1. Burch ideals and Burch rings
- Author
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Hailong Dao, Ryo Takahashi, and Toshinori Kobayashi
- Subjects
Pure mathematics ,(weakly) m-full ideal ,Generalization ,Gorenstein ring ,singularity category ,Commutative Algebra (math.AC) ,hypersurface ,singular locus ,01 natural sciences ,fiber product ,Integrally closed ,syzygy ,0103 physical sciences ,FOS: Mathematics ,Representation Theory (math.RT) ,0101 mathematics ,Categorical variable ,Mathematics ,13C13 ,Algebra and Number Theory ,Hilbert's syzygy theorem ,Mathematics::Commutative Algebra ,13H10 ,Burch ring ,010102 general mathematics ,Multiplicity (mathematics) ,Mathematics - Commutative Algebra ,13C13, 13D09, 13H10 ,Hypersurface ,Burch ideal ,direct summand ,thick subcategory ,13D09 ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen--Macaulay rings of minimal multiplicity. We give several characterizations of these objects. We show that they satisfy many interesting and desirable properties: ideal-theoretic, homological, categorical. We relate them to other classes of ideals and rings in the literature., 23 pages, add Example 2.2, Prop 5.5 and Example 5.6
- Published
- 2019