{"title":"Non-vanishing and cofiniteness of generalized local cohomology modules","authors":"Tran Tuan Nam, Nguyen Minh Tri","doi":"10.1007/s10998-023-00567-w","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we show some results on the non-vanishing of the generalized local cohomology modules <span>\\(H^i_I(M,N)\\)</span>. In a Cohen–Macaulay local ring <span>\\((R,\\mathop {\\mathfrak {m}})\\)</span>, we prove, by using induction on <span>\\(\\dim N\\)</span>, that if <i>M</i>, <i>N</i> are two finitely generated <i>R</i>-modules with <span>\\({\\text {id}}\\,M<\\infty \\)</span> and <span>\\({\\text {Gid}}\\,N<\\infty \\)</span>, then <span>\\(H^{\\dim R-grade _R({\\text {Ann}}_RN,M)}_{\\mathop {\\mathfrak {m}}}(M,N)\\ne 0\\)</span>. We also study the <i>I</i>-cofiniteness of the generalized local cohomology module <span>\\(H^i_{\\mathop {\\mathfrak {m}}}(M,N)\\)</span>.\n</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":"6 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Periodica Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10998-023-00567-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we show some results on the non-vanishing of the generalized local cohomology modules \(H^i_I(M,N)\). In a Cohen–Macaulay local ring \((R,\mathop {\mathfrak {m}})\), we prove, by using induction on \(\dim N\), that if M, N are two finitely generated R-modules with \({\text {id}}\,M<\infty \) and \({\text {Gid}}\,N<\infty \), then \(H^{\dim R-grade _R({\text {Ann}}_RN,M)}_{\mathop {\mathfrak {m}}}(M,N)\ne 0\). We also study the I-cofiniteness of the generalized local cohomology module \(H^i_{\mathop {\mathfrak {m}}}(M,N)\).
期刊介绍:
Periodica Mathematica Hungarica is devoted to publishing research articles in all areas of pure and applied mathematics as well as theoretical computer science. To be published in the Periodica, a paper must be correct, new, and significant. Very strong submissions (upon the consent of the author) will be redirected to Acta Mathematica Hungarica.
Periodica Mathematica Hungarica is the journal of the Hungarian Mathematical Society (János Bolyai Mathematical Society). The main profile of the journal is in pure mathematics, being open to applied mathematical papers with significant mathematical content.