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  "body": "\n## Definition\n\nLet $R$ be a [[Noetherian ring|0670T59HGE]].\nA Noetherian ring $A$ is *regular* if $A_{\\mathfrak{q}}$ is a [[regular local ring|63XXTF6B5M]] for every prime ideal $\\mathfrak{q}$ of $A$.\nAn algebra $A$ over a field $\\kappa$ is *geometrically regular* if $A \\otimes_\\kappa L$ is regular for every finite field extension $L$ of $\\kappa$.\nThe ring $R$ is *excellent* if:\n\n1. $R$ is [[universally catenary|5WAFPTH09V]].\n2. $R$ is a G-ring: let $\\mathfrak{p}$ be a prime ideal of $R$ and let $\\widehat{R_{\\mathfrak{p}}}$ be the [[completion|KBN13NPYJQ]] of the [[localization|X6JD4AG973]] $R_{\\mathfrak{p}}$.\nThen for every prime ideal $\\mathfrak{q}$ of $R_{\\mathfrak{p}}$ with residue field $\\kappa(\\mathfrak{q})$, the ring $\\widehat{R_{\\mathfrak{p}}} \\otimes_{R_{\\mathfrak{p}}} \\kappa(\\mathfrak{q})$ is geometrically regular over $\\kappa(\\mathfrak{q})$.\n3. $R$ is J-2: let $S$ be a [[finitely generated|AC5TFFQT9M]] $R$-algebra.\nThen the set of prime ideals $\\mathfrak{q}$ of $S$ with $S_{\\mathfrak{q}}$ regular is open in the [[spectrum|FB45BHHA2J]] of $S$ with the [[Zariski topology|8SQK7RTB63]].\n\n## Remarks\n\nExample: every field $k$ is excellent.\nIt is universally catenary because finitely generated $k$-algebras are catenary.\nClause 2 holds because the only prime of $k$ is $0$, the ring $k$ is its own completion, and $k \\otimes_k k = k$ is geometrically regular over $k$.\nThe regular locus of a finitely generated $k$-algebra is open, which is clause 3.\nMore generally, $\\Z$, every complete Noetherian local ring, and every Dedekind domain of characteristic $0$ are excellent.\nLocalizations and finitely generated algebras of excellent rings are excellent.\n\nNon-example: there are Noetherian local domains whose completion has nonzero nilpotent elements.\nSuch a ring fails clause 2, because if a reduced Noetherian local ring is a G-ring, then its completion is reduced.\n\nSome authors call a ring that satisfies only clauses 2 and 3 quasi-excellent.\n",
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