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

Comment #298 by arp on

Another funny typo I spotted: in the second sentence the boundary maps should increase degree.

Comment #654 by Fan Zheng on

I think so, too (should be $A[k]^n \to A[k]^{n+1}$. This typo has somehow escaped attention for so long.

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• 2 comment(s) on Section 12.14: Homotopy and the shift functor

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