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

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

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 $\epsilon _*H^ m(\epsilon ^*F) = \epsilon _*\epsilon ^*H^ m(F) = H^ m(F)$. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)