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; 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.
quasi-excellentring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellentring if it...
pages with titles containing Excellent Excellence (disambiguation) Excellentring, in commutative algebra Most Excellent Majesty, a form of address in...
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of a quasi-excellent ring that is not an excellentring. Formally catenary ring (which is the same as a universally catenary ring). Hochster, Mel (Winter...
An engagement ring, also known as a betrothal ring, is a ring indicating that the person wearing it is engaged to be married, especially in Western cultures...
discrete valuation ring is not necessarily Japanese. Any quasi-excellentring is a Nagata ring, so in particular almost all Noetherian rings that occur in...
functor Excellentring Fibred category – Concept in category theory Formally smooth map Fundamental groupoid Fundamental group scheme Gorenstein ring – Local...
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geometrically regular local ring. acceptable ring Acceptable rings are generalizations of excellentrings, with the conditions about regular rings in the definition...
benzene molecule is composed of six carbon atoms joined in a planar hexagonal ring with one hydrogen atom attached to each. Because it contains only carbon...
Wu and his companions on a mission to the Ringworld, an enormous rotating ring, an alien construct in space 186 million miles (299 million kilometres) in...
PPEs that contain only o- and p-substituted rings have the highest melting points. PPEs have excellent high temperature properties and good oxidation...
an excellent heat and acid resistance. Lessing rings withstand corrosion and are used in regenerative oxide systems and transfer systems. Pall rings are...
ordinary city street up to motorway level. An excellent example of this is London's North Circular/South Circular ring roads, which are largely made up of (mainly...
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The Ring and the Book is a long dramatic narrative poem, and, more specifically, a verse novel, of 21,000 lines, written by Robert Browning. It was published...
Ring culture is a method of cultivating tomato plants. Tomato plants are grown in a bottomless pot, a "ring", and the pot is partially submerged in a...
Noetherian, reduced, normal, regular, or excellent then so is its Henselization. For example, the Henselization of the ring of polynomials k[x,y,...] localized...
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