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Given a complex $M^{\bullet}$ (say in an abelian category, in particular a module category) with differential $d^{i}: M^{i} \rightarrow M^{i+1}$, we can form another complex $\widetilde{M}^{\bullet}$ with differential $\tilde{d}^{i}=(-1)^{i}d^{i}$.

How are these complexes related?

They certainly have the same cohomology (since the signs don't change kernels and cokernels), so that suggests that they could be quasi-isomorphic, but I don't see how to write down the quasi-isomorphism. The identity is not a cochain map (outside of characteristic $2$), nor is alternating the signs on the identity.

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up vote 2 down vote accepted

There is a map going from $M$ to $\tilde M$ whose components are $$\dots, 1, 1, -1, -1, \dots$$ repeating with period $4$.

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