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

is exact. This can be verified by hand, and holds even if $J$ is empty. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)