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.
Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; Hironaka (1964) 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.
It is called excellent[3]pg 214 if it is quasi-excellent and universally catenary.
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 it is universally catenary, every maximal chain of prime ideals has the same length.
In practice, this means infinite-dimensional Noetherian rings[4] which have an inductive definition of maximal chains of prime ideals, giving an infinite-dimensional ring, cannot be constructed.
Most naturally occurring commutative rings in number theory or algebraic geometry are excellent.
In particular: 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.
Here kp denotes the image of k under the Frobenius morphism a → ap.
Quasi-excellent rings are closely related to the problem of resolution of singularities, and this seems to have been Grothendieck's motivation[3]pg 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.