Chain complex

Definition

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)

such that

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

The associated homology modules are

see homology module .

Cross-links

• 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

1. 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 $n\neq 0$.

2. 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 .

3. “Multiplication by $x$” complex.
   For $x\in R$, 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 $H_1=0$.