Compare chaotic with Zariski topology

Thanks to Bhargav Bhatt

Kind of fun and kind of idiotic at the same time. But the principle is
useful and it makes sense to state it in its starkest form...

etale-cohomology.tex

@@ -2134,6 +2134,29 @@ \section{The \'etale topos}
 Hence the result follows from Lemma \ref{lemma-alternative}.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-alternative-zariski}
+Let $S$ be a scheme. Let $S_{affine, Zar}$ denote the
+full subcategory of $S_{Zar}$ consisting of affine objects.
+A covering of $S_{affine, Zar}$ will be a standard
+Zariski covering, see
+Topologies, Definition \ref{topologies-definition-standard-Zariski}.
+Then restriction
+$$
+\mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, Zar}}
+$$
+defines an equivalence of topoi
+$\Sh(S_{Zar}) \cong \Sh(S_{affine, Zar})$.
+\end{lemma}
+
+\begin{proof}
+Please skip the proof of this lemma. It follows immediately from
+Sites, Lemma \ref{sites-lemma-equivalence} by checking that the
+inclusion functor $S_{affine, Zar} \to S_{Zar}$
+is a special cocontinuous functor (see
+Sites, Definition \ref{sites-definition-special-cocontinuous-functor}).
+\end{proof}
+
 
 
 
@@ -17773,7 +17796,64 @@ \section{K\"unneth in \'etale cohomology}
 
 
 
+\section{Comparing chaotic and Zariski topologies}
+\label{section-compare-chaotic-Zariski}
 
+\noindent
+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 usefull 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 Lemma \ref{lemma-alternative-zariski}.
+Let's denote $X_{affine, chaotic}$ the same underlying
+category with the chaotic topology, i.e., such that sheaves
+agree with presheaves. We obtain a morphisms of sites
+$$
+\epsilon : X_{affine, Zar} \longrightarrow X_{affine, chaotic}
+$$
+as in Cohomology on Sites, Section \ref{sites-cohomology-section-compare}.
+
+\begin{lemma}
+\label{lemma-check-zar}
+In the situation above let $K$ be an object of $D^+(X_{affine, chaotic})$.
+Then $K$ is in the essential image of the (fully faithful) functor
+$R\epsilon_* ; D(X_{affine, Zar}) \to D(X_{affine, chaotic})$ if and only
+if the following two conditions hold
+\begin{enumerate}
+\item $R\Gamma(\emptyset, K)$ is zero in $D(\textit{Ab})$, and
+\item if $U = V \cup W$ with $U, V, W \subset X$ affine open and
+$V, W \subset U$ standard open
+(Algebra, Definition \ref{algebra-definition-Zariski-topology}), then
+the map $c^K_{U, V, W, V \cap W}$ of
+Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-c-square}
+is a quasi-isomorphism.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Except for a snafu having to do with the empty set,
+this follows from the very general Cohomology on Sites, Lemma
+\ref{sites-cohomology-lemma-descent-squares} whose hypotheses hold by
+Schemes, Lemma \ref{schemes-lemma-sheaf-on-affines} and
+Cohomology on Sites, Lemma
+\ref{sites-cohomology-lemma-descent-squares-helper}.
+
+\medskip\noindent
+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 \ref{sites-example-site-topological} for a
+discussion). Then we have morphisms of sites
+$$
+X_{affine, Zar} \to X_{affine, almost-chaotic} \to X_{affine, chaotic}
+$$
+The argument above works for the first arrow. Then we leave it
+to the reader to see that an object $K$ of $D^+(X_{affine, chaotic})$
+is in the essential image of the (fully faithful) functor
+$D(X_{affine, almost-chaotic}) \to D(X_{affine, chaotic})$ if and only
+if $R\Gamma(\emptyset, K)$ is zero in $D(\textit{Ab})$.
+\end{proof}