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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].\nA *projective resolution* of $M$ is an [[exact sequence|JB7ZBFBA9C]] of $R$-modules\n$$\\cdots \\to P_2 \\xrightarrow{d_2} P_1 \\xrightarrow{d_1} P_0 \\xrightarrow{\\varepsilon} M \\to 0$$\nin which every $P_i$ is a [[projective module|BFFA03GE3V]].\nIt is a *free resolution* if every $P_i$ is a [[free module|0DZ9DKAKZR]].\nIt has *length* $n$ if $P_n \\ne 0$ and $P_i = 0$ for all $i \u003e n$.\n\n## Remarks\n\nEvery module has a free resolution: take a free module $F_0$ mapping onto $M$, then a free module mapping onto the kernel, and repeat.\nExample: over $\\Z$, the sequence $0 \\to \\Z \\xrightarrow{\\cdot n} \\Z \\to \\Z/n\\Z \\to 0$ is a free resolution of length $1$, since multiplication by $n \\ne 0$ is injective and its image $n\\Z$ is the kernel of the reduction map.\nOver $R = \\Z/4\\Z$, the module $\\Z/2\\Z$ has the free resolution $\\cdots \\to R \\xrightarrow{\\cdot 2} R \\xrightarrow{\\cdot 2} R \\to \\Z/2\\Z \\to 0$, which never stops.\nNon-example: $0 \\to \\Z \\xrightarrow{\\cdot 4} \\Z \\to \\Z/2\\Z \\to 0$ is not a resolution, because it is not exact at the middle $\\Z$.\nThe kernel of the reduction is $2\\Z$, but the image of multiplication by $4$ is $4\\Z$.\nOne often writes the resolution as the complex $P_\\bullet$ together with the map $\\varepsilon \\colon P_0 \\to M$; then $H_0(P_\\bullet) \\cong M$ and $H_i(P_\\bullet) = 0$ for $i \u003e 0$.\n",
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