Lemma 10.8.10. Let $\mathcal{I}$ be an index category satisfying the assumptions of Categories, Lemma 4.19.8. Then taking colimits of diagrams of abelian groups over $\mathcal{I}$ is exact (i.e., the analogue of Lemma 10.8.8 holds in this situation).
Proof. By Categories, Lemma 4.19.8 we may write $\mathcal{I} = \coprod _{j \in J} \mathcal{I}_ j$ with each $\mathcal{I}_ j$ a filtered category, and $J$ possibly empty. By Categories, Lemma 4.21.5 taking colimits over the index categories $\mathcal{I}_ j$ is the same as taking the colimit over some directed set. Hence Lemma 10.8.8 applies to these colimits. This reduces the problem to showing that coproducts in the category of $R$-modules over the set $J$ are exact. In other words, exact sequences $L_ j \to M_ j \to N_ j$ of $R$ modules we have to show that
\[ \bigoplus \nolimits _{j \in J} L_ j \longrightarrow \bigoplus \nolimits _{j \in J} M_ j \longrightarrow \bigoplus \nolimits _{j \in J} N_ j \]
is exact. This can be verified by hand, and holds even if $J$ is empty. $\square$
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