Definition:Homomorphism of Complexes

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 family of module homomorphisms $\phi_i: M_i \to N_i$.

Then $\phi$ is a homomorphism of complexes if and only if for each $i \in \Z$:

$\phi_{i + 1} \circ d_i = \phi_i \circ d'_i$

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}$

Linguistic Note

The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.

Thus homomorphism means similar structure.

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