Chain complex

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.