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  "body": "\n## Statement\n\nLet $A \\subseteq B$ be commutative rings with $B$ [[integral over|B7X0KFY1S1]] $A$.\nLet $\\mathfrak{p}_1 \\subseteq \\mathfrak{p}_2$ 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 $\\mathfrak{q}_1 \\subseteq \\mathfrak{q}_2$ and $\\mathfrak{q}_2 \\cap A = \\mathfrak{p}_2$.\n\n## Remarks\n\nThe ring $B/\\mathfrak{q}_1$ is integral over its subring $A/\\mathfrak{p}_1$.\nThe [[lying over theorem|YZ97GKYY3P]] gives a prime of $B/\\mathfrak{q}_1$ over $\\mathfrak{p}_2/\\mathfrak{p}_1$, and its preimage in $B$ is $\\mathfrak{q}_2$.\n\nBy induction, every chain $\\mathfrak{p}_1 \\subseteq \\cdots \\subseteq \\mathfrak{p}_n$ of primes of $A$ lifts to a chain $\\mathfrak{q}_1 \\subseteq \\cdots \\subseteq \\mathfrak{q}_n$ of primes of $B$ with $\\mathfrak{q}_i \\cap A = \\mathfrak{p}_i$.\nSo the [[Krull dimension|MVGBS3J1P7]] of $B$ is at least that of $A$.\n\nExample: for $\\Z \\subseteq \\Z[i]$, the chain $0 \\subseteq 5\\Z$ lifts to $0 \\subseteq (2 + i)$.\n\nIntegrality is needed: for $k[x] \\subseteq k[x, x^{-1}]$, the prime $0$ lies under $0$, but no prime of $k[x, x^{-1}]$ lies over $(x)$, since $x$ is a unit there.\n\n## Explanations\n\n### L1 human\n\nIf a prime of the big ring sits over a prime of the small ring, then one can climb up along any longer chain downstairs and stay above it upstairs.\nIn geometry: the image of a closed subset under an integral map is closed, and specializations of points can be followed upstairs.\n\n### L2 human\n\nGoing-up and [[incomparability|Q2XBHJ96GV]] together give $\\dim B = \\dim A$ for every integral extension $A \\subseteq B$.\nThis is the key step in computing the dimension of a finitely generated algebra over a field through [[Noether normalization|7Z28JY8QW4]].\nThe companion result in the other direction, [[going-down|YE52MQ13MV]], needs extra hypotheses.\n",
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  "name": "Going-up theorem",
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