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  "body": "\n## Summary\n\nA projective resolution of a module $M$ is an [[exact sequence|JB7ZBFBA9C]] of [[projective modules|BFFA03GE3V]] that ends in $M$.\nIt replaces $M$ by a complex of simple pieces that has the same homological information.\nEvery module has a projective resolution, and the derived functors such as [[Ext|4P7GBWJ4RR]] and [[Tor|DDQRERAQCV]] are computed from it.\n\n## Explanations\n\n### L1 human\n\nWrite $M$ as a quotient of a free module, then describe the relations by another free module, then the relations among the relations, and so on.\nThe chain of free modules that you get is a free resolution, a special kind of projective resolution.\nFor example, $\\Z/n\\Z$ is the quotient of $\\Z$ by the multiples of $n$, and there are no further relations.\nSo $0 \\to \\Z \\xrightarrow{n} \\Z \\to \\Z/n\\Z \\to 0$ is a free resolution of $\\Z/n\\Z$.\n",
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