The Stacks project

Lemma 13.33.5. Let $\mathcal{A}$ be an abelian category. If $\mathcal{A}$ has exact countable direct sums, then $D(\mathcal{A})$ has countable direct sums. In fact given a collection of complexes $K_ i^\bullet $ indexed by a countable index set $I$ the termwise direct sum $\bigoplus K_ i^\bullet $ is the direct sum of $K_ i^\bullet $ in $D(\mathcal{A})$.

Proof. Let $L^\bullet $ be a complex. Suppose given maps $\alpha _ i : K_ i^\bullet \to L^\bullet $ in $D(\mathcal{A})$. This means there exist quasi-isomorphisms $s_ i : M_ i^\bullet \to K_ i^\bullet $ of complexes and maps of complexes $f_ i : M_ i^\bullet \to L^\bullet $ such that $\alpha _ i = f_ is_ i^{-1}$. By assumption the map of complexes

\[ s : \bigoplus M_ i^\bullet \longrightarrow \bigoplus K_ i^\bullet \]

is a quasi-isomorphism. Hence setting $f = \bigoplus f_ i$ we see that $\alpha = fs^{-1}$ is a map in $D(\mathcal{A})$ whose composition with the coprojection $K_ i^\bullet \to \bigoplus K_ i^\bullet $ is $\alpha _ i$. We omit the verification that $\alpha $ is unique. $\square$

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