# Is the chain map linear?

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.

• 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. – William Feb 22 '20 at 14:56
• 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 $B_n$ – asuuuka Feb 22 '20 at 19:12
• 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 – William Feb 22 '20 at 19:18

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".