Definition 12.13.7. 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:

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

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 $f[k]^ n = f^{k + n}$.

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