Definition:Kernel of Homomorphism of Differential Complexes

This page is about Kernel in the context of Homological Algebra. For other uses, see Kernel.

Definition

Let $\struct {R, +, \cdot}$ 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 $\phi = \set {\phi_i : i \in \Z}$ 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