Excellent ring

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 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; Hironaka (1964) showed this in characteristic 0, but the positive characteristic case is (as of 2016) 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.

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. 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.

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