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  "body": "\n## Statement\n\nLet $R$ be a commutative ring and let $S$ be a commutative $R$-algebra of [[finite type|AC5TFFQT9M]].\nLet $S'$ be the [[integral closure|G0V86SJBQW]] of $R$ in $S$.\nSo $S'$ is the set of elements of $S$ that are integral over the image of $R$.\nLet $\\mathfrak{q}$ be a [[prime ideal|06HC6HK0Z8]] of $S$ and let $\\mathfrak{p}$ be its preimage in $R$.\nAssume that $\\mathfrak{q}$ is both minimal and maximal among the prime ideals of $S$ whose preimage in $R$ is $\\mathfrak{p}$.\nThen there is an element $g \\in S'$ with $g \\notin \\mathfrak{q}$ such that the inclusion $S' \\to S$ induces an isomorphism of [[localizations|4C1AE84N5N]] $S'_g \\cong S_g$.\n\n## Remarks\n\nThe hypothesis says that $S$ is quasi-finite at $\\mathfrak{q}$: the point $\\mathfrak{q}$ is isolated in its fibre over $\\mathfrak{p}$.\nThis is Grothendieck's algebraic form of the theorem.\nIn words, near a point where it is quasi-finite, a finite type algebra looks like a localization of an integral algebra.\nIn geometric language, a quasi-finite separated morphism of finite type between Noetherian schemes factors as an open immersion followed by a finite morphism.\nZariski's original form follows: a birational morphism of varieties with finite fibres onto a normal variety is an open immersion.\nFor example, $k[x, x^{-1}]$ is quasi-finite over $k[x]$ at every prime, and indeed it is the localization of $k[x]$ at $g = x$.\nWithout the hypothesis the conclusion fails: for $R = k$ a field, $S = k[x]$ and $\\mathfrak{q} = (x)$, the prime $\\mathfrak{q}$ contains $(0)$, which also lies over $(0)$; here $S' = k$, and $k_g = k$ is never $k[x]_g$.\n\n## Explanations\n\n### L1 human\n\nA map of algebraic objects is quasi-finite at a point when, near that point, the point is alone in its fibre.\nThe theorem says that such a map is, near that point, just an open piece of a finite map.\nSo finiteness of fibres, which looks like a weak condition, forces strong structure.\n\n### L2 human\n\nThe theorem has many forms: the algebraic one stated here, the factorisation of quasi-finite separated morphisms, and Zariski's statement on birational maps to normal varieties.\nIt is a basic tool in algebraic geometry, for example in the study of étale and unramified morphisms.\nThe role of [[integral closure|G0V86SJBQW]] is central: the finite algebra in the factorisation is built from the integral closure of $R$ in $S$.\n",
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