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  "body": "\n## Summary\n\nA faithfully flat module is a [[flat module|73RYYEVDKA]] $M$ such that tensoring with $M$ kills no nonzero module.\nThen a sequence of modules is exact if and only if it stays exact after tensoring with $M$.\nA ring map $R \\to S$ is faithfully flat when $S$ is faithfully flat as an $R$-module; such maps are the setting of faithfully flat descent.\nExamples are nonzero free modules, polynomial rings $R[x]$ over $R$, and the completion of a Noetherian local ring.\n\n## Explanations\n\n### L1 human\n\nA flat module never destroys injectivity, but it can still make a whole module vanish.\nFor example, $\\Q$ is flat over $\\Z$, yet $\\Z/2\\Z \\otimes_\\Z \\Q = 0$.\nA faithfully flat module never does this: it reflects all the information.\nSo a statement can be checked after a faithfully flat change of rings, where it may be easier to prove.\n",
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