{"title":"拉伸理想的第一希尔伯特系数","authors":"Kazuho Ozeki","doi":"10.1007/s40306-021-00470-x","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we explore the almost Cohen-Macaulayness of the associated graded ring of stretched <span>\\({\\mathfrak m}\\)</span>-primary ideals with small first Hilbert coefficient in a Cohen-Macaulay local ring <span>\\((A,{\\mathfrak m})\\)</span>. In particular, we explore the structure of stretched <span>\\({\\mathfrak m}\\)</span>-primary ideals satisfying the equality e<sub>1</sub>(<i>I</i>) = e<sub>0</sub>(<i>I</i>) − <i>ℓ</i><sub><i>A</i></sub>(<i>A</i>/<i>I</i>) + 4, where e<sub>0</sub>(<i>I</i>) and e<sub>1</sub>(<i>I</i>) denote the multiplicity and the first Hilbert coefficient, respectively.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The First Hilbert Coefficient of Stretched Ideals\",\"authors\":\"Kazuho Ozeki\",\"doi\":\"10.1007/s40306-021-00470-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we explore the almost Cohen-Macaulayness of the associated graded ring of stretched <span>\\\\({\\\\mathfrak m}\\\\)</span>-primary ideals with small first Hilbert coefficient in a Cohen-Macaulay local ring <span>\\\\((A,{\\\\mathfrak m})\\\\)</span>. In particular, we explore the structure of stretched <span>\\\\({\\\\mathfrak m}\\\\)</span>-primary ideals satisfying the equality e<sub>1</sub>(<i>I</i>) = e<sub>0</sub>(<i>I</i>) − <i>ℓ</i><sub><i>A</i></sub>(<i>A</i>/<i>I</i>) + 4, where e<sub>0</sub>(<i>I</i>) and e<sub>1</sub>(<i>I</i>) denote the multiplicity and the first Hilbert coefficient, respectively.</p></div>\",\"PeriodicalId\":45527,\"journal\":{\"name\":\"Acta Mathematica Vietnamica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2022-02-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Vietnamica\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40306-021-00470-x\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Vietnamica","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40306-021-00470-x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
In this paper, we explore the almost Cohen-Macaulayness of the associated graded ring of stretched \({\mathfrak m}\)-primary ideals with small first Hilbert coefficient in a Cohen-Macaulay local ring \((A,{\mathfrak m})\). In particular, we explore the structure of stretched \({\mathfrak m}\)-primary ideals satisfying the equality e1(I) = e0(I) − ℓA(A/I) + 4, where e0(I) and e1(I) denote the multiplicity and the first Hilbert coefficient, respectively.
期刊介绍：
Acta Mathematica Vietnamica is a peer-reviewed mathematical journal. The journal publishes original papers of high quality in all branches of Mathematics with strong focus on Algebraic Geometry and Commutative Algebra, Algebraic Topology, Complex Analysis, Dynamical Systems, Optimization and Partial Differential Equations.
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