估价环上的格罗登第克-塞雷猜想

IF 1.3 1区 数学 Q1 MATHEMATICS Compositio Mathematica Pub Date : 2024-01-05 DOI:10.1112/s0010437x23007583
Ning Guo
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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. 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引用次数: 0

摘要

在这篇文章中,我们建立了估价环上的格罗thendieck-Serre 猜想:对于估价环 $V$ 上带分数域 $K$ 的还原群方案 $G$,如果在 $K$ 上是微不足道的,那么在 $V$ 上的 $G$-torsor 就是微不足道的。最初的格罗内迪克-塞雷猜想和奇点解析预示了这一结果。我们证明的新颖之处在于克服了一般非离散估值环带来的微妙之处。通过使用 flasque 解析和局部同调归纳,我们证明了 Colliot-Thélène-Sansuc 关于环的非诺特对应情况。然后,利用代数化技术,我们获得了亨塞尔秩一情况的通道。最后,我们归纳了 Levi 子群,并利用各向异性群有理点的积分性还原到半简单各向异性的情况,在此我们求助于布鲁哈特-蒂茨理论中的准子群的性质来得出结论。在最后一节中,我们利用形式幂级数在估值环上的反身剪的扩展性质和簇的修补,证明了尼斯涅维奇纯度猜想的一个变体。
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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.

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来源期刊
Compositio Mathematica
Compositio Mathematica 数学-数学
CiteScore
2.10
自引率
0.00%
发文量
62
审稿时长
6-12 weeks
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.
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