Definition:Homomorphism of Complexes

Definition
Let $\left({R, +, \cdot}\right)$ be a ring.

Let:
 * $M: \quad \cdots \longrightarrow M_i \stackrel{d_i}{\longrightarrow} M_{i+1} \stackrel{d_{i+1}}{\longrightarrow} M_{i+2} \stackrel{d_{i+2}}{\longrightarrow} \cdots$

and
 * $N: \quad \cdots \longrightarrow N_i \stackrel{d'_i}{\longrightarrow} N_{i+1} \stackrel{d'_{i+1}}{\longrightarrow} N_{i+2} \stackrel{d'_{i+2}}{\longrightarrow} \cdots$

be two differential complexes of $R$-modules.

Let $\phi = \left\{ \phi_i : i \in \Z \right\}$ be a family of module homomorphisms $\phi_i : M_i \to N_i$.

Then $\phi$ is a homomorphism of complexes if for each $i \in \Z$ we have $\phi_{i+1} \circ d_i = \phi_i \circ d'_i$. That is, for each $i \in \Z$ we have a commutative diagram:


 * $\begin{xy}\xymatrix@L+2mu@+1em{

M_i \ar[r]^*{d_i} \ar[d]^*{\phi_i} & M_{i+1} \ar[d]^*{\phi_{i+1}} \\ N_i \ar[r]^*{d'_i} & N_{i+1}

}\end{xy}$