Chain complex - Knowlpedia

Let $R$ be a ring and let $\{C_n\}_{n\in\mathbb Z}$ be a family of (left) R-modules. A chain complex $(C_\bullet, d)$ is a collection of $R$ -linear maps (called differentials)

d_n : C_n \longrightarrow C_{n-1}\qquad (n\in\mathbb Z)

such that

d_{n-1}\circ d_n = 0\quad\text{for all }n.

The associated homology modules are

H_n(C_\bullet)=\ker(d_n)/\operatorname{im}(d_{n+1}),

see homology module.

Equivalent characterizations

Equivalently, $\operatorname{im}(d_n)\subseteq \ker(d_{n-1})$ for all $n$ .

Remarks

Morphisms between complexes: chain map.
When all homology vanishes: exact complex.
Categorical setting: in an abelian category, a chain complex is defined the same way using kernels and images.

Examples

Complex concentrated in degree 0. For an $R$ -module $M$ , the diagram $\cdots \to 0 \to M \to 0 \to \cdots$ with $M$ in degree $0$ and all differentials $0$ , is a chain complex. Its homology is $H_0\cong M$ and $H_n=0$ for $n\neq 0$ .

A map as a 2-term complex. Any $R$ -linear map $f:M\to N$ gives a chain complex $0 \longrightarrow M \xrightarrow{\,f\,} N \longrightarrow 0$ with $M$ in degree $1$ and $N$ in degree $0$ . Then $H_1 \cong \ker(f),\qquad H_0 \cong \operatorname{coker}(f),$ using kernel / cokernel.

“Multiplication by $x$ ” complex. For a commutative ring $R$ and $x\in R$ , the 2-term complex $0 \to R \xrightarrow{\,\cdot x\,} R \to 0$ (degrees $1\to 0$ ) has $H_1 \cong \{r\in R: xr=0\} = \operatorname{Ann}_R(x), \qquad H_0 \cong R/xR.$ If $R$ is a domain and $x\neq 0$ , then $H_1=0$ .