Chain map

Definition

The notion of chain map is the notion of morphism in the category of chain complexes with chain maps. Below, the definitions are given in the context of chain complexes of abelian groups and the more general context of chain complexes of modules over a commutative unital ring.

For chain complexes of abelian groups

Suppose $C$ and $D$ are both chain complexes of abelian groups. A chain map $f : C \to D$ from $C$ to $D$ is defined as the following data subject to the specified compatibility condition:

Data: For each integer $n$ , it specifies a homomorphism of groups $f_{n} : C_{n} \to D_{n}$ .
Compatibility condition: For each integer $n$ , it must be true that $\partial_{n} \circ f_{n} = f_{n - 1} \circ \partial_{n}$ , where the $\partial_{n}$ on the left side denotes the boundary map from $D_{n}$ to $D_{n - 1}$ and the $\partial_{n}$ on the right side denotes the boundary map from $C_{n}$ to $C_{n - 1}$ .

For chain complexes of modules

Suppose $C$ and $D$ are both chain complexes of modules over a commutative unital ring $R$ . A chain map $f : C \to D$ from $C$ to $D$ is defined as the following data subject to the specified compatibility condition:

Data: For each integer $n$ , it specifies a homomorphism of $R$ -modules $f_{n} : C_{n} \to D_{n}$ .
Compatibility condition: For each integer $n$ , it must be true that $\partial_{n} \circ f_{n} = f_{n - 1} \circ \partial_{n}$ , where the $\partial_{n}$ on the left side denotes the boundary map from $D_{n}$ to $D_{n - 1}$ and the $\partial_{n}$ on the right side denotes the boundary map from $C_{n}$ to $C_{n - 1}$ .