The Stacks project

Definition 22.4.3. Let $(A, \text{d})$ be a differential graded algebra. Let $M$ be a differential graded module whose underlying complex of $R$-modules is $M^\bullet $. For any $k \in \mathbf{Z}$ we define the $k$-shifted module $M[k]$ as follows

1. the underlying complex of $R$-modules of $M[k]$ is $M^\bullet [k]$, i.e., we have $M[k]^ n = M^{n + k}$ and $\text{d}_{M[k]} = (-1)^ k\text{d}_ M$ and

2. as $A$-module the multiplication

   \[ (M[k])^ n \times A^ m \longrightarrow (M[k])^{n + m} \]

   is equal to the given multiplication $M^{n + k} \times A^ m \to M^{n + k + m}$.

For a morphism $f : M \to N$ of differential graded $A$-modules we let $f[k] : M[k] \to N[k]$ be the map equal to $f$ on underlying $A$-modules. This defines a functor $[k] : \text{Mod}_{(A, \text{d})} \to \text{Mod}_{(A, \text{d})}$.