Definition:Kernel of Homomorphism of Differential 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 homomorphism $M \to N$.

For each $i \in \Z$ let $K_i$ be the kernel of $\phi_i$.

For each $i \in \Z$ let $f_i$ be the restriction of $d_i$ to $K_i$.

Then the kernel of $\phi$ is:


 * $\ker \phi : \quad \cdots \longrightarrow K_i \stackrel{f_i}{\longrightarrow} K_{i+1} \stackrel{f_{i+1}}{\longrightarrow} K_{i+2} \stackrel{f_{i+2}}{\longrightarrow} \cdots$

Also see

 * Kernel of Homomorphism of Differential Complexes is Differential Complex