The Stacks project

Lemma 59.98.1. In the situation above let $K$ be an object of $D^+(X_{affine})$. Then $K$ is in the essential image of the (fully faithful) functor $R\epsilon _* ; D(X_{affine, Zar}) \to D(X_{affine})$ if and only if the following two conditions hold

  1. $R\Gamma (\emptyset , K)$ is zero in $D(\textit{Ab})$, and

  2. if $U = V \cup W$ with $U, V, W \subset X$ affine open and $V, W \subset U$ standard open (Algebra, Definition 10.17.3), then the map $c^ K_{U, V, W, V \cap W}$ of Cohomology on Sites, Lemma 21.26.1 is a quasi-isomorphism.

Proof. (The functor $R\epsilon _*$ is fully faithful by the discussion in Cohomology on Sites, Section 21.27.) Except for a snafu having to do with the empty set, this follows from the very general Cohomology on Sites, Lemma 21.29.2 whose hypotheses hold by Schemes, Lemma 26.11.7 and Cohomology on Sites, Lemma 21.29.3.

To get around the snafu, denote $X_{affine, almost-chaotic}$ the site where the empty object $\emptyset $ has two coverings, namely, $\{ \emptyset \to \emptyset \} $ and the empty covering (see Sites, Example 7.6.4 for a discussion). Then we have morphisms of sites

\[ X_{affine, Zar} \to X_{affine, almost-chaotic} \to X_{affine} \]

The argument above works for the first arrow. Then we leave it to the reader to see that an object $K$ of $D^+(X_{affine})$ is in the essential image of the (fully faithful) functor $D(X_{affine, almost-chaotic}) \to D(X_{affine})$ if and only if $R\Gamma (\emptyset , K)$ is zero in $D(\textit{Ab})$. $\square$

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