The Stacks project

Lemma 13.31.7. Let $\mathcal{A}$ be an abelian category. Let $L_ n^\bullet $ be a system of complexes of $\mathcal{A}$. Assume colimits over $\mathbf{N}$ exist and are exact in $\mathcal{A}$. Then the termwise colimit $L^\bullet = \mathop{\mathrm{colim}}\nolimits L_ n^\bullet $ is a homotopy colimit of the system in $D(\mathcal{A})$.

Proof. We have an exact sequence of complexes

\[ 0 \to \bigoplus L_ n^\bullet \to \bigoplus L_ n^\bullet \to L^\bullet \to 0 \]

by Lemma 13.31.6. The direct sums are direct sums in $D(\mathcal{A})$ by Lemma 13.31.5. Thus the result follows from the definition of derived colimits in Definition 13.31.1 and the fact that a short exact sequence of complexes gives a distinguished triangle (Lemma 13.12.1). $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0949. Beware of the difference between the letter 'O' and the digit '0'.