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  "body": "\n## Summary\n\nThe projective dimension of a module $M$ is the length of a shortest [[projective resolution|HK2AA1YSQ2]] of $M$.\nIt is $0$ exactly for the nonzero [[projective modules|BFFA03GE3V]], so it measures how far $M$ is from being projective.\nOver a Noetherian local ring it is linked to the [[depth|M2RADBFW99]] by the [[Auslander–Buchsbaum formula|SQB74J5E8Q]].\n\n## Explanations\n\n### L1 human\n\nSome modules are free or projective; they need no relations at all.\nOthers need relations, then relations among relations, and so on.\nThe projective dimension counts how many steps this takes in the best case, or is infinite if it never stops.\nOver $\\Z$, the module $\\Z/6\\Z$ has projective dimension $1$: one relation, $6 = 0$, and nothing more.\n",
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