Lemma 13.31.6. Let $\mathcal{A}$ be an abelian category. Assume colimits over $\mathbf{N}$ exist and are exact. Then countable direct sums exists and are exact. Moreover, if $(A_ n, f_ n)$ is a system over $\mathbf{N}$, then there is a short exact sequence

$0 \to \bigoplus A_ n \to \bigoplus A_ n \to \mathop{\mathrm{colim}}\nolimits A_ n \to 0$

where the first map in degree $n$ is given by $1 - f_ n$.

Proof. The first statement follows from $\bigoplus A_ n = \mathop{\mathrm{colim}}\nolimits (A_1 \oplus \ldots \oplus A_ n)$. For the second, note that for each $n$ we have the short exact sequence

$0 \to A_1 \oplus \ldots \oplus A_{n - 1} \to A_1 \oplus \ldots \oplus A_ n \to A_ n \to 0$

where the first map is given by the maps $1 - f_ i$ and the second map is the sum of the transition maps. Take the colimit to get the sequence of the lemma. $\square$

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