U$-module $\mathcal{F}$ is quasi-coherent if and only if it is cartesian and $\mathcal{F}_ n$ is quasi-coherent for all $n$. See Lemma 85.12.10. By Lemma 85.12.6 (and flatness of the maps $d^ n_ i : U_ n \to U_{n - 1}$ shown above) the cartesian modules for a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_ U)$. On the other hand $\mathit{QCoh}(\mathcal{O}_{U_ n}) \subset \textit{Mod}(\mathcal{O}_{U_ n})$ is a weak Serre subcategory for each $n$ (Properties of Spaces, Lemma 66.29.7). Combined we see that $\mathit{QCoh}(\mathcal{O}_ U) \subset \textit{Mod}(\mathcal{O}_ U)$ is a weak Serre subcategory.

To finish the proof we check the conditions (1) – (5) of Cohomology on Sites, Lemma 21.28.6 one by one.

Ad (1). This holds since $a_ n$ flat (seen above) implies $a$ is flat by Lemma 85.11.1.

Ad (2). This is the content of Lemma 85.34.2.

Ad (3). This is the content of Lemma 85.34.3.

Ad (4). Recall that we can use either the site $U_{\acute{e}tale}$ or $U_{spaces, {\acute{e}tale}}$ to define the small étale topos $\mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$, see Section 85.32. The assumption of Cohomology on Sites, Situation 21.25.1 holds for the triple $(U_{spaces, {\acute{e}tale}}, \mathcal{O}_ U, \mathit{QCoh}(\mathcal{O}_ U))$ and by the same reasoning for the triple $(U_{\acute{e}tale}, \mathcal{O}_ U, \mathit{QCoh}(\mathcal{O}_ U))$. Namely, take

\[ \mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (U_{\acute{e}tale}) \subset \mathop{\mathrm{Ob}}\nolimits (U_{spaces, {\acute{e}tale}}) \]

to be the set of affine objects. For $V/U_ n \in \mathcal{B}$ take $d_{V/U_ n} = 0$ and take $\text{Cov}_{V/U_ n}$ to be the set of étale coverings $\{ V_ i \to V\} $ with $V_ i$ affine. Then we get the desired vanishing because for $\mathcal{F} \in \mathit{QCoh}(\mathcal{O}_ U)$ and any $V/U_ n \in \mathcal{B}$ we have

\[ H^ p(V/U_ n, \mathcal{F}) = H^ p(V, \mathcal{F}_ n) \]

by Lemma 85.10.4. Here on the right hand side we have the cohomology of the quasi-coherent sheaf $\mathcal{F}_ n$ on $U_ n$ over the affine obect $V$ of $U_{n, {\acute{e}tale}}$. This vanishes for $p > 0$ by the discussion in Cohomology of Spaces, Section 69.3 and Cohomology of Schemes, Lemma 30.2.2.

Ad (5). Follows by taking $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$ the set of affine objects and the references given above. $\square$

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