The Stacks project

Definition 12.14.7. Let $\mathcal{A}$ be an additive category. Let $A^\bullet $ be a cochain complex with boundary maps $d_ A^ n : A^ n \to A^{n + 1}$. For any $k \in \mathbf{Z}$ we define the $k$-shifted cochain complex $A[k]^\bullet $ as follows:

1. we set $A[k]^ n = A^{n + k}$, and

2. we set $d_{A[k]}^ n : A[k]^ n \to A[k]^{n + 1}$ equal to $d_{A[k]}^ n = (-1)^ k d_ A^{n + k}$.

If $f : A^\bullet \to B^\bullet $ is a morphism of cochain complexes, then we let $f[k] : A[k]^\bullet \to B[k]^\bullet $ be the morphism of cochain complexes with $f[k]^ n = f^{k + n}$.