# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Example 2.5.1.2. Let $\operatorname{\mathcal{A}}$ be an additive category, let $C \in \operatorname{\mathcal{A}}$ be an object, and let $n$ be an integer. We will write $C[n]$ for the chain complex given by

$C[n]_{\ast } = \begin{cases} C & \text{ if } \ast = n \\ 0 & \text{ otherwise, } \end{cases}$

where each differential is the zero morphism. Note that a chain complex $M_{\ast }$ is isomorphic to $C[n]$ (for some object $C \in \operatorname{\mathcal{A}}$) if and only if it is concentrated both in degrees $\geq n$ and in degrees $\leq n$.