Reference: Weilber - Homological algebra
Let $R$ be a ring. Let $C$ and $D$ be chain complexes of $R$-module and $f:C\rightarrow D$ be a chain map.
With this notation, there is a lemma given in the text:
If $f$ is chain null-homotopic, then $f_*:H_n(C)\rightarrow H_n(D)$ is zero.
What is $f_*$ here? I cannot find the definition in the text. And what is this called?
Say, $C=\{(C_n,d_n)\}$ and $D=\{(D_n,d'_n)\}$. Then, does this mean a map $g:Ker(d_n)/im(d_{n+1})\rightarrow ker(d'_n)/im(d'_{n+1}):x + im(d_{n+1})\mapsto f(x)+ im(d'_{n+1})$?
Each chain map $$f:C\to D$$ induces a homomorphism $$f_*:H_n(C)\to H_n(D)$$ for all $n$ via $$f_*([x]) \equiv [f(x)]$$It is an exercise to show that this is well-defined (independent of the choice of representative $x$).