{"title":"On Rees algebras of linearly presented ideals and modules","authors":"","doi":"10.1007/s13348-024-00440-0","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>Let <em>I</em> be a perfect ideal of height two in <span> <span>\\(R=k[x_1, \\ldots , x_d]\\)</span> </span> and let <span> <span>\\(\\varphi \\)</span> </span> denote its Hilbert–Burch matrix. When <span> <span>\\(\\varphi \\)</span> </span> has linear entries, the algebraic structure of the Rees algebra <span> <span>\\({\\mathcal {R}}(I)\\)</span> </span> is well-understood under the additional assumption that the minimal number of generators of <em>I</em> is bounded locally up to codimension <span> <span>\\(d-1\\)</span> </span>. In the first part of this article, we determine the defining ideal of <span> <span>\\({\\mathcal {R}}(I)\\)</span> </span> under the weaker assumption that such condition holds only up to codimension <span> <span>\\(d-2\\)</span> </span>, generalizing previous work of P. H. L. Nguyen. In the second part, we use generic Bourbaki ideals to extend our findings to Rees algebras of linearly presented modules of projective dimension one. </p>","PeriodicalId":50993,"journal":{"name":"Collectanea Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Collectanea Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s13348-024-00440-0","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let I be a perfect ideal of height two in \(R=k[x_1, \ldots , x_d]\) and let \(\varphi \) denote its Hilbert–Burch matrix. When \(\varphi \) has linear entries, the algebraic structure of the Rees algebra \({\mathcal {R}}(I)\) is well-understood under the additional assumption that the minimal number of generators of I is bounded locally up to codimension \(d-1\). In the first part of this article, we determine the defining ideal of \({\mathcal {R}}(I)\) under the weaker assumption that such condition holds only up to codimension \(d-2\), generalizing previous work of P. H. L. Nguyen. In the second part, we use generic Bourbaki ideals to extend our findings to Rees algebras of linearly presented modules of projective dimension one.
期刊介绍：
Collectanea Mathematica publishes original research peer reviewed papers of high quality in all fields of pure and applied mathematics. It is an international journal of the University of Barcelona and the oldest mathematical journal in Spain. It was founded in 1948 by José M. Orts. Previously self-published by the Institut de Matemàtica (IMUB) of the Universitat de Barcelona, as of 2011 it is published by Springer.
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