{"title":"关于正则半局部环的点状谱的维特群","authors":"Stefan Gille, Ivan Panin","doi":"10.1112/jlms.70042","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math> be a regular semilocal ring of dimension <span></span><math>\n <semantics>\n <mrow>\n <mn>4</mn>\n <mi>q</mi>\n <mo>+</mo>\n <mn>1</mn>\n <mo>⩾</mo>\n <mn>5</mn>\n </mrow>\n <annotation>$4q+1\\geqslant 5$</annotation>\n </semantics></math> which contains <span></span><math>\n <semantics>\n <mfrac>\n <mn>1</mn>\n <mn>2</mn>\n </mfrac>\n <annotation>$\\frac{1}{2}$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mrow>\n <mi>l</mi>\n <mo>⩾</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$l\\geqslant 1$</annotation>\n </semantics></math> the number of maximal ideals of <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math> which are assumed to be all of the same height, and <span></span><math>\n <semantics>\n <mi>U</mi>\n <annotation>$U$</annotation>\n </semantics></math> the punctured spectrum of <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math>, that is, <span></span><math>\n <semantics>\n <mrow>\n <mo>Spec</mo>\n <mi>R</mi>\n </mrow>\n <annotation>$\\operatorname{Spec}R$</annotation>\n </semantics></math> without the maximal ideals. We show that the Witt ring <span></span><math>\n <semantics>\n <mrow>\n <mi>W</mi>\n <mo>(</mo>\n <mi>U</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathrm{W}(U)$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mi>U</mi>\n <annotation>$U$</annotation>\n </semantics></math> has <span></span><math>\n <semantics>\n <mi>l</mi>\n <annotation>$l$</annotation>\n </semantics></math> non-trivial generators <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>E</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <mtext>…</mtext>\n <mo>,</mo>\n <msub>\n <mi>E</mi>\n <mi>l</mi>\n </msub>\n </mrow>\n <annotation>$\\mathfrak {E}_{1},\\ldots ,\\mathfrak {E}_{l}$</annotation>\n </semantics></math> as <span></span><math>\n <semantics>\n <mrow>\n <mi>W</mi>\n <mo>(</mo>\n <mi>R</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathrm{W}(R)$</annotation>\n </semantics></math>-algebra, which satisfy <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>E</mi>\n <mi>i</mi>\n </msub>\n <msub>\n <mi>E</mi>\n <mi>j</mi>\n </msub>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <annotation>$\\mathfrak {E}_{i}\\mathfrak {E}_{j}=0$</annotation>\n </semantics></math> for all <span></span><math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>⩽</mo>\n <mi>i</mi>\n <mo>,</mo>\n <mi>j</mi>\n <mo>⩽</mo>\n <mi>l</mi>\n </mrow>\n <annotation>$1\\leqslant i,j\\leqslant l$</annotation>\n </semantics></math>. If <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math> is integral, then these generators become trivial over the fraction field <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>$K$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math>. In particular, the natural morphism <span></span><math>\n <semantics>\n <mrow>\n <mi>W</mi>\n <mo>(</mo>\n <mi>U</mi>\n <mo>)</mo>\n <mo>⟶</mo>\n <mi>W</mi>\n <mo>(</mo>\n <mi>K</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathrm{W}(U)\\longrightarrow \\mathrm{W}(K)$</annotation>\n </semantics></math> is not injective.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70042","citationCount":"0","resultStr":"{\"title\":\"On the Witt group of the punctured spectrum of a regular semilocal ring\",\"authors\":\"Stefan Gille, Ivan Panin\",\"doi\":\"10.1112/jlms.70042\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math> be a regular semilocal ring of dimension <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>4</mn>\\n <mi>q</mi>\\n <mo>+</mo>\\n <mn>1</mn>\\n <mo>⩾</mo>\\n <mn>5</mn>\\n </mrow>\\n <annotation>$4q+1\\\\geqslant 5$</annotation>\\n </semantics></math> which contains <span></span><math>\\n <semantics>\\n <mfrac>\\n <mn>1</mn>\\n <mn>2</mn>\\n </mfrac>\\n <annotation>$\\\\frac{1}{2}$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>l</mi>\\n <mo>⩾</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$l\\\\geqslant 1$</annotation>\\n </semantics></math> the number of maximal ideals of <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math> which are assumed to be all of the same height, and <span></span><math>\\n <semantics>\\n <mi>U</mi>\\n <annotation>$U$</annotation>\\n </semantics></math> the punctured spectrum of <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math>, that is, <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>Spec</mo>\\n <mi>R</mi>\\n </mrow>\\n <annotation>$\\\\operatorname{Spec}R$</annotation>\\n </semantics></math> without the maximal ideals. We show that the Witt ring <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>W</mi>\\n <mo>(</mo>\\n <mi>U</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathrm{W}(U)$</annotation>\\n </semantics></math> of <span></span><math>\\n <semantics>\\n <mi>U</mi>\\n <annotation>$U$</annotation>\\n </semantics></math> has <span></span><math>\\n <semantics>\\n <mi>l</mi>\\n <annotation>$l$</annotation>\\n </semantics></math> non-trivial generators <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>E</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <mtext>…</mtext>\\n <mo>,</mo>\\n <msub>\\n <mi>E</mi>\\n <mi>l</mi>\\n </msub>\\n </mrow>\\n <annotation>$\\\\mathfrak {E}_{1},\\\\ldots ,\\\\mathfrak {E}_{l}$</annotation>\\n </semantics></math> as <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>W</mi>\\n <mo>(</mo>\\n <mi>R</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathrm{W}(R)$</annotation>\\n </semantics></math>-algebra, which satisfy <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>E</mi>\\n <mi>i</mi>\\n </msub>\\n <msub>\\n <mi>E</mi>\\n <mi>j</mi>\\n </msub>\\n <mo>=</mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$\\\\mathfrak {E}_{i}\\\\mathfrak {E}_{j}=0$</annotation>\\n </semantics></math> for all <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>⩽</mo>\\n <mi>i</mi>\\n <mo>,</mo>\\n <mi>j</mi>\\n <mo>⩽</mo>\\n <mi>l</mi>\\n </mrow>\\n <annotation>$1\\\\leqslant i,j\\\\leqslant l$</annotation>\\n </semantics></math>. If <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math> is integral, then these generators become trivial over the fraction field <span></span><math>\\n <semantics>\\n <mi>K</mi>\\n <annotation>$K$</annotation>\\n </semantics></math> of <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math>. In particular, the natural morphism <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>W</mi>\\n <mo>(</mo>\\n <mi>U</mi>\\n <mo>)</mo>\\n <mo>⟶</mo>\\n <mi>W</mi>\\n <mo>(</mo>\\n <mi>K</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathrm{W}(U)\\\\longrightarrow \\\\mathrm{W}(K)$</annotation>\\n </semantics></math> is not injective.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":\"110 6\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-12-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70042\",\"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.70042\",\"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.70042","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the Witt group of the punctured spectrum of a regular semilocal ring
Let be a regular semilocal ring of dimension which contains , the number of maximal ideals of which are assumed to be all of the same height, and the punctured spectrum of , that is, without the maximal ideals. We show that the Witt ring of has non-trivial generators as -algebra, which satisfy for all . If is integral, then these generators become trivial over the fraction field of . In particular, the natural morphism is not injective.
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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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