In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal.
The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; showed this in characteristic 0, but the positive characteristic case is (as of 2024) still a major open problem. Essentially all Noetherian rings that occur naturally in algebraic geometry or number theory are excellent; in fact it is quite hard to construct examples of Noetherian rings that are not excellent.
The definition of excellent rings is quite involved, so we recall the definitions of the technical conditions it satisfies. Although it seems like a long list of conditions, most rings in practice are excellent, such as fields, polynomial rings, complete Noetherian rings, Dedekind domains over characteristic 0 (such as
Z
R
k
k
K
k
R ⊗ kK
R\toS
ak{p}\inSpec(R)
S ⊗ R\kappa(ak{p})
\kappa(ak{p})
ak{p}
R
ak{p}\inSpec(R)
Rak{p}\to\hat{Rak{p}}
R
S
Reg(Spec(S))\subsetSpec(S)
A ring
R
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.
Because an excellent ring
R
Given an excellent scheme
X
f:X'\toX
X'
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:
R[x1,\ldots,xn]/(f1,\ldots,fk)
R
Here is an example of a discrete valuation ring of dimension and characteristic which is but not a -ring and so is not quasi-excellent. If is any field of characteristic with and is the ring of power series such that is finite then the formal fibers of are not all geometrically regular so is not a -ring. It is a ring as all Noetherian local rings of dimension at most are rings. It is also universally catenary as it is a Dedekind domain. Here denotes the image of under the Frobenius morphism .
Here is an example of a ring that is a G-ring but not a J-2 ring and so not quasi-excellent. If is the subring of the polynomial ring in infinitely many generators generated by the squares and cubes of all generators, and is obtained from by adjoining inverses to all elements not in any of the ideals generated by some, then is a 1-dimensional Noetherian domain that is not a ring as 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 . So it is a quasi-excellent catenary local ring that is not excellent.
Quasi-excellent rings are closely related to the problem of resolution of singularities, and this seems to have been Grothendieck's motivationpg 218 for defining them. Grothendieck (1965) observed that if it is possible to resolve singularities of all complete integral local Noetherian rings, then it is possible to resolve the singularities of all reduced quasi-excellent rings. Hironaka (1964) proved this for all complete integral Noetherian local rings over a field of characteristic 0, which implies his theorem that all singularities of excellent schemes over a field of characteristic 0 can be resolved. Conversely if it is possible to resolve all singularities of the spectra of all integral finite algebras over a Noetherian ring R then the ring R is quasi-excellent.