# stacks / stacks-project

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
 @@ -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