{"title":"模0-环的Bloch公式与高维类场论","authors":"F. Binda, A. Krishna, S. Saito","doi":"10.1090/jag/792","DOIUrl":null,"url":null,"abstract":"<p>We prove Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q overbar Subscript script l\">\n <mml:semantics>\n <mml:msub>\n <mml:mover>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo>\n </mml:mover>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\overline {\\mathbb {Q}}_{\\ell }</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-sheaves. This was originally solved by Kerz and Saito in characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"not-equals 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo>≠<!-- ≠ --></mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\neq 2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2020-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"11","resultStr":"{\"title\":\"Bloch’s formula for 0-cycles with modulus and higher-dimensional class field theory\",\"authors\":\"F. Binda, A. Krishna, S. Saito\",\"doi\":\"10.1090/jag/792\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper Q overbar Subscript script l\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mover>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">Q</mml:mi>\\n </mml:mrow>\\n <mml:mo accent=\\\"false\\\">¯<!-- ¯ --></mml:mo>\\n </mml:mover>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\overline {\\\\mathbb {Q}}_{\\\\ell }</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-sheaves. This was originally solved by Kerz and Saito in characteristic <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"not-equals 2\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo>≠<!-- ≠ --></mml:mo>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\neq 2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\",\"PeriodicalId\":54887,\"journal\":{\"name\":\"Journal of Algebraic Geometry\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2020-02-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"11\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Algebraic Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/jag/792\",\"RegionNum\":1,\"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":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jag/792","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Bloch’s formula for 0-cycles with modulus and higher-dimensional class field theory
We prove Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse Q¯ℓ\overline {\mathbb {Q}}_{\ell }-sheaves. This was originally solved by Kerz and Saito in characteristic ≠2\neq 2.
期刊介绍：
The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology.
This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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