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  "body": "\n## Summary\n\nA regular local ring is a Noetherian [[local ring|MZ02BKPHJA]] whose maximal ideal can be generated by as many elements as its [[Krull dimension|MVGBS3J1P7]].\nEquivalently, its [[embedding dimension|DP3KSCRKFP]] equals its dimension.\nRegular local rings are the algebraic form of smooth points of varieties.\nThey are [[integral domains|V3ZBMRKC08]] and even [[unique factorization domains|1HD3EK5392]], and by [[Serre's theorem|93RQRG9R1P]] they are the Noetherian local rings of finite global dimension.\n\n## Explanations\n\n### L1 human\n\nAt a smooth point of a surface, two coordinates describe everything nearby, and the surface has dimension two.\nAt a singular point, like the tip of a cone, you need more coordinates than the dimension.\nA regular local ring is the local ring of a point where the number of needed coordinates equals the dimension.\nExamples are fields, the ring $\\Z_{(p)}$, and power series rings $k[\\![x_1, \\ldots, x_n]\\!]$.\nThe [[definition|KTQC33ZJTN]] gives the condition.\n",
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