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  "body": "\n## Summary\n\nThe height of a prime ideal $P$ is the largest length of a strictly increasing chain of prime ideals that ends at $P$.\nIt equals the [[Krull dimension|MVGBS3J1P7]] of the [[localization|X6JD4AG973]] at $P$, and it is the algebraic notion of codimension.\n[[Krull's principal ideal theorem|SJZW2HN1SC]] and [[Krull's height theorem|TBRTZMHFTV]] bound the height by the number of generators of an ideal.\n\n## Explanations\n\n### L1 human\n\nThe dimension of a ring counts the longest staircase of prime ideals anywhere.\nThe height of a prime $P$ counts only the staircases that end at $P$, looking down from $P$.\nIn $k[x, y, z]$, the prime $(x)$ is a surface and has height $1$, while $(x, y)$ is a line and has height $2$: it has codimension $2$ in space.\n",
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