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Missing assumptions in 2 lemmas and 1 proposition

This I should have caught much earlier myself since the third part of
the statement of the lemma somehow really doesn't make sense without the
assumption that \mathcal{P} is closed under direct sums...! Moreover,
the condition that \mathcal{P} contains 0 is necessary for the
application of lemma 05T7... Argh!

Thanks to awllower
https://stacks.math.columbia.edu/tag/06XX#comment-4827
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aisejohan committed Dec 12, 2019
1 parent d271fed commit 7397e8efe8c422bbcadf036f562d86c1c79cbd58
Showing with 9 additions and 5 deletions.
  1. +9 −5 derived.tex
@@ -8744,7 +8744,8 @@ \section{Unbounded complexes}
\label{lemma-special-direct-system}
Let $\mathcal{A}$ be an abelian category. Let
$\mathcal{P} \subset \Ob(\mathcal{A})$ be a subset.
Assume that every object of $\mathcal{A}$ is a quotient of an
Assume $\mathcal{P}$ contains $0$, is closed under (finite) direct sums,
and every object of $\mathcal{A}$ is a quotient of an
element of $\mathcal{P}$. Let $K^\bullet$ be a complex.
There exists a commutative diagram
$$
@@ -8805,7 +8806,8 @@ \section{Unbounded complexes}
$$
in the triangulated category $K(\mathcal{A})$.
Set $P_{n + 1}^\bullet = C_2^\bullet[-1]$.
Note that (3) holds by construction.
Then (2) holds because $\mathcal{P}$ is closed
under direct sums. Note that (3) holds by construction.
Choose an actual morphism of complexes
$f : P_{n + 1}^\bullet \to \tau_{\leq n + 1}K^\bullet$.
The left square of the diagram above commutes up to homotopy, but as
@@ -8825,8 +8827,9 @@ \section{Unbounded complexes}
of abelian categories. Let $\mathcal{P} \subset \Ob(\mathcal{A})$ be a
subset. Assume
\begin{enumerate}
\item every object of $\mathcal{A}$ is a quotient of an element of
$\mathcal{P}$,
\item $\mathcal{P}$ contains $0$, is closed under (finite) direct sums,
and every object of $\mathcal{A}$ is a quotient of an
element of $\mathcal{P}$,
\item for any bounded above acyclic complex $P^\bullet$ of
$\mathcal{A}$ with $P^n \in \mathcal{P}$ for all $n$ the
complex $F(P^\bullet)$ is exact,
@@ -8934,7 +8937,8 @@ \section{Unbounded complexes}
\label{lemma-special-inverse-system}
Let $\mathcal{A}$ be an abelian category. Let
$\mathcal{I} \subset \Ob(\mathcal{A})$ be a subset.
Assume that every object of $\mathcal{A}$ is a subobject of an
Assume $\mathcal{I}$ contains $0$, is closed under (finite) products,
and every object of $\mathcal{A}$ is a subobject of an
element of $\mathcal{I}$. Let $K^\bullet$ be a complex.
There exists a commutative diagram
$$

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