In commutative algebra, a J-0 ring is a ring such that the set of regular points, that is, points of the spectrum at which the localization is a regular local ring, contains a non-empty open subset, a J-1 ring is a ring such that the set of regular points is an open subset, and a J-2 ring is a ring such that any finitely generated algebra over the ring is a J-1 ring.
Examplesedit
Most rings that occur in algebraic geometry or number theory are J-2 rings, and in fact it is not trivial to construct any examples of rings that are not. In particular all excellent rings are J-2 rings; in fact this is part of the definition of an excellent ring.
All Dedekind domains of characteristic 0 and all local Noetherian rings of dimension at most 1 are J-2 rings. The family of J-2 rings is closed under taking localizations and finitely generated algebras.
For an example of a Noetherian domain that is not a J-0 ring, take R to be the subring of the polynomial ringk[x1,x2,...] in infinitely many generators generated by the squares and cubes of all generators, and form the ring S from R by adjoining inverses to all elements not in any of the ideals generated by some xn. Then S is a 1-dimensional Noetherian domain that is not a J-0 ring. More precisely S has a cusp singularity at every closed point, so the set of non-singular points consists of just the ideal (0) and contains no nonempty open sets.
ring, this, article, includes, list, references, related, reading, external, links, sources, remain, unclear, because, lacks, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, december, 2015, learn, when, remove, t. This article includes a list of references related reading or external links but its sources remain unclear because it lacks inline citations Please help improve this article by introducing more precise citations December 2015 Learn how and when to remove this template message In commutative algebra a J 0 ring is a ring R displaystyle R such that the set of regular points that is points p displaystyle p of the spectrum at which the localization R p displaystyle R p is a regular local ring contains a non empty open subset a J 1 ring is a ring such that the set of regular points is an open subset and a J 2 ring is a ring such that any finitely generated algebra over the ring is a J 1 ring Examples editMost rings that occur in algebraic geometry or number theory are J 2 rings and in fact it is not trivial to construct any examples of rings that are not In particular all excellent rings are J 2 rings in fact this is part of the definition of an excellent ring All Dedekind domains of characteristic 0 and all local Noetherian rings of dimension at most 1 are J 2 rings The family of J 2 rings is closed under taking localizations and finitely generated algebras For an example of a Noetherian domain that is not a J 0 ring take R to be the subring of the polynomial ring k x1 x2 in infinitely many generators generated by the squares and cubes of all generators and form the ring S from R by adjoining inverses to all elements not in any of the ideals generated by some xn Then S is a 1 dimensional Noetherian domain that is not a J 0 ring More precisely S has a cusp singularity at every closed point so the set of non singular points consists of just the ideal 0 and contains no nonempty open sets See also editExcellent ringReferences editH Matsumura Commutative algebra ISBN 0 8053 7026 9 chapter 12 nbsp This commutative algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title J 2 ring amp oldid 1170051035, wikipedia, wiki, book, books, library,
