The Stacks project

Proof. For the underlying diagram of topoi we refer to the discussion in the proof of Lemma 85.33.1. The sheaf $\mathcal{O}_ U$ is the structure sheaf of the simplicial algebraic space $U$ as defined in Section 85.32. The sheaf $\mathcal{O}_ X$ is the usual structure sheaf of the algebraic space $X$. The sheaves of rings $\mathcal{O}_{big, total}$ and $\mathcal{O}_{big}$ come from the structure sheaf on $(\textit{Spaces}/S)_{fppf}$ in the manner explained in Section 85.22 which also constructs $a_{fppf}$ as a morphism of ringed topoi. The component morphisms $h_ n = a_{U_ n}$ and $h_{-1} = a_ X$ are morphisms of ringed topoi by More on Cohomology of Spaces, Section 84.7. Finally, since the continuous functor $u : U_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/U)_{fppf, total}$ used to define $h$¹ is given by $V/U_ n \mapsto V/U_ n$ we see that $h_*\mathcal{O}_{big, total} = \mathcal{O}_ U$ which is how we endow $h$ with the structure of a morphism of ringed simplicial sites as in Remark 85.7.1. Then we obtain $h$ as a morphism of ringed topoi by Lemma 85.7.2. Please observe that the morphisms $h_ n$ indeed agree with the morphisms $a_{U_ n}$ described above. We omit the verification that the diagram is commutative (as a diagram of ringed topoi – we already know it is commutative as a diagram of topoi). $\square$

Proof. Consider the diagram of Lemma 85.34.1. In the proof of this lemma we have seen that $h_{-1}$ is the morphism $a_ X$ of More on Cohomology of Spaces, Section 84.7. Thus it follows from More on Cohomology of Spaces, Lemma 84.7.1 that

is an equivalence with quasi-inverse $h_{-1, *}$. The same holds true for the components $h_ n$ of $h$. Recall that $\mathit{QCoh}(\mathcal{O}_ U)$ and $\mathit{QCoh}(\mathcal{O}_{big, total})$ consist of cartesian modules whose components are quasi-coherent, see Lemma 85.12.10. Since the functors $h^*$ and $h_*$ of Lemma 85.7.2 agree with the functors $h_ n^*$ and $h_{n, *}$ on components we conclude that

is an equivalence with quasi-inverse $h_*$. Observe that $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{fppf}$ as defined in Section 85.21. By Lemma 85.22.1 we see that $a_{fppf}^*$ is fully faithful with quasi-inverse $a_{fppf, *}$ and with essential image the cartesian sheaves of $\mathcal{O}_{fppf, total}$-modules. Thus, by the description of $\mathit{QCoh}(\mathcal{O}_{big})$ and $\mathit{QCoh}(\mathcal{O}_{big, total})$ of Lemma 85.12.10, we get an equivalence

with quasi-inverse given by $a_{fppf, *}$. A formal argument (chasing around the diagram) now shows that $a^*$ is fully faithful on $\mathit{QCoh}(\mathcal{O}_ X)$ and has image contained in $\mathit{QCoh}(\mathcal{O}_ U)$.

Finally, suppose that $\mathcal{G}$ is in $\mathit{QCoh}(\mathcal{O}_ U)$. Then $h^*\mathcal{G}$ is in $\mathit{QCoh}(\mathcal{O}_{big, total})$. Hence $h^*\mathcal{G} = a_{fppf}^*\mathcal{H}$ with $\mathcal{H} = a_{fppf, *}h^*\mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_{big})$ (see above). In turn we see that $\mathcal{H} = (h_{-1})^*\mathcal{F}$ with $\mathcal{F} = h_{-1, *}\mathcal{H}$ in $\mathit{QCoh}(\mathcal{O}_ X)$. Going around the diagram we deduce that $h^*\mathcal{G} \cong h^*a^*\mathcal{F}$. By fully faithfulness of $h^*$ we conclude that $a^*\mathcal{F} \cong \mathcal{G}$. Since $\mathcal{F} = h_{-1, *}a_{fppf, *}h^*\mathcal{G} = a_*h_*h^*\mathcal{G} = a_*\mathcal{G}$ we also obtain the statement that the quasi-inverse is given by $a_*$. $\square$

Proof. Consider the diagram of Lemma 85.33.1. Let $\mathcal{F}_ n = a_ n^*\mathcal{F}$ be the $n$th component of $a^*\mathcal{F}$. This is a quasi-coherent $\mathcal{O}_{U_ n}$-module. Then $\mathcal{F}_ n = Rh_{n, *}h_ n^*\mathcal{F}_ n$ by More on Cohomology of Spaces, Lemma 84.7.2. Hence $a^*\mathcal{F} = Rh_*h^*a^*\mathcal{F}$ by Lemma 85.7.3. We have

The first equality by the discussion above, the second equality because of the commutativity of the diagram in Lemma 85.25.1, the third equality by Lemma 85.22.2 as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{fppf}$ and $La_{fppf}^* = a_{fppf}^*$ as $a_{fppf}$ is flat (namely $a_{fppf}^{-1}\mathcal{O}_{big} = \mathcal{O}_{big, total}$, see Remark 85.16.5), and the last equality by the already used More on Cohomology of Spaces, Lemma 84.7.2. $\square$

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

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

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. $\square$

Proof. This follows from Lemma 85.34.4 because $R\Gamma (U_{\acute{e}tale}, -) = R\Gamma (X_{\acute{e}tale}, -) \circ Ra_*$ by Cohomology on Sites, Remark 21.14.4. $\square$

Proof. Immediate consequence of Lemmas 85.34.5 and 85.10.3. $\square$