From 8f452f6228a78feb03ab96c3863400cf10d9b4ed Mon Sep 17 00:00:00 2001
From: Aise Johan de Jong <aise.johan.de.jong@gmail.com>
Date: Sun, 14 Apr 2024 09:33:50 -0400
Subject: [PATCH] Add remark on boundedness perfect complexes
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Thanks to Nicolás
https://stacks.math.columbia.edu/tag/08CM#comment-8373
---
 cohomology.tex       | 5 +++++
 sites-cohomology.tex | 6 ++++++
 2 files changed, 11 insertions(+)

diff --git a/cohomology.tex b/cohomology.tex
index ff8cb713..02cd3318 100644
--- a/cohomology.tex
+++ b/cohomology.tex
@@ -12482,6 +12482,11 @@ \section{Perfect complexes}
 if it can be represented by a perfect complex of $\mathcal{O}_X$-modules.
 \end{definition}
 
+\noindent
+If $X$ is quasi-compact, then a perfect object of $D(\mathcal{O}_X)$
+is in $D^b(\mathcal{O}_X)$. But this need not be the case if
+$X$ is not quasi-compact.
+
 \begin{lemma}
 \label{lemma-perfect-independent-representative}
 Let $(X, \mathcal{O}_X)$ be a ringed space.
diff --git a/sites-cohomology.tex b/sites-cohomology.tex
index b9385a54..a728043f 100644
--- a/sites-cohomology.tex
+++ b/sites-cohomology.tex
@@ -12707,6 +12707,12 @@ \section{Perfect complexes}
 if it can be represented by a perfect complex of $\mathcal{O}$-modules.
 \end{definition}
 
+\noindent
+If $\Sh(\mathcal{C})$ is quasi-compact
+(Sites, Section \ref{sites-section-quasi-compact}),
+then a perfect object of $D(\mathcal{O})$
+is in $D^b(\mathcal{O})$. But this need not be the case otherwise.
+
 \begin{lemma}
 \label{lemma-perfect-independent-representative}
 Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.