上上同调的生成

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-06-24 DOI:10.1112/blms.70132

Manoj Kummini, Mohit Upmanyu

{"title":"上上同调的生成","authors":"Manoj Kummini, Mohit Upmanyu","doi":"10.1112/blms.70132","DOIUrl":null,"url":null,"abstract":"Let <math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math> be a commutative Noetherian ring and <math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>:</mo>\n <mi>X</mi>\n <mo>⟶</mo>\n <mo>Spec</mo>\n <mi>R</mi>\n </mrow>\n <annotation>$f: X \\longrightarrow \\operatorname{Spec}R$</annotation>\n </semantics></math> a proper smooth morphism, of relative dimension <math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>. From Hartshorne, Residues and Duality, Springer, 1966, one knows that the trace map <math>\n <semantics>\n <mrow>\n <msub>\n <mo>Tr</mo>\n <mi>f</mi>\n </msub>\n <mo>:</mo>\n <msup>\n <mo>H</mo>\n <mi>n</mi>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>X</mi>\n <mo>,</mo>\n <msub>\n <mi>ω</mi>\n <mrow>\n <mi>X</mi>\n <mo>/</mo>\n <mi>R</mi>\n </mrow>\n </msub>\n <mo>)</mo>\n </mrow>\n <mo>⟶</mo>\n <mi>R</mi>\n </mrow>\n <annotation>$\\operatorname{Tr}_f: \\operatorname{H}^n(X, \\omega _{X/R}) \\longrightarrow R$</annotation>\n </semantics></math> is an isomorphism when <math>\n <semantics>\n <mi>f</mi>\n <annotation>$f$</annotation>\n </semantics></math> has geometrically connected fibres. We construct an exact sequence that generates <math>\n <semantics>\n <mrow>\n <msubsup>\n <mo>Ext</mo>\n <mi>X</mi>\n <mi>n</mi>\n </msubsup>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>O</mi>\n <mi>X</mi>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>ω</mi>\n <mrow>\n <mi>X</mi>\n <mo>/</mo>\n <mi>R</mi>\n </mrow>\n </msub>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <msup>\n <mo>H</mo>\n <mi>n</mi>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>X</mi>\n <mo>,</mo>\n <msub>\n <mi>ω</mi>\n <mrow>\n <mi>X</mi>\n <mo>/</mo>\n <mi>R</mi>\n </mrow>\n </msub>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\operatorname{Ext}_X^n(\\mathcal O_X, \\omega _{X/R}) = \\operatorname{H}^n(X, \\omega _{X/R})$</annotation>\n </semantics></math> as an <math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math>-module in the following cases:\n This partially answers a question raised by Lipman.","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 9","pages":"2865-2884"},"PeriodicalIF":0.9000,"publicationDate":"2025-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Generators of top cohomology\",\"authors\":\"Manoj Kummini, Mohit Upmanyu\",\"doi\":\"10.1112/blms.70132\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math> be a commutative Noetherian ring and <math>\\n <semantics>\\n <mrow>\\n <mi>f</mi>\\n <mo>:</mo>\\n <mi>X</mi>\\n <mo>⟶</mo>\\n <mo>Spec</mo>\\n <mi>R</mi>\\n </mrow>\\n <annotation>$f: X \\\\longrightarrow \\\\operatorname{Spec}R$</annotation>\\n </semantics></math> a proper smooth morphism, of relative dimension <math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math>. From Hartshorne, Residues and Duality, Springer, 1966, one knows that the trace map <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mo>Tr</mo>\\n <mi>f</mi>\\n </msub>\\n <mo>:</mo>\\n <msup>\\n <mo>H</mo>\\n <mi>n</mi>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>,</mo>\\n <msub>\\n <mi>ω</mi>\\n <mrow>\\n <mi>X</mi>\\n <mo>/</mo>\\n <mi>R</mi>\\n </mrow>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <mo>⟶</mo>\\n <mi>R</mi>\\n </mrow>\\n <annotation>$\\\\operatorname{Tr}_f: \\\\operatorname{H}^n(X, \\\\omega _{X/R}) \\\\longrightarrow R$</annotation>\\n </semantics></math> is an isomorphism when <math>\\n <semantics>\\n <mi>f</mi>\\n <annotation>$f$</annotation>\\n </semantics></math> has geometrically connected fibres. We construct an exact sequence that generates <math>\\n <semantics>\\n <mrow>\\n <msubsup>\\n <mo>Ext</mo>\\n <mi>X</mi>\\n <mi>n</mi>\\n </msubsup>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>O</mi>\\n <mi>X</mi>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>ω</mi>\\n <mrow>\\n <mi>X</mi>\\n <mo>/</mo>\\n <mi>R</mi>\\n </mrow>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <mo>=</mo>\\n <msup>\\n <mo>H</mo>\\n <mi>n</mi>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mi>X</mi>\\n <mo>,</mo>\\n <msub>\\n <mi>ω</mi>\\n <mrow>\\n <mi>X</mi>\\n <mo>/</mo>\\n <mi>R</mi>\\n </mrow>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\operatorname{Ext}_X^n(\\\\mathcal O_X, \\\\omega _{X/R}) = \\\\operatorname{H}^n(X, \\\\omega _{X/R})$</annotation>\\n </semantics></math> as an <math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math>-module in the following cases:\\n This partially answers a question raised by Lipman.\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"57 9\",\"pages\":\"2865-2884\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2025-06-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.70132\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.70132","RegionNum":3,"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

摘要

设R$ R$是一个交换诺瑟环，并且f: X {Spec $R} f: X \ lonightarrow \operatorname{Spec}R$是一个相对维数为n$ n$的光滑态射。从Hartshorne，残数与对偶性，b施普林格，1966，可知迹映射Tr f：H n (X, ω X / R) {R $\operatorname{Tr}_f：\operatorname{H}^n(X, \ ω _{X/R}) \ lonightarrow R$是一个同构，当f$ f$具有几何连接的纤维。我们构造一个精确的序列生成Ext X n （O X）ω X / R = H n (X，ω X/R)$ \operatorname{Ext}_X^n(\mathcal O_X, \omega _{X/R}) = \operatorname{H}^n(X, \omega _{X/R})$在以下情况下作为R$ R$ -模块：这部分地回答了利普曼提出的一个问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

Generators of top cohomology

查看原文本刊更多论文

Generators of top cohomology

Let $R$ be a commutative Noetherian ring and $f : X ⟶ Spec R$ a proper smooth morphism, of relative dimension $n$ . From Hartshorne, Residues and Duality, Springer, 1966, one knows that the trace map ${Tr}_{f} : H^{n} (X, ω_{X / R}) ⟶ R$ is an isomorphism when $f$ has geometrically connected fibres. We construct an exact sequence that generates ${Ext}_{X}^{n} (O_{X}, ω_{X / R}) = H^{n} (X, ω_{X / R})$ as an $R$ -module in the following cases: