{"title":"有效通用自由性及其在局部同调中的应用","authors":"Yairon Cid-Ruiz, Ilya Smirnov","doi":"10.1112/jlms.12995","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> be a Noetherian domain and <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math> be a finitely generated <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>-algebra. We study several features regarding the generic freeness over <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> of an <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math>-module. For an ideal <span></span><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 <span></span><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 <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> under certain settings where <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math> is a smooth <span></span><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 <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> of a finitely generated <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math>-module.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":\"<p>Let <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math> be a Noetherian domain and <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math> be a finitely generated <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math>-algebra. We study several features regarding the generic freeness over <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math> of an <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math>-module. For an ideal <span></span><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 <span></span><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 <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math> under certain settings where <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math> is a smooth <span></span><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 <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math> of a finitely generated <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math>-module.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"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}","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
摘要
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。
Effective generic freeness and applications to local cohomology
Let be a Noetherian domain and be a finitely generated -algebra. We study several features regarding the generic freeness over of an -module. For an ideal , we show that the local cohomology modules are generically free over under certain settings where is a smooth -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 of a finitely generated -module.
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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.
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