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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[additive category|B2VP96NJSK]], for example the category of [[modules|0NZ8JA25WK]] over a ring.\nA *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\n$$d_{n-1} \\circ d_n = 0 \\quad \\text{for all } n \\in \\Z,$$\nwhere $0$ is the [[zero morphism|YF9F9EG6M3]] $C_n \\to C_{n-2}$.\n\n## Remarks\n\nA *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.\nA complex is *bounded* if only finitely many $C_n$ are nonzero.\nOne often writes $d$ for every $d_n$, so that the condition reads $d^2 = 0$.\n\nExample: let $C_n = \\Z/4\\Z$ and let $d_n$ be multiplication by $2$ for all $n$.\nThen $d_{n-1} \\circ d_n$ is multiplication by $4$, which is $0$ on $\\Z/4\\Z$.\n\nNon-example: let $C_2 = C_1 = C_0 = \\Z$, with $d_2$ and $d_1$ the identity of $\\Z$ and all other objects $0$.\nThen $d_1 \\circ d_2 = 1_{\\Z} \\ne 0$, so this is not a chain complex.\n",
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