From 9daea869e75e2144f5d20b284a169049acb08302 Mon Sep 17 00:00:00 2001 From: Aise Johan de Jong Date: Fri, 7 Jun 2024 08:42:59 -0400 Subject: [PATCH] Fix example Argh... not sure what I was thinking before... --- sites.tex | 24 +++++++++++++----------- 1 file changed, 13 insertions(+), 11 deletions(-) diff --git a/sites.tex b/sites.tex index 20392e15..dfdddb81 100644 --- a/sites.tex +++ b/sites.tex @@ -4950,17 +4950,19 @@ \section{Cocontinuous functors which have a right adjoint} Lemma \ref{lemma-continuous-with-continuous-left-adjoint} shows that if $v$ is continuous, then $u$ is cocontinous. Conversely, if $u$ is cocontinuous, then we can't conclude that $v$ -is continuous. Namely, consider a topological space $X$ and -the site $X_{Zar}$ of Example \ref{example-site-topological}. -On the other hand, given an open covering $X = W \cup V$ -we can consider the site $X'_{Zar}$ defined in exactly the -same manner, except that we declare $U = \bigcup U_i$ to be -a covering if and only if for each $i$ either $U_i \subset W$ -or $U_i \subset V$. Set $u = v = \text{id}$. Then -$v$ viewed as functor $X'_{Zar} \to X_{Zar}$ is continuous -(as any covering in $X_{Zar}$ can be refined by a covering in -$X'_{Zar}$) but $u$ viewed as a functor $X_{Zar} \to X'_{Zar}$ -is not continuous (provided neither $W \not = X$ and $V \not = X$). +is continuous. We will give an example of this phenomenon using +the big \'etale and smooth sites of a scheme, but presumably there is an +elementary example as well. Namely, consider a scheme $S$ and the sites +$(\Sch/S)_\etale$ and $(\Sch/S)_{smooth}$. We may assume these +sites have the same underlying category, see +Topologies, Remark \ref{topologies-remark-choice-sites}. +Let $u = v = \text{id}$. Then $u$ as a functor +from $(\Sch/S)_\etale$ to $(\Sch/S)_{smooth}$ is cocontinuous +as every smooth covering of a scheme can be refined by an \'etale +covering, see More on Morphisms, Lemma +\ref{more-morphisms-lemma-etale-dominates-smooth}. +Conversely, the functor $v$ from $(\Sch/S)_{smooth}$ to $(\Sch/S)_\etale$ +is not continuous as a smooth covering is not an \'etale covering in general. \end{example}