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  "body": "\n## Statement\n\nLet $Y$ be a locally [[Noetherian scheme|ND6XEY68P4]] and let $f \\colon X \\to Y$ be a [[proper morphism|9537RKBZPP]].\nThen $f = g \\circ f'$ for [[morphisms|JXRKH6P1Z4]] $f' \\colon X \\to Y'$ and $g \\colon Y' \\to Y$ with the following properties.\n\n1. $g$ is a [[finite morphism|N9C4X7XKE4]].\n2. $f'$ is proper, and the map $\\mathcal{O}_{Y'} \\to f'_* \\mathcal{O}_X$ is an isomorphism.\n3. Every fibre of $f'$ is nonempty and connected.\n\nOne may take $Y'$ to be the relative spectrum over $Y$ of the sheaf of $\\mathcal{O}_Y$-algebras $f_* \\mathcal{O}_X$.\n\n## Remarks\n\nThe sheaf $f_* \\mathcal{O}_X$ is coherent because $f$ is proper, so $g$ is finite.\nClause 3 follows from clause 2 by Zariski's connectedness theorem, which in turn rests on the theorem on formal functions.\nFor example, if $f$ is finite, then $f$ is affine and $X$ is the relative spectrum of $f_* \\mathcal{O}_X$.\nSo $Y' = X$, $f'$ is the identity and $g = f$.\nAt the other extreme, for $X$ proper and connected over $k$ with $\\mathcal{O}(X) = k$, the map $X \\to \\operatorname{Spec} k$ already has $Y' = Y$.\nThe properness hypothesis is necessary: for a non-proper $f$, the sheaf $f_* \\mathcal{O}_X$ need not be coherent.\nFor $\\mathbb{A}^1_k \\to \\operatorname{Spec} k$, the algebra $f_* \\mathcal{O}_X = k[t]$ is not finite over $k$.\n\n## Explanations\n\n### L1 human\n\nEvery proper map splits into two simpler maps.\nThe first has connected fibres, so it only collapses connected pieces.\nThe second is finite, so each point of the target has finitely many preimages.\nOne can picture it as first squashing each connected part of each fibre to a point, and then mapping the result finitely onto the target.\n\n### L2 human\n\nThe Stein factorization reduces many questions about proper morphisms to two cases: finite morphisms and morphisms with connected fibres.\nIt is used in the study of contractions in birational geometry.\nThe contractions of the minimal model program are by definition proper morphisms $f$ with $f_* \\mathcal{O}_X = \\mathcal{O}_Y$, exactly the maps $f'$ of the factorization.\nThe name comes from the analogous result of Karl Stein for proper holomorphic maps of complex spaces.\n",
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