The Stacks project

59.98 Comparing chaotic and Zariski topologies

When constructing the structure sheaf of an affine scheme, we first construct the values on affine opens, and then we extend to all opens. A similar construction is often useful for constructing complexes of abelian groups on a scheme $X$. Recall that $X_{affine, Zar}$ denotes the category of affine opens of $X$ with topology given by standard Zariski coverings, see Topologies, Definition 34.3.7. We remind the reader that the topos of $X_{affine, Zar}$ is the small Zariski topos of $X$, see Topologies, Lemma 34.3.11. In this section we denote $X_{affine}$ the same underlying category with the chaotic topology, i.e., such that sheaves agree with presheaves. We obtain a morphisms of sites

\[ \epsilon : X_{affine, Zar} \longrightarrow X_{affine} \]

as in Cohomology on Sites, Section 21.27.

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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