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  "body": "\n## Statement\n\nLet $R$ be a [[local ring|MZ02BKPHJA]] with maximal ideal $\\mathfrak{m}$ and [[residue field|NJ3APQ8BHH]] $k = R/\\mathfrak{m}$.\nLet $P$ be a [[finitely generated|05QY1VEBYD]] [[projective|BFFA03GE3V]] $R$-module.\nThen $P$ is [[free|0DZ9DKAKZR]], of rank $n = \\dim_k P/\\mathfrak{m}P$.\n\n## Remarks\n\nChoose $x_1, \\ldots, x_n \\in P$ whose classes form a basis of the $k$-vector space $P/\\mathfrak{m}P$.\nBy [[Nakayama's lemma|15JHBYR6KH]], they generate $P$, so there is a surjection $\\pi \\colon R^n \\to P$; it splits because $P$ is projective, so $R^n \\cong P \\oplus K$ with $K = \\ker \\pi$.\nReducing modulo $\\mathfrak{m}$ gives $k^n \\cong k^n \\oplus K/\\mathfrak{m}K$, so $K/\\mathfrak{m}K = 0$.\nAs a direct summand of $R^n$, $K$ is finitely generated, and Nakayama's lemma gives $K = 0$; thus $P \\cong R^n$.\n\nThe local hypothesis is necessary.\nOver $R = \\Z/6\\Z$, the module $\\Z/2\\Z$ is a direct summand of $R$, hence projective, but it has $2$ elements, while every finitely generated free $R$-module has $6^n$ elements for some $n \\ge 0$.\nKaplansky's theorem removes the finiteness hypothesis: every projective module over a local ring is free.\nMore generally, a finitely presented flat module over a local ring is free.\n\n## Explanations\n\n### L1 human\n\nProjective modules are pieces of free modules, and in general they can be twisted.\nOver a local ring there is no room for twisting: every finitely generated projective module has a basis.\nIts size is read off from the vector space you get by reducing modulo the maximal ideal.\n\n### L2 human\n\nThis is why finitely generated projective modules over any commutative ring are called locally free: their localizations at prime ideals are free.\nThe rank at each prime gives a locally constant function on the spectrum, and rank one projectives form the [[Picard group|DZW58R6T2Q]].\nGeometrically, finitely generated projective modules are vector bundles on $\\operatorname{Spec} R$, and the theorem says that a vector bundle is trivial near each point.\nWhether such modules are globally free over polynomial rings is the content of the [[Quillen–Suslin theorem|7N50GA46W1]].\n",
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