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  "body": "\n## Statement\n\nLet $A \\subseteq B$ be [[integral domains|0QVD7WBFGN]] such that $A$ is [[integrally closed|H20P600HQS]] and $B$ is [[integral over|B7X0KFY1S1]] $A$.\nLet $\\mathfrak{p}_1 \\supseteq \\mathfrak{p}_2$ be [[prime ideals|00MHHZ9SNE]] of $A$, and let $\\mathfrak{q}_1$ be a prime ideal of $B$ with $\\mathfrak{q}_1 \\cap A = \\mathfrak{p}_1$.\nThen there is a prime ideal $\\mathfrak{q}_2$ of $B$ with $\\mathfrak{q}_2 \\subseteq \\mathfrak{q}_1$ and $\\mathfrak{q}_2 \\cap A = \\mathfrak{p}_2$.\n\n## Remarks\n\nThe proof shows that $\\mathfrak{p}_2 B_{\\mathfrak{q}_1} \\cap A = \\mathfrak{p}_2$; a prime of $B_{\\mathfrak{q}_1}$ over $\\mathfrak{p}_2$ then exists, and it contracts to the wanted $\\mathfrak{q}_2$ in $B$.\nThe key step uses that $A$ is integrally closed.\nIf an element of $B$ is integral over the ideal $\\mathfrak{p}_2$, then the coefficients of its minimal polynomial over the fraction field of $A$ lie in $\\mathfrak{p}_2$.\nHere one also uses that $B$ is a domain, so that nonzero elements of $A$ are not zero divisors on $B$.\n\nExample: $\\Z \\subseteq \\Z[i]$ satisfies the hypotheses, and for $\\mathfrak{p}_1 = 5\\Z \\supseteq \\mathfrak{p}_2 = 0$ and $\\mathfrak{q}_1 = (2 + i)$, the prime $\\mathfrak{q}_2 = 0$ works.\n\nThe hypothesis that $A$ is integrally closed cannot be dropped: there are integral extensions of domains in which going-down fails.\nA different sufficient condition is flatness: going-down holds for every flat ring extension.\n\n## Explanations\n\n### L1 human\n\nGoing-down is the reverse of [[going-up|GHKYR5PBQC]]: given a prime upstairs, one wants to descend along a chain of smaller primes downstairs.\nUnlike going-up, this needs the small ring to be integrally closed.\n\n### L2 human\n\nGoing-down is used to prove that heights behave well in integral extensions over a normal base: $\\operatorname{ht} \\mathfrak{q} = \\operatorname{ht}(\\mathfrak{q} \\cap A)$ for every prime $\\mathfrak{q}$ of $B$.\nIt is a key ingredient in dimension theory of finitely generated algebras over a field, for example to show that such domains are catenary.\nIn geometry it gives that a finite surjective map from an irreducible variety onto a normal variety is an open map.\n",
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