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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $X$ be an object of $\\mathcal{A}$.\nAn *injective resolution* of $X$ is a sequence of objects and morphisms\n$$0 \\to X \\xrightarrow{\\eta} I^0 \\xrightarrow{d^0} I^1 \\xrightarrow{d^1} I^2 \\to \\cdots$$\nsuch that:\n\n1. every $I^n$ is an [[injective object|5QRFTBF4PG]];\n2. the sequence is [[exact|JB7ZBFBA9C]].\n\n## Remarks\n\nExample: over $\\Z$, the sequence $0 \\to \\Z \\to \\Q \\to \\Q/\\Z \\to 0$ is an injective resolution of $\\Z$.\nIt is exact, and $\\Q$ and $\\Q/\\Z$ are divisible groups, which are [[injective modules|85H7PRHDPZ]] over $\\Z$.\nNon-example: $0 \\to \\Z \\xrightarrow{\\cdot 2} \\Z \\to \\Z/2\\Z \\to 0$ is exact, but $\\Z$ is not an injective $\\Z$-module, since it is not divisible, so clause 1 fails.\nOne writes the resolution as the cochain complex $I^\\bullet$ with the map $\\eta$; then $H^0(I^\\bullet) \\cong X$ and $H^n(I^\\bullet) = 0$ for $n \u003e 0$.\nIf $\\mathcal{A}$ has enough injectives, every object has an injective resolution; this holds for modules over any ring, since every module embeds into an injective module.\n",
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