Lemma 75.11.3 (08H0)—The Stacks project

Lemma 75.11.3. Let $S$ be a scheme. Let $X$ be an affine algebraic space over $S$ . Set $A = \Gamma (X, \mathcal{O}_ X)$ . Then

$Q_ X : \textit{Mod}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ X)$ is the functor which sends $\mathcal{F}$ to the quasi-coherent $\mathcal{O}_ X$ -module associated to the $A$ -module $\Gamma (X, \mathcal{F})$ ,
$RQ_ X : D(\mathcal{O}_ X) \to D(\mathit{QCoh}(\mathcal{O}_ X))$ is the functor which sends $E$ to the complex of quasi-coherent $\mathcal{O}_ X$ -modules associated to the object $R\Gamma (X, E)$ of $D(A)$ ,
restricted to $D_\mathit{QCoh}(\mathcal{O}_ X)$ the functor $RQ_ X$ defines a quasi-inverse to (75.5.1.1).

Proof. Let $X_0 = \mathop{\mathrm{Spec}}(A)$ be the affine scheme representing $X$ . Recall that there is a morphism of ringed sites $\epsilon : X_{\acute{e}tale}\to X_{0, Zar}$ which induces equivalences

$\xymatrix{ \mathit{QCoh}(\mathcal{O}_ X) \ar@<1ex>[r]^{{\epsilon _*}} & \mathit{QCoh}(\mathcal{O}_{X_0}) \ar@<1ex>[l]^{{\epsilon ^*}} }$

see Lemma 75.4.2. Hence we see that $Q_ X = \epsilon ^* \circ Q_{X_0} \circ \epsilon _*$ by uniqueness of adjoint functors. Hence (1) follows from the description of $Q_{X_0}$ in Derived Categories of Schemes, Lemma 36.7.3 and the fact that $\Gamma (X_0, \epsilon _*\mathcal{F}) = \Gamma (X, \mathcal{F})$ . Part (2) follows from (1) and the fact that the functor from $A$ -modules to quasi-coherent $\mathcal{O}_ X$ -modules is exact. The third assertion now follows from the result for schemes (Derived Categories of Schemes, Lemma 36.7.3) and Lemma 75.4.2. $\square$

Comments (1)

Comment #1652 by Pieter Belmans on September 24, 2015 at 12:11

The diagram in this proof doesn't parse. I don't know whether it is due to some XyJax syntax voodoo (which you can often fix by magically adding or removing spaces) or some issue with the HTML that I output. If the former doesn't work I'll put it in the issues list on GitHub.

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