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Simple question, but if $f_n$ is a chain map between $A_n$ and $B_n$ is $f_n$ linear as well? Since $f_n$ induces a homomorphism between homology groups, the induced map must be linear.

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    $\begingroup$ It depends on what you mean by "linear", and what structure we assume on $A_n$ and $B_n$. If they are only abelian groups, a chain map is just a group homomorphism. In general if they are from chain complexes of $R$-modules for some ring $R$ then in our definition of chain map typically requires it to be an $R$-module morphism i.e. it is "$R$-linear"; if $R$ is a field then "$R$-module" means "vector space" and a chain map will be a linear function of $R$-vector spaces. $\endgroup$ – William Feb 22 '20 at 14:56
  • $\begingroup$ Assuming our groups are just abelian, can you explain why the chain maps are group homomorphisms? I though we only have $f_n \circ \delta_A = \delta_B \circ f_{n+1}$ where $\delta_{A/B}$ are the group homomorphisms for $A_n$ and $B_n$ $\endgroup$ – asuuuka Feb 22 '20 at 19:12
  • $\begingroup$ If $A_\bullet$ and $B_\bullet$ are chain complexes of abelian groups then by definition a chain map $f\colon A_\bullet \to B_\bullet$ is a sequence of group homomorphisms $\{f_n \colon A_n \to B_n \mid n\in \mathbb{Z} \}$ with the extra property that $f_n\circ \delta_A = \delta_B \circ f_{n+1}$. If $A_\bullet$ and $B_\bullet$ are chain complexes of $R$-modules then a chain map is again by definition a sequence of $R$-module homomorphisms. en.wikipedia.org/wiki/Chain_complex#Chain_maps $\endgroup$ – William Feb 22 '20 at 19:18
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If by "linear" you mean "a homomorphism" then this the case by any standard definition. If $(A_\bullet, \delta_A)$ and $(B_\bullet, \delta_B)$ are chain complexes of $R$-modules for some ring $R$, then a chain map $f\colon A_\bullet \to B_\bullet$ is defined to be a sequence of $R$-module homomorphisms (aka $R$-linear functions) $\{f_n \colon A_n \to B_n \mid n \in \mathbb{Z}\}$ such that $f_n\circ \delta_A = \delta_B \circ f_{n+1}$ for all $n$, see for example the Wikipedia article. If $R=\mathbb{Z}$ then "$R$-module" just means "abelian group" and "$R$-linear" just means "group homomorphism"; if $R$ is a field then "$R$-module" $\equiv$ "vector space" and "$R$-linear" $\equiv$ "linear".

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