The Stacks project

Lemma 22.9.2. Let $(A, \text{d})$ be a differential graded algebra. Let $\alpha : K \to L$ and $\beta : L \to M$ define an admissible short exact sequence

\[ 0 \to K \to L \to M \to 0 \]

of differential graded $A$-modules. Let $(K, L, M, \alpha , \beta , \delta )$ be the associated triangle. Then the triangles

\[ (M[-1], K, L, \delta [-1], \alpha , \beta ) \quad \text{and}\quad (M[-1], K, C(\delta [-1]), \delta [-1], i, p) \]

are isomorphic.

Proof. Using a choice of splittings we write $L = K \oplus M$ and we identify $\alpha $ and $\beta $ with the natural inclusion and projection maps. By construction of $\delta $ we have

\[ d_ B = \left( \begin{matrix} d_ K & \delta \\ 0 & d_ M \end{matrix} \right) \]

On the other hand the cone of $\delta [-1] : M[-1] \to K$ is given as $C(\delta [-1]) = K \oplus M$ with differential identical with the matrix above! Whence the lemma. $\square$

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Your email address will not be published.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 09KC. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 22.9.2 as pdf