The Stacks project

Proof. Denote $X_{affine}$ the category of affine schemes étale over $X$ endowed with the chaotic topology and its structure sheaf $\mathcal{O}_ X$, see Derived Categories of Spaces, Section 75.30. The functor $u : X_{\acute{e}tale}\to \mathcal{X}_{\acute{e}tale}$ of Lemma 96.10.1 gives rise to a functor $X_{affine} \to \mathcal{X}_{affine}$. This is compatible with structure sheaves and produces a functor

See Cohomology on Sites, Lemma 21.43.10. By Derived Categories of Spaces, Lemma 75.30.1 the triangulated category $\mathit{QC}(X_{affine}, \mathcal{O}_ X)$ is equivalent to $D_\mathit{QCoh}(\mathcal{O}_ X)$. Hence it suffices to prove that $G$ is an equivalence.

Consider the flat comparison morphisms $\epsilon _\mathcal {X} : \mathcal{X}_{affine, {\acute{e}tale}} \to \mathcal{X}_{affine}$ and $\epsilon _ X : X_{affine, {\acute{e}tale}} \to X_{affine}$ of ringed sites. Lemma 96.26.3 and (the proof of) Derived Categories of Spaces, Lemma 75.30.1 show that the functors $\epsilon _\mathcal {X}^*$ and $\epsilon _ X^*$ identify $\mathit{QC}(\mathcal{X}_{affine}, \mathcal{O})$ and $\mathit{QC}(X_{affine}, \mathcal{O}_ X)$ with subcategories $Q_\mathcal {X} \subset D(\mathcal{X}_{affine, {\acute{e}tale}}, \mathcal{O})$ and $Q_ X \subset D(X_{affine, {\acute{e}tale}}, \mathcal{O}_ X)$. With these identifications the functor $G$ in the first paragraph is induced by the functor

where $i_ X$ and $\pi _ X$ are the morphisms from Lemma 96.10.1 but with the étale sites replaced by the corresponding affine ones. The reader can show that this replacement is permissible either by reproving the lemma for the affine sites directly or by using the equivalences of topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{affine, {\acute{e}tale}}) = \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{\acute{e}tale})$ and $\mathop{\mathit{Sh}}\nolimits (X_{affine, {\acute{e}tale}}) = \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$. The lemma also tells us $Li_ X^*$ has a left adjoint

and moreover we have $Li_ X^* \circ L\pi _ X^* = \text{id}$ since $\pi _ X \circ i_ X$ is the identity. Thus it suffices to show that (a) $L\pi _ X^*$ sends $Q_ X$ into $Q_\mathcal {X}$ and (b) the kernel of $Li_ X^*$ is $0$. See Derived Categories, Lemma 13.7.2.

Proof of (a). By Derived Categories of Spaces, Lemma 75.30.1 we have $Q_ X = D_\mathit{QCoh}(X_{affine, {\acute{e}tale}}, \mathcal{O}_ X)$. Let $K$ be an object of $Q_ X$. Let $x$ be an object of $\mathcal{X}_{affine, {\acute{e}tale}}$ lying over the affine scheme $U = p(x)$. Denote $f : U \to X$ the morphism corresponding to $x$. Then we see that

This follows from transitivity of pullbacks; see discussion in Section 96.10. Next, suppose that $x \to x'$ is a morphism of $\mathcal{X}_{affine, {\acute{e}tale}}$ lying over the morphism $h : U \to U'$ of affine schemes. As before denote $f : U \to X$ and $f' : U' \to X$ the morphisms corresponding to $x$ and $x'$ so that we have $f = f' \circ h$. Then

and hence we have (a) by the footnote in the statement of Cohomology on Sites, Lemma 21.43.12. The third equality is Derived Categories of Schemes, Lemma 36.3.8.

Proof of (b). Let $M$ be an object of $Q_\mathcal {X}$ such that $Li_ X^*M = 0$. Let $x'$ be an object of $\mathcal{X}_{affine, {\acute{e}tale}}$ lying over the affine scheme $U' = p(x')$ and assume that the corresponding morphism $f' : U' \to X$ is étale. Then $f' : U' \to X$ is an object of $X_{affine, {\acute{e}tale}}$ and the condition $Li_ X^*M = 0$ implies that $M|_{U'_{\acute{e}tale}} = 0$. In particular, we see that $R\Gamma (x', M) = 0$. However, for an arbitrary object $x$ of the site $\mathcal{X}_{affine, {\acute{e}tale}}$ there exists a covering $\{ x_ i \to x\} $ such that for each $i$ there is a morphism $x_ i \to x'_ i$ with $x'_ i$ corresponding to an object of $X_{affine, {\acute{e}tale}}$. Now since $M$ is in $Q_\mathcal {X}$ we have

and we conclude that $M$ is zero as desired. $\square$