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  "body": "\n## Summary\n\nA regular scheme is a locally Noetherian [[scheme|5A35TYZD02]] whose local rings are all [[regular local rings|63XXTF6B5M]].\nIt is the algebraic notion of a space without singularities that works over any base, including $\\Z$.\nOver a perfect field, a scheme of finite type is regular exactly when it is smooth.\n\n## Explanations\n\n### L1 human\n\nAt a smooth point of a surface, two coordinates are enough to describe everything nearby.\nAt a singular point, such as the tip of a cone, more are needed than the dimension.\nRegularity says exactly this, in algebra: near each point, the number of local generators equals the dimension.\nThe plane is regular, and so is $\\operatorname{Spec} \\Z$, but the cusp $y^2 = x^3$ is not regular at the origin.\n",
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