Excellent ring

• A ring ${\displaystyle R}$ containing a field ${\displaystyle k}$ is called geometrically regular over ${\displaystyle k}$ if for any finite extension ${\displaystyle K}$ of ${\displaystyle k}$ the ring ${\displaystyle R\otimes _{k}K}$ is regular.
• A homomorphism of rings from ${\displaystyle R\to S}$ is called regular if it is flat and for every ${\displaystyle {\mathfrak {p}}\in {\text{Spec}}(R)}$ the fiber ${\displaystyle S\otimes _{R}\kappa ({\mathfrak {p}})}$ is geometrically regular over the residue field ${\displaystyle \kappa ({\mathfrak {p}})}$ of ${\displaystyle {\mathfrak {p}}}$.
• A ring ${\displaystyle R}$ is called a G-ring[1] (or Grothendieck ring) if it is Noetherian and its formal fibers are geometrically regular; this means that for any ${\displaystyle {\mathfrak {p}}\in {\text{Spec}}(R)}$, the map from the local ring ${\displaystyle R_{\mathfrak {p}}\to {\hat {R_{\mathfrak {p}}}}}$ to its completion is regular in the sense above.

Finally, a ring is J-2[2] if any finite type ${\displaystyle R}$-algebra ${\displaystyle S}$ is J-1, meaning the regular subscheme ${\displaystyle {\text{Reg}}({\text{Spec}}(S))\subset {\text{Spec}}(S)}$ is open.

A ring ${\displaystyle R}$ is called quasi-excellent if it is a G-ring and J-2 ring. It is called excellent[3]^{pg 214} if it is quasi-excellent and universally catenary. In practice almost all Noetherian rings are universally catenary, so there is little difference between excellent and quasi-excellent rings.

A scheme is called excellent or quasi-excellent if it has a cover by open affine subschemes with the same property, which implies that every open affine subscheme has this property.

Given an excellent scheme ${\displaystyle X}$ and a locally finite type morphism ${\displaystyle f:X'\to X}$, then ${\displaystyle X'}$ is excellent[3]^{pg 217}.

Any quasi-excellent ring is a Nagata ring.

Any quasi-excellent reduced local ring is analytically reduced.

Any quasi-excellent normal local ring is analytically normal.

Most naturally occurring commutative rings in number theory or algebraic geometry are excellent. In particular:

• All complete Noetherian local rings, for instance all fields and the ring Z_p of p-adic integers, are excellent.
• All Dedekind domains of characteristic 0 are excellent. In particular the ring Z of integers is excellent. Dedekind domains over fields of characteristic greater than 0 need not be excellent.
• The rings of convergent power series in a finite number of variables over R or C are excellent.
• Any localization of an excellent ring is excellent.
• Any finitely generated algebra over an excellent ring is excellent. This includes all polynomial algebras ${\displaystyle R[x_{1},\ldots ,x_{n}]/(f_{1},\ldots ,f_{k})}$ with ${\displaystyle R}$ excellent. This means most rings considered in algebraic geometry are excellent.

Here is an example of a discrete valuation ring A of dimension 1 and characteristic p > 0 which is J-2 but not a G-ring and so is not quasi-excellent. If k is any field of characteristic p with [k : k^p] = ∞ and A is the ring of power series Σa_ixⁱ such that [k^p(a₀, a₁, ...) : k^p] is finite then the formal fibers of A are not all geometrically regular so A is not a G-ring. It is a J-2 ring as all Noetherian local rings of dimension at most 1 are J-2 rings. It is also universally catenary as it is a Dedekind domain. Here k^p denotes the image of k under the Frobenius morphism a → a^p.

Here is an example of a ring that is a G-ring but not a J-2 ring and so not quasi-excellent. If R is the subring of the polynomial ring k[x₁,x₂,...] in infinitely many generators generated by the squares and cubes of all generators, and S is obtained from R by adjoining inverses to all elements not in any of the ideals generated by some x_n, then S is a 1-dimensional Noetherian domain that is not a J-1 ring as S has a cusp singularity at every closed point, so the set of singular points is not closed, though it is a G-ring. This ring is also universally catenary, as its localization at every prime ideal is a quotient of a regular ring.

Nagata's example of a 2-dimensional Noetherian local ring that is catenary but not universally catenary is a G-ring, and is also a J-2 ring as any local G-ring is a J-2 ring (Matsumura 1980, p.88, 260). So it is a quasi-excellent catenary local ring that is not excellent.