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  "body": "\n## Definition\n\nA [[scheme|5A35TYZD02]] $X$ is *normal* if for every point $x \\in X$ the [[stalk|JTAKAPTT8H]] $\\mathcal{O}_{X,x}$ is an [[integrally closed domain|H20P600HQS]].\n\n## Remarks\n\nAn affine scheme $\\operatorname{Spec} A$ is normal if and only if $A_{\\mathfrak p}$ is an integrally closed domain for every prime $\\mathfrak{p}$, that is, $A$ is a [[normal ring|KR7SWBYHRQ]].\nFor an integral domain $A$, this holds if and only if $A$ is integrally closed.\nExample: $\\operatorname{Spec} \\Z$ is normal, since $\\Z$ is a unique factorization domain and hence integrally closed.\nNon-example: let $A = k[x, y]/(y^2 - x^3)$ for a field $k$; it is a domain.\nThe element $t = y/x$ of its field of fractions satisfies $t^2 - x = 0$, which is monic over $A$, but $t \\notin A$.\nSo $A$ is not integrally closed, and the cusp $\\operatorname{Spec} A$ is not normal; the failure is at the origin.\nA normal scheme is reduced, but it need not be irreducible: a disjoint union of two normal schemes is normal.\n",
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