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  "body": "\n## Statement\n\nLet $p$ be a prime and let $R$ be a [[Noetherian ring|0670T59HGE]] of [[characteristic|0F6ES3BRYP]] $p$.\nLet $F \\colon R \\to R$, $F(r) = r^p$, be the [[Frobenius endomorphism|0T3WVTD675]], and view $R$ as an $R$-module through $F$.\nThen the following are equivalent:\n\n1. $R$ is a [[flat|73RYYEVDKA]] $R$-module through $F$.\n2. $R$ is regular, that is, for every [[prime ideal|00MHHZ9SNE]] $\\mathfrak p$ of $R$, the [[localization|4C1AE84N5N]] $R_{\\mathfrak p}$ is a [[regular local ring|63XXTF6B5M]].\n\n## Remarks\n\nBoth conditions are local, so one may assume that $R$ is local with maximal ideal $\\mathfrak m$.\nIf $R$ is regular, $F$ is a local homomorphism from a regular local ring to a Cohen–Macaulay ring with $\\dim R/\\mathfrak m^{[p]} = 0$, and such a map is flat (\"miracle flatness\").\nThe converse is the deeper direction, and it was the main new point of Kunz's work.\n\nExample: for $R = \\F_p[x]$, the image of $F$ is $\\F_p[x^p]$.\nThe ring $R$ is a free module over it with basis $1, x, \\ldots, x^{p-1}$, so $F$ is flat, as the theorem predicts for this regular ring.\nFor $R = \\F_p[x, y]/(xy)$, which is not regular at $(x, y)$, the Frobenius map is not flat.\n\n## Explanations\n\n### L1 human\n\nIn characteristic $p$, taking $p$-th powers is a ring homomorphism.\nKunz's theorem says that this map is as nice as possible, namely flat, exactly when the ring has no singular points.\nSo a purely algebraic property of one canonical map detects regularity, the algebraic form of having no singular points.\n\n### L2 human\n\nThe theorem is the starting point of the use of Frobenius to study singularities in prime characteristic.\nIt shows, for instance, that in a regular ring every ideal is tightly closed: from $c x^q \\in I^{[q]}$ and flatness one gets $c \\in (I^{[q]} : x^q) = (I : x)^{[q]}$, which forces $x \\in I$.\nWeakening flatness of Frobenius leads to the classes of F-regular, F-rational, F-pure and F-injective rings, see [[tight closure|KJ9MMP99G6]].\n",
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