The Stacks project

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 $C(f) \to K$.

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