The Stacks project

Definition 24.22.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $f : \mathcal{K} \to \mathcal{L}$ be a homomorphism of differential graded $\mathcal{A}$-modules. The cone of $f$ is the differential graded $\mathcal{A}$-module $C(f)$ defined as follows:

  1. the underlying complex of $\mathcal{O}$-modules is the cone of the corresponding map $f : \mathcal{K}^\bullet \to \mathcal{L}^\bullet $ of complexes of $\mathcal{A}$-modules, i.e., we have $C(f)^ n = \mathcal{L}^ n \oplus \mathcal{K}^{n + 1}$ and differential

    \[ d_{C(f)} = \left( \begin{matrix} \text{d}_\mathcal {L} & f \\ 0 & -\text{d}_\mathcal {K} \end{matrix} \right) \]
  2. the multiplication map

    \[ C(f)^ n \times \mathcal{A}^ m \to C(f)^{n + m} \]

    is the direct sum of the multiplication map $\mathcal{L}^ n \times \mathcal{A}^ m \to \mathcal{L}^{n + m}$ and the multiplication map $\mathcal{K}^{n + 1} \times \mathcal{A}^ m \to \mathcal{K}^{n + 1 + m}$.

It comes equipped with canonical hommorphisms of differential graded $\mathcal{A}$-modules $i : \mathcal{L} \to C(f)$ and $p : C(f) \to \mathcal{K}[1]$ induced by the obvious maps.

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