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  "body": "\n## Definition\n\nA scheme $X$ is *regular* if it is [[locally Noetherian|ND6XEY68P4]] and for every point $x \\in X$ the [[stalk|JTAKAPTT8H]] $\\mathcal{O}_{X,x}$ is a [[regular local ring|63XXTF6B5M]].\nHere a Noetherian local ring $R$ with maximal ideal $\\mathfrak{m}$ and residue field $\\kappa$ is regular if $\\dim_\\kappa \\mathfrak{m}/\\mathfrak{m}^2 = \\dim R$, the [[Krull dimension|MVGBS3J1P7]] of $R$.\n\n## Remarks\n\nA regular local ring is an integrally closed domain, so a regular scheme is normal.\nExample: $\\operatorname{Spec} \\Z$ is regular.\nIts local rings are $\\Q$, of dimension $0$ with $\\mathfrak{m} = 0$, and $\\Z_{(p)}$, of dimension $1$ with $\\mathfrak{m}/\\mathfrak{m}^2$ spanned by $p$.\nNon-example: the cusp $\\operatorname{Spec} k[x, y]/(y^2 - x^3)$ over a field $k$ is not regular at the origin.\nThere the local ring has dimension $1$, but $\\mathfrak{m}/\\mathfrak{m}^2$ has basis $x, y$, since $y^2 - x^3 \\in \\mathfrak{m}^2$; so its dimension is $2 \\ne 1$.\n",
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