Gorenstein代数和模的第一希尔伯特系数的上界

IF 0.3 4区 数学 Q4 MATHEMATICS
Sabine El Khoury, Manoj Kummini, H. Srinivasan
{"title":"Gorenstein代数和模的第一希尔伯特系数的上界","authors":"Sabine El Khoury, Manoj Kummini, H. Srinivasan","doi":"10.1090/conm/773/15542","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\n <mml:semantics>\n <mml:mi>R</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a polynomial ring over a field and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M equals circled-plus Underscript n Endscripts upper M Subscript n\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>M</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:munder>\n <mml:mo>⨁<!-- ⨁ --></mml:mo>\n <mml:mi>n</mml:mi>\n </mml:munder>\n <mml:msub>\n <mml:mi>M</mml:mi>\n <mml:mi>n</mml:mi>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">M= \\bigoplus _n M_n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a finitely generated graded <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\n <mml:semantics>\n <mml:mi>R</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-module, minimally generated by homogeneous elements of degree zero with a graded <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\n <mml:semantics>\n <mml:mi>R</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-minimal free resolution <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper F\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"bold\">F</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbf {F}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. A Cohen-Macaulay module <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\n <mml:semantics>\n <mml:mi>M</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"e 1\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>e</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">e_1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in terms of the shifts in the graded resolution of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\n <mml:semantics>\n <mml:mi>M</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. When <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M equals upper R slash upper I\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>M</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mi>R</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mi>I</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">M = R/I</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, a Gorenstein algebra, this bound agrees with the bound obtained in <bold>[ES09]</bold> in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.</p>","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"21 1","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2020-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"An upper bound for the first Hilbert coefficient of Gorenstein algebras and modules\",\"authors\":\"Sabine El Khoury, Manoj Kummini, H. Srinivasan\",\"doi\":\"10.1090/conm/773/15542\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper R\\\">\\n <mml:semantics>\\n <mml:mi>R</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">R</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> be a polynomial ring over a field and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper M equals circled-plus Underscript n Endscripts upper M Subscript n\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>M</mml:mi>\\n <mml:mo>=</mml:mo>\\n <mml:munder>\\n <mml:mo>⨁<!-- ⨁ --></mml:mo>\\n <mml:mi>n</mml:mi>\\n </mml:munder>\\n <mml:msub>\\n <mml:mi>M</mml:mi>\\n <mml:mi>n</mml:mi>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">M= \\\\bigoplus _n M_n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> be a finitely generated graded <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper R\\\">\\n <mml:semantics>\\n <mml:mi>R</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">R</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-module, minimally generated by homogeneous elements of degree zero with a graded <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper R\\\">\\n <mml:semantics>\\n <mml:mi>R</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">R</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-minimal free resolution <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"bold upper F\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"bold\\\">F</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbf {F}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. A Cohen-Macaulay module <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper M\\\">\\n <mml:semantics>\\n <mml:mi>M</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">M</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"e 1\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>e</mml:mi>\\n <mml:mn>1</mml:mn>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">e_1</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> in terms of the shifts in the graded resolution of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper M\\\">\\n <mml:semantics>\\n <mml:mi>M</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">M</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. When <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper M equals upper R slash upper I\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>M</mml:mi>\\n <mml:mo>=</mml:mo>\\n <mml:mi>R</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mi>I</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">M = R/I</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, a Gorenstein algebra, this bound agrees with the bound obtained in <bold>[ES09]</bold> in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.</p>\",\"PeriodicalId\":49037,\"journal\":{\"name\":\"Journal of Commutative Algebra\",\"volume\":\"21 1\",\"pages\":\"\"},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2020-12-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Commutative Algebra\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/conm/773/15542\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Commutative Algebra","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/conm/773/15542","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

设R R是一个域上的多项式环,M= φ n M= φ n M= \bigo + _n M n是一个有限生成的梯度R R -模,最小由零次齐次元素生成,具有梯度R R -最小自由分辨率F \mathbf {F}。当梯度分辨率对称时,Cohen-Macaulay模M M为Gorenstein模。我们给出了第一希尔伯特系数e1e_1的上界,这是根据M M的梯度分辨率的偏移。当M = R/I M = R/I为Gorenstein代数时,该界与[ES09]在准纯分辨的Gorenstein代数中得到的界一致。对于较高的系数,我们推测出一个类似的界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An upper bound for the first Hilbert coefficient of Gorenstein algebras and modules

Let R R be a polynomial ring over a field and M = n M n M= \bigoplus _n M_n be a finitely generated graded R R -module, minimally generated by homogeneous elements of degree zero with a graded R R -minimal free resolution F \mathbf {F} . A Cohen-Macaulay module M M is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, e 1 e_1 in terms of the shifts in the graded resolution of M M . When M = R / I M = R/I , a Gorenstein algebra, this bound agrees with the bound obtained in [ES09] in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
0.80
自引率
16.70%
发文量
28
审稿时长
>12 weeks
期刊介绍: Journal of Commutative Algebra publishes significant results in the area of commutative algebra and closely related fields including algebraic number theory, algebraic geometry, representation theory, semigroups and monoids. The journal also publishes substantial expository/survey papers as well as conference proceedings. Any person interested in editing such a proceeding should contact one of the managing editors.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信