Effective generic freeness and applications to local cohomology

IF 1 2区数学 Q1 MATHEMATICS Journal of the London Mathematical Society-Second Series Pub Date : 2024-09-20 DOI:10.1112/jlms.12995

Yairon Cid-Ruiz, Ilya Smirnov

{"title":"Effective generic freeness and applications to local cohomology","authors":"Yairon Cid-Ruiz, Ilya Smirnov","doi":"10.1112/jlms.12995","DOIUrl":null,"url":null,"abstract":"Let <math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> be a Noetherian domain and <math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math> be a finitely generated <math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>-algebra. We study several features regarding the generic freeness over <math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> of an <math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math>-module. For an ideal <math>\n <semantics>\n <mrow>\n <mi>I</mi>\n <mo>⊂</mo>\n <mi>R</mi>\n </mrow>\n <annotation>$I \\subset R$</annotation>\n </semantics></math>, we show that the local cohomology modules <math>\n <semantics>\n <mrow>\n <msubsup>\n <mi>H</mi>\n <mi>I</mi>\n <mi>i</mi>\n </msubsup>\n <mrow>\n <mo>(</mo>\n <mi>R</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\normalfont \\text{H}_I^i(R)$</annotation>\n </semantics></math> are generically free over <math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> under certain settings where <math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math> is a smooth <math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>-algebra. By utilizing the theory of Gröbner bases over arbitrary Noetherian rings, we provide an effective method to b make explicit the generic freeness over <math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> of a finitely generated <math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math>-module.","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 4","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12995","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let $A$ be a Noetherian domain and $R$ be a finitely generated $A$ -algebra. We study several features regarding the generic freeness over $A$ of an $R$ -module. For an ideal $I \subset R$ , we show that the local cohomology modules $H_{I}^{i} (R)$ are generically free over $A$ under certain settings where $R$ is a smooth $A$ -algebra. By utilizing the theory of Gröbner bases over arbitrary Noetherian rings, we provide an effective method to b make explicit the generic freeness over $A$ of a finitely generated $R$ -module.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

有效通用自由性及其在局部同调中的应用

假设 A $A$ 是诺特域，R $R$ 是有限生成的 A $A$ -代数。我们将研究 R $R$ 模块在 A $A$ 上的泛自由性的几个特征。对于一个理想 I ⊂ R $I （子集 R$），我们证明了局部同调模块 H I i ( R ) $\normalfont \text{H}_I^i(R)$ 在 R $R$ 是光滑的 A $A$ -代数的特定情况下在 A $A$ 上是泛自由的。通过利用任意诺特环上的格氏基理论，我们提供了一种有效的方法来明确有限生成的 R $R$ 模块在 A $A$ 上的泛自由性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.