The Stacks project

Example 13.37.2. Let $\mathcal{A}$ be an abelian category. Let $\ldots \to A_2 \to A_1 \to A_0$ be a chain complex in $\mathcal{A}$. Then we can consider the objects

\[ X_ n = A_ n \quad \text{and}\quad Y_ n = (A_ n \to A_{n - 1} \to \ldots \to A_0)[-n] \]

of $D(\mathcal{A})$. With the evident canonical maps $Y_ n \to X_ n$ and $Y_0 \to Y_1[1] \to Y_2[2] \to \ldots $ the distinguished triangles $Y_ n \to X_ n \to Y_{n - 1} \to Y_ n[1]$ define a Postnikov system as in Definition 13.37.1 for $\ldots \to X_2 \to X_1 \to X_0$. Here we are using the obvious extension of Postnikov systems for an infinite complex of $D(\mathcal{A})$. Finally, if colimits over $\mathbf{N}$ exist and are exact in $\mathcal{A}$ then

\[ \text{hocolim} Y_ n[n] = (\ldots \to A_2 \to A_1 \to A_0 \to 0 \to \ldots ) \]

in $D(\mathcal{A})$. This follows immediately from Lemma 13.31.7.


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Your email address will not be published.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 0D8Z. Beware of the difference between the letter 'O' and the digit '0'.