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  "body": "\n## Summary\n\nA projective module is a module $P$ such that every linear map from $P$ lifts along every surjective linear map.\nEquivalently, $P$ is a direct summand of a [[free module|0DZ9DKAKZR]].\nProjective modules keep the most useful property of free modules, lifting of maps, but need not have a basis.\nThey are the building blocks of projective resolutions in homological algebra, and finitely generated projective modules correspond to vector bundles in geometry.\n\n## Explanations\n\n### L1 human\n\nFree modules are the easiest modules: you can send a basis anywhere you like.\nA projective module is a piece cut off from a free module, so it shares this freedom when you map out of it.\nEvery free module is projective.\nOver the ring $\\Z/6\\Z \\cong \\Z/2\\Z \\times \\Z/3\\Z$, the module $\\Z/2\\Z$ is a piece of the free module $\\Z/6\\Z$, so it is projective, but it is too small to be free.\n",
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