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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