Definition 12.14.1. Let $\mathcal{A}$ be an additive category. Let $A_\bullet $ be a chain complex with boundary maps $d_{A, n} : A_ n \to A_{n - 1}$. For any $k \in \mathbf{Z}$ we define the *$k$-shifted chain 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 chain complexes, then we let $f[k] : A[k]_\bullet \to B[k]_\bullet $ be the morphism of chain complexes with $f[k]_ n = f_{k + n}$.

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