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  "body": "\n## Summary\n\nA flat module is a module $M$ such that tensoring with $M$ keeps injective linear maps injective.\nEquivalently, the functor $- \\otimes_R M$ preserves [[exact sequences|JB7ZBFBA9C]].\nFree and [[projective modules|BFFA03GE3V]] are flat, and so are [[localizations|4C1AE84N5N]] such as $\\Q$ over $\\Z$.\nFlatness is the algebraic form of \"varying continuously in a family\".\nFlat ring maps are the basic tool of descent and base change in algebraic geometry.\n\n## Explanations\n\n### L1 human\n\nTensoring with a module is a way to change the ring of scalars or to combine two modules.\nIn general this operation can collapse things: a nonzero map may become zero.\nA flat module never causes such a collapse of injective maps.\nFor example, $\\Z/2\\Z$ is not flat over $\\Z$: the injective map \"multiply by $2$\" on $\\Z$ becomes the zero map on $\\Z/2\\Z$.\n",
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