{"title":"模块不变环的局部同调","authors":"Kriti Goel, Jack Jeffries, Anurag K. Singh","doi":"10.1007/s00031-024-09851-6","DOIUrl":null,"url":null,"abstract":"<p>For <i>K</i> a field, consider a finite subgroup <i>G</i> of <span>\\({\\text {GL}}_n(K)\\)</span> with its natural action on the polynomial ring <span>\\(R:= K[x_1,\\dots ,x_n]\\)</span>. Let <span>\\(\\mathfrak {n}\\)</span> denote the homogeneous maximal ideal of the ring of invariants <span>\\(R^G\\)</span>. We study how the local cohomology module <span>\\(H^n_{\\mathfrak {n}}(R^G)\\)</span> compares with <span>\\(H^n_{\\mathfrak {n}}(R)^G\\)</span>. Various results on the <i>a</i>-invariant and on the Hilbert series of <span>\\(H^n_\\mathfrak {n}(R^G)\\)</span> are obtained as a consequence.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"36 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Local Cohomology of Modular Invariant Rings\",\"authors\":\"Kriti Goel, Jack Jeffries, Anurag K. Singh\",\"doi\":\"10.1007/s00031-024-09851-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For <i>K</i> a field, consider a finite subgroup <i>G</i> of <span>\\\\({\\\\text {GL}}_n(K)\\\\)</span> with its natural action on the polynomial ring <span>\\\\(R:= K[x_1,\\\\dots ,x_n]\\\\)</span>. Let <span>\\\\(\\\\mathfrak {n}\\\\)</span> denote the homogeneous maximal ideal of the ring of invariants <span>\\\\(R^G\\\\)</span>. We study how the local cohomology module <span>\\\\(H^n_{\\\\mathfrak {n}}(R^G)\\\\)</span> compares with <span>\\\\(H^n_{\\\\mathfrak {n}}(R)^G\\\\)</span>. Various results on the <i>a</i>-invariant and on the Hilbert series of <span>\\\\(H^n_\\\\mathfrak {n}(R^G)\\\\)</span> are obtained as a consequence.</p>\",\"PeriodicalId\":49423,\"journal\":{\"name\":\"Transformation Groups\",\"volume\":\"36 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-03-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transformation Groups\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00031-024-09851-6\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transformation Groups","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00031-024-09851-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
For K a field, consider a finite subgroup G of \({\text {GL}}_n(K)\) with its natural action on the polynomial ring \(R:= K[x_1,\dots ,x_n]\). Let \(\mathfrak {n}\) denote the homogeneous maximal ideal of the ring of invariants \(R^G\). We study how the local cohomology module \(H^n_{\mathfrak {n}}(R^G)\) compares with \(H^n_{\mathfrak {n}}(R)^G\). Various results on the a-invariant and on the Hilbert series of \(H^n_\mathfrak {n}(R^G)\) are obtained as a consequence.
期刊介绍:
Transformation Groups will only accept research articles containing new results, complete Proofs, and an abstract. Topics include: Lie groups and Lie algebras; Lie transformation groups and holomorphic transformation groups; Algebraic groups; Invariant theory; Geometry and topology of homogeneous spaces; Discrete subgroups of Lie groups; Quantum groups and enveloping algebras; Group aspects of conformal field theory; Kac-Moody groups and algebras; Lie supergroups and superalgebras.