Chain map - Knowlpedia

Let $(C_\bullet,d^C)$ and $(D_\bullet,d^D)$ be chain complexes of $R$ -modules. A chain map $f:C_\bullet\to D_\bullet$ is a family of $R$ -linear maps

f_n: C_n \longrightarrow D_n \qquad (n\in\mathbb Z)

such that for every $n$ ,

d^D_n\circ f_n = f_{n-1}\circ d^C_n.

Induced maps on homology

A chain map induces maps on homology:

H_n(f): H_n(C_\bullet)\to H_n(D_\bullet),

see homology module.

Equivalent characterizations

Equivalently, the squares commute:

\begin{array}{ccc} C_n & \xrightarrow{d^C_n} & C_{n-1}\\ \downarrow f_n && \downarrow f_{n-1}\\ D_n & \xrightarrow{d^D_n} & D_{n-1}. \end{array}

Remarks

Two chain maps may be equivalent “up to homotopy”: chain homotopy.
Chain maps are morphisms in the category of complexes; more generally in an abelian category.
In degree 0, chain maps recover ordinary module homomorphisms.

Examples

A module homomorphism as a chain map. If $M,N$ are modules viewed as complexes concentrated in degree $0$ (see chain complex examples), then a chain map $M\to N$ is exactly an $R$ -linear map $M\to N$ .

Inclusion of a subcomplex. If $C_\bullet\subseteq D_\bullet$ degreewise and $d^D$ restricts to $C_\bullet$ , then the inclusions $i_n:C_n\hookrightarrow D_n$ form a chain map $i:C_\bullet\to D_\bullet$ .

Multiplication on a fixed complex. Let $C_\bullet$ be any chain complex of $R$ -modules and fix a central element $r\in Z(R)$ (in particular, this holds for every $r$ when $R$ is commutative). Define $f_n:C_n\to C_n$ by $f_n(c)=rc$ . Centrality ensures $f_n$ is $R$ -linear, since $f_n(ac)=r(ac)=(ra)c=(ar)c=a(rc)$ ; then $R$ -linearity of the differentials gives $d_n(rc)=r\,d_n(c)$ , so $f=(f_n)$ is a chain endomorphism $C_\bullet\to C_\bullet$ .