{"title":"估价环上的格罗登第克-塞雷猜想","authors":"Ning Guo","doi":"10.1112/s0010437x23007583","DOIUrl":null,"url":null,"abstract":"<p>In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> over a valuation ring <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$V$</span></span></img></span></span> with fraction field <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$K$</span></span></img></span></span>, a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span>-torsor over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$V$</span></span></img></span></span> is trivial if it is trivial over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$K$</span></span></img></span></span>. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Grothendieck–Serre conjecture over valuation rings\",\"authors\":\"Ning Guo\",\"doi\":\"10.1112/s0010437x23007583\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline1.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$G$</span></span></img></span></span> over a valuation ring <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$V$</span></span></img></span></span> with fraction field <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$K$</span></span></img></span></span>, a <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$G$</span></span></img></span></span>-torsor over <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline5.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$V$</span></span></img></span></span> is trivial if it is trivial over <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104180628938-0323:S0010437X23007583:S0010437X23007583_inline6.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$K$</span></span></img></span></span>. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.</p>\",\"PeriodicalId\":55232,\"journal\":{\"name\":\"Compositio Mathematica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-01-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Compositio Mathematica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1112/s0010437x23007583\",\"RegionNum\":1,\"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":"Compositio Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/s0010437x23007583","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The Grothendieck–Serre conjecture over valuation rings
In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture.
期刊介绍:
Compositio Mathematica is a prestigious, well-established journal publishing first-class research papers that traditionally focus on the mainstream of pure mathematics. Compositio Mathematica has a broad scope which includes the fields of algebra, number theory, topology, algebraic and differential geometry and global analysis. Papers on other topics are welcome if they are of broad interest. All contributions are required to meet high standards of quality and originality. The Journal has an international editorial board reflected in the journal content.