Blowup algebras of determinantal ideals in prime characteristic

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-07-23 DOI:10.1112/jlms.12969

Alessandro De Stefani, Jonathan Montaño, Luis Núñez-Betancourt

{"title":"Blowup algebras of determinantal ideals in prime characteristic","authors":"Alessandro De Stefani, Jonathan Montaño, Luis Núñez-Betancourt","doi":"10.1112/jlms.12969","DOIUrl":null,"url":null,"abstract":"We study when blowup algebras are <math>\n <semantics>\n <mi>F</mi>\n <annotation>$F$</annotation>\n </semantics></math>-split or strongly <math>\n <semantics>\n <mi>F</mi>\n <annotation>$F$</annotation>\n </semantics></math>-regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew-symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results, we develop the notion of <math>\n <semantics>\n <mi>F</mi>\n <annotation>$F$</annotation>\n </semantics></math>-split filtrations and symbolic <math>\n <semantics>\n <mi>F</mi>\n <annotation>$F$</annotation>\n </semantics></math>-split ideals.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12969","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 0

Abstract

We study when blowup algebras are $F$ -split or strongly $F$ -regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew-symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results, we develop the notion of $F$ -split filtrations and symbolic $F$ -split ideals.

查看原文本刊更多论文

素特征中行列式理想的吹胀代数

我们研究的是吹胀代数是 F $F$ 分裂的还是强 F $F$ 不规则的。我们的主要研究重点是一般矩阵、对称矩阵和汉克尔矩阵的小数理想的符号幂和普通幂所给出的数组。我们还研究了倾斜对称矩阵的普法因子的理想。我们利用这些结果获得了这些代数方程定义方程的度数边界。我们还证明了这些理想的符号幂的归一化正则极限是存在的，而且它们的深度是稳定的。最后，我们证明，对于行列式理想，存在一个取初始理想与取符号幂相乘的单项式阶。为了获得这些结果，我们提出了 F $F$ 分裂过滤和符号 F $F$ 分裂理想的概念。

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分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.