find a cochain map $ψ: A• → B•$ with a condition that $ψ^i:A^i→B^i$ is an inijective

Question

I try to find an example for a cochain map $ψ: A• → B•$ that exists : $ψ^i:A^i→B^i$ is an injective map (for i>=0) but $ψ∗ :H^k(A∙)→H^k(B∙)$ is not a injective map (for k>=0)? I use the fact that a short exact sequences of cochain complexes induces a long exact sequence on the cohomology. I will be happy if someone give me a hint. thank you!

Answer 1

Let $A^•$ be the cochain complex $$0 → ℤ → ℤ → ℤ → …,$$ where every horizontal map is trivial, and $B^•$ be the cochain complex $$0 \oplus ℤ → ℤ\oplus ℤ → ℤ \oplus ℤ → ℤ\oplus ℤ → …,$$ wherey every horizontal map projects the right component onto the left component.

Can you construct a cochain map $ψ^• \colon A^• → B^•$ yourself that is injective, but doesn’t induce an injective map $ψ^•_\ast$ on the cohomologies?