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- Excellent_ring abstract "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.".
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- Excellent_ring wikiPageLength "7141".
- Excellent_ring wikiPageOutDegree "21".
- Excellent_ring wikiPageRevisionID "705704898".
- Excellent_ring wikiPageWikiLink Alexander_Grothendieck.
- Excellent_ring wikiPageWikiLink Algebraic_geometry.
- Excellent_ring wikiPageWikiLink Analytically_normal_ring.
- Excellent_ring wikiPageWikiLink Analytically_unramified_ring.
- Excellent_ring wikiPageWikiLink Category:Algebraic_geometry.
- Excellent_ring wikiPageWikiLink Category:Commutative_algebra.
- Excellent_ring wikiPageWikiLink Catenary_ring.
- Excellent_ring wikiPageWikiLink Commutative_algebra.
- Excellent_ring wikiPageWikiLink Frobenius_endomorphism.
- Excellent_ring wikiPageWikiLink G-ring.
- Excellent_ring wikiPageWikiLink Geometrically_regular_ring.
- Excellent_ring wikiPageWikiLink J-2_ring.
- Excellent_ring wikiPageWikiLink Nagata_ring.
- Excellent_ring wikiPageWikiLink Noetherian_ring.
- Excellent_ring wikiPageWikiLink Number_theory.
- Excellent_ring wikiPageWikiLink Resolution_of_singularities.
- Excellent_ring wikiPageWikiLinkText "Excellent ring".
- Excellent_ring wikiPageWikiLinkText "excellent ring".
- Excellent_ring wikiPageWikiLinkText "excellent".
- Excellent_ring author "V.I. Danilov".
- Excellent_ring id "e/e036760".
- Excellent_ring title "Excellent ring".
- Excellent_ring wikiPageUsesTemplate Template:Brackets.
- Excellent_ring wikiPageUsesTemplate Template:Harv.
- Excellent_ring wikiPageUsesTemplate Template:Harvtxt.
- Excellent_ring wikiPageUsesTemplate Template:Springer.
- Excellent_ring subject Category:Algebraic_geometry.
- Excellent_ring subject Category:Commutative_algebra.
- Excellent_ring hypernym Ring.
- Excellent_ring type AnatomicalStructure.
- Excellent_ring comment "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.".
- Excellent_ring label "Excellent ring".
- Excellent_ring sameAs Q5419439.
- Excellent_ring sameAs m.02pzx75.
- Excellent_ring sameAs Q5419439.
- Excellent_ring sameAs 優環.
- Excellent_ring wasDerivedFrom Excellent_ring?oldid=705704898.
- Excellent_ring isPrimaryTopicOf Excellent_ring.