Definition 22.6.1. Let $(A, \text{d})$ be a differential graded algebra. Let $f : K \to L$ be a homomorphism of differential graded $A$-modules. The cone of $f$ is the differential graded $A$-module $C(f)$ given by $C(f) = L \oplus K$ with grading $C(f)^ n = L^ n \oplus K^{n + 1}$ and differential

$d_{C(f)} = \left( \begin{matrix} \text{d}_ L & f \\ 0 & -\text{d}_ K \end{matrix} \right)$

It comes equipped with canonical morphisms of complexes $i : L \to C(f)$ and $p : C(f) \to K[1]$ induced by the obvious maps $L \to C(f)$ and $C(f) \to K$.

There are also:

• 2 comment(s) on Section 22.6: Cones

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.

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 09KA. Beware of the difference between the letter 'O' and the digit '0'.