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75.4 Derived category of quasi-coherent modules on the small étale site

Let $X$ be a scheme. In this section we show that $D_\mathit{QCoh}(\mathcal{O}_ X)$ can be defined in terms of the small étale site $X_{\acute{e}tale}$ of $X$. Denote $\mathcal{O}_{\acute{e}tale}$ the structure sheaf on $X_{\acute{e}tale}$. Consider the morphism of ringed sites
\begin{equation} \label{spaces-perfect-equation-epsilon} \epsilon : (X_{\acute{e}tale}, \mathcal{O}_{\acute{e}tale}) \longrightarrow (X_{Zar}, \mathcal{O}_ X). \end{equation}

denoted $\text{id}_{small, {\acute{e}tale}, Zar}$ in Descent, Lemma 35.8.5.

Lemma 75.4.1. The morphism $\epsilon $ of ( is a flat morphism of ringed sites. In particular the functor $\epsilon ^* : \textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_{\acute{e}tale})$ is exact. Moreover, if $\epsilon ^*\mathcal{F} = 0$, then $\mathcal{F} = 0$.

Proof. The flatness of the morphism $\epsilon $ is Descent, Lemma 35.10.1. Here is another proof. We have to show that $\mathcal{O}_{\acute{e}tale}$ is a flat $\epsilon ^{-1}\mathcal{O}_ X$-module. To do this it suffices to check $\mathcal{O}_{X, x} \to \mathcal{O}_{{\acute{e}tale}, \overline{x}}$ is flat for any geometric point $\overline{x}$ of $X$, see Modules on Sites, Lemma 18.39.3, Sites, Lemma 7.34.2, and Étale Cohomology, Remarks 59.29.11. By Étale Cohomology, Lemma 59.33.1 we see that $\mathcal{O}_{{\acute{e}tale}, \overline{x}}$ is the strict henselization of $\mathcal{O}_{X, x}$. Thus $\mathcal{O}_{X, x} \to \mathcal{O}_{{\acute{e}tale}, \overline{x}}$ is faithfully flat by More on Algebra, Lemma 15.45.1.

The exactness of $\epsilon ^*$ follows from the flatness of $\epsilon $ by Modules on Sites, Lemma 18.31.2.

Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. If $\epsilon ^*\mathcal{F} = 0$, then with notation as above

\[ 0 = \epsilon ^*\mathcal{F}_{\overline{x}} = \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \mathcal{O}_{{\acute{e}tale}, \overline{x}} \]

(Modules on Sites, Lemma 18.36.4) for all geometric points $\overline{x}$. By faithful flatness of $\mathcal{O}_{X, x} \to \mathcal{O}_{{\acute{e}tale}, \overline{x}}$ we conclude $\mathcal{F}_ x = 0$ for all $x \in X$. $\square$

Let $X$ be a scheme. Notation as in ( Recall that $\epsilon ^* : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_{\acute{e}tale})$ is an equivalence by Descent, Proposition 35.8.9 and Remark 35.8.6. Moreover, $\mathit{QCoh}(\mathcal{O}_{\acute{e}tale})$ forms a Serre subcategory of $\textit{Mod}(\mathcal{O}_{\acute{e}tale})$ by Descent, Lemma 35.10.2. Hence we can let $D_\mathit{QCoh}(\mathcal{O}_{\acute{e}tale})$ be the triangulated subcategory of $D(\mathcal{O}_{\acute{e}tale})$ whose objects are the complexes with quasi-coherent cohomology sheaves, see Derived Categories, Section 13.17. The functor $\epsilon ^*$ is exact (Lemma 75.4.1) hence induces $\epsilon ^* : D(\mathcal{O}_ X) \to D(\mathcal{O}_{\acute{e}tale})$ and since pullbacks of quasi-coherent modules are quasi-coherent also $\epsilon ^* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_{\acute{e}tale})$.

Lemma 75.4.2. Let $X$ be a scheme. The functor $\epsilon ^* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_{\acute{e}tale})$ defined above is an equivalence.

Proof. We will prove this by showing the functor $R\epsilon _* : D(\mathcal{O}_{\acute{e}tale}) \to D(\mathcal{O}_ X)$ induces a quasi-inverse. We will use freely that $\epsilon _*$ is given by restriction to $X_{Zar} \subset X_{\acute{e}tale}$ and the description of $\epsilon ^* = \text{id}_{small, {\acute{e}tale}, Zar}^*$ in Descent, Lemma 35.8.5.

For a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the adjunction map $\mathcal{F} \to \epsilon _*\epsilon ^*\mathcal{F}$ is an isomorphism by the fact that $\mathcal{F}^ a$ (Descent, Definition 35.8.2) is a sheaf as proved in Descent, Lemma 35.8.1. Conversely, every quasi-coherent $\mathcal{O}_{\acute{e}tale}$-module $\mathcal{H}$ is of the form $\epsilon ^*\mathcal{F}$ for some quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$, see Descent, Proposition 35.8.9. Then $\mathcal{F} = \epsilon _*\mathcal{H}$ by what we just said and we conclude that the adjunction map $\epsilon ^*\epsilon _*\mathcal{H} \to \mathcal{H}$ is an isomorphism for all quasi-coherent $\mathcal{O}_{\acute{e}tale}$-modules $\mathcal{H}$.

Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_{\acute{e}tale})$ and denote $\mathcal{H}^ q = H^ q(E)$ its $q$th cohomology sheaf. Let $\mathcal{B}$ be the set of affine objects of $X_{\acute{e}tale}$. Then $H^ p(U, \mathcal{H}^ q) = 0$ for all $p > 0$, all $q \in \mathbf{Z}$, and all $U \in \mathcal{B}$, see Descent, Proposition 35.9.3 and Cohomology of Schemes, Lemma 30.2.2. By Cohomology on Sites, Lemma 21.23.11 this means that

\[ H^ q(U, E) = H^0(U, \mathcal{H}^ q) \]

for all $U \in \mathcal{B}$. In particular, we find that this holds for affine opens $U \subset X$. It follows that the $q$th cohomology of $R\epsilon _*E$ over $U$ is the value of the sheaf $\epsilon _*\mathcal{H}^ q$ over $U$. Applying sheafification we obtain

\[ H^ q(R\epsilon _*E) = \epsilon _*\mathcal{H}^ q \]

which in particular shows that $R\epsilon _*$ induces a functor $D_\mathit{QCoh}(\mathcal{O}_{\acute{e}tale}) \to D_\mathit{QCoh}(\mathcal{O}_ X)$. Since $\epsilon ^*$ is exact we then obtain $H^ q(\epsilon ^*R\epsilon _*E) = \epsilon ^*\epsilon _*\mathcal{H}^ q = \mathcal{H}^ q$ (by discussion above). Thus the adjunction map $\epsilon ^*R\epsilon _*E \to E$ is an isomorphism.

Conversely, for $F \in D_\mathit{QCoh}(\mathcal{O}_ X)$ the adjunction map $F \to R\epsilon _*\epsilon ^*F$ is an isomorphism for the same reason, i.e., because the cohomology sheaves of $R\epsilon _*\epsilon ^*F$ are isomorphic to $\epsilon _*H^ m(\epsilon ^*F) = \epsilon _*\epsilon ^*H^ m(F) = H^ m(F)$. $\square$

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