---
tag: P6AJ840RV4
kind: definition
name: Chain complex
msc: [18G35]
defines: WQ82C0M3QB
uses: [B2VP96NJSK, YF9F9EG6M3, 0NZ8JA25WK]
sources:
  - url: https://en.wikipedia.org/wiki/Chain_complex
---

## Definition

Let $\mathcal{A}$ be an [[additive category|B2VP96NJSK]], for example the category of [[modules|0NZ8JA25WK]] over a ring.
A *chain complex* $C_\bullet$ in $\mathcal{A}$ consists of objects $C_n$ and morphisms $d_n \colon C_n \to C_{n-1}$, for $n \in \Z$, with
$$d_{n-1} \circ d_n = 0 \quad \text{for all } n \in \Z,$$
where $0$ is the [[zero morphism|YF9F9EG6M3]] $C_n \to C_{n-2}$.

## Remarks

A *cochain complex* has objects $C^n$ and differentials $d^n \colon C^n \to C^{n+1}$ with $d^{n+1} \circ d^n = 0$; setting $C_n = C^{-n}$ turns it into a chain complex.
A complex is *bounded* if only finitely many $C_n$ are nonzero.
One often writes $d$ for every $d_n$, so that the condition reads $d^2 = 0$.

Example: let $C_n = \Z/4\Z$ and let $d_n$ be multiplication by $2$ for all $n$.
Then $d_{n-1} \circ d_n$ is multiplication by $4$, which is $0$ on $\Z/4\Z$.

Non-example: let $C_2 = C_1 = C_0 = \Z$, with $d_2$ and $d_1$ the identity of $\Z$ and all other objects $0$.
Then $d_1 \circ d_2 = 1_{\Z} \ne 0$, so this is not a chain complex.