Definition 22.4.3. Let $(A, \text{d})$ be a differential graded algebra. Let $M$ be a differential graded module. For any $k \in \mathbf{Z}$ we define the *$k$-shifted module* $M[k]$ as follows

as $A$-module $M[k] = M$,

$M[k]^ n = M^{n + k}$,

$\text{d}_{M[k]} = (-1)^ k\text{d}_ M$.

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 $[k] : \text{Mod}_{(A, \text{d})} \to \text{Mod}_{(A, \text{d})}$.

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