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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], for example the category of modules over a ring, and let $(C_\\bullet, d)$ be a [[chain complex|WQ82C0M3QB]] in $\\mathcal{A}$.\nFor each integer $n$, let $Z_n = \\ker d_n$ be the [[kernel|S417YNEJ6M]] of $d_n$, called the object of *$n$-cycles*.\nSince $d_n \\circ d_{n+1} = 0$, the morphism $d_{n+1}$ factors through a morphism $\\bar d_{n+1} \\colon C_{n+1} \\to Z_n$.\nIts image $B_n = \\im \\bar d_{n+1}$ is a subobject of $Z_n$, called the object of *$n$-boundaries*.\nThe *$n$-th homology* of $C_\\bullet$ is the [[cokernel|H8W38933NC]] of $\\bar d_{n+1}$:\n$$H_n(C) = Z_n / B_n = \\ker d_n / \\im d_{n+1}.$$\n\n## Remarks\n\nThe complex is exact at $C_n$ if and only if $H_n(C) = 0$.\nA [[chain map|HTK9WQZJ1G]] $f \\colon C_\\bullet \\to D_\\bullet$ sends cycles to cycles and boundaries to boundaries, so it induces morphisms $H_n(f) \\colon H_n(C) \\to H_n(D)$, and $H_n$ is a functor.\nFor a cochain complex $(C^\\bullet, d)$ the same construction gives the *cohomology* $H^n(C) = \\ker d^n / \\im d^{n-1}$.\n\nExample: let $C_1 = C_0 = \\Z$ with $d_1$ multiplication by $2$, and $C_n = 0$ otherwise.\nThen $Z_1 = \\ker d_1 = 0$, so $H_1(C) = 0$; and $Z_0 = \\Z$, $B_0 = 2\\Z$, so $H_0(C) = \\Z/2\\Z$.\n\nNon-example: for the sequence $\\Z \\xrightarrow{1} \\Z \\xrightarrow{1} \\Z$, the image $\\Z$ of the first map is not contained in the kernel $0$ of the second.\nSo $\\ker / \\im$ is not defined; the condition $d^2 = 0$ is what makes homology make sense.\n",
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