{"title":"戈伦斯坦理想的修剪的托尔代数","authors":"Luigi Ferraro, Alexis Hardesty","doi":"10.1007/s40306-023-00512-6","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\((R,\\mathfrak m,\\Bbbk )\\)</span> be a regular local ring of dimension 3. Let <i>I</i> be a Gorenstein ideal of <i>R</i> of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that <i>I</i> is generated by the sub-maximal pfaffians of this matrix. Let <i>J</i> be the ideal obtained by multiplying some of the pfaffian generators of <i>I</i> by <span>\\(\\mathfrak m\\)</span>; we say that <i>J</i> is a trimming of <i>I</i>. Building on a recent paper of Vandebogert, we construct an explicit free resolution of <i>R/J</i> and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class <span>\\(\\textbf{G}\\)</span> hold true in our context. Furthermore, we address the realizability question for ideals of class <span>\\(\\textbf{G}\\)</span>.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Tor Algebra of Trimmings of Gorenstein Ideals\",\"authors\":\"Luigi Ferraro, Alexis Hardesty\",\"doi\":\"10.1007/s40306-023-00512-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span>\\\\((R,\\\\mathfrak m,\\\\Bbbk )\\\\)</span> be a regular local ring of dimension 3. Let <i>I</i> be a Gorenstein ideal of <i>R</i> of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that <i>I</i> is generated by the sub-maximal pfaffians of this matrix. Let <i>J</i> be the ideal obtained by multiplying some of the pfaffian generators of <i>I</i> by <span>\\\\(\\\\mathfrak m\\\\)</span>; we say that <i>J</i> is a trimming of <i>I</i>. Building on a recent paper of Vandebogert, we construct an explicit free resolution of <i>R/J</i> and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class <span>\\\\(\\\\textbf{G}\\\\)</span> hold true in our context. Furthermore, we address the realizability question for ideals of class <span>\\\\(\\\\textbf{G}\\\\)</span>.</p></div>\",\"PeriodicalId\":45527,\"journal\":{\"name\":\"Acta Mathematica Vietnamica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2023-12-13\",\"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/s40306-023-00512-6\",\"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/s40306-023-00512-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
让((R,\mathfrak m,\Bbbk )\) 是维数为 3 的正则局部环。让 I 是 R 的 3 级戈伦斯坦理想。布赫斯鲍姆和艾森布德证明了有一个奇数大小的偏斜对称矩阵,使得 I 是由这个矩阵的次最大 pfaffians 生成的。让 J 成为 I 的一些 pfaffian 生成器乘以 \(\mathfrak m\) 所得到的理想;我们说 J 是 I 的修剪。在范德博格特(Vandebogert)最近一篇论文的基础上,我们构建了 R/J 的显式自由解析,并计算了这个解析上的部分 DG 代数结构。我们在附录中提供了完整的 DG 代数结构。我们利用此解析上的乘积来研究此类修剪理想的 Tor 代数,并利用所获得的信息证明克里斯滕森、维利切和韦曼最近关于类 \(\textbf{G}\) 理想的猜想在我们的上下文中成立。此外,我们还讨论了类(\textbf{G}\)理想的可实现性问题。
Let \((R,\mathfrak m,\Bbbk )\) be a regular local ring of dimension 3. Let I be a Gorenstein ideal of R of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that I is generated by the sub-maximal pfaffians of this matrix. Let J be the ideal obtained by multiplying some of the pfaffian generators of I by \(\mathfrak m\); we say that J is a trimming of I. Building on a recent paper of Vandebogert, we construct an explicit free resolution of R/J and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class \(\textbf{G}\) hold true in our context. Furthermore, we address the realizability question for ideals of class \(\textbf{G}\).
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.