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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...
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aisejohan committed Oct 27, 2018
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  1. +80 −0 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}

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