离散Hodge代数上的各向同性约化群

IF 0.5 4区 数学
Anastasia Stavrova
{"title":"离散Hodge代数上的各向同性约化群","authors":"Anastasia Stavrova","doi":"10.1007/s40062-018-0221-7","DOIUrl":null,"url":null,"abstract":"<p>Let <i>G</i> be a reductive group over a commutative ring <i>R</i>. We say that <i>G</i> has isotropic rank <span>\\(\\ge n\\)</span>, if every normal semisimple reductive <i>R</i>-subgroup of <i>G</i> contains <span>\\(({{\\mathrm{{{\\mathbf {G}}}_m}}}_{,R})^n\\)</span>. We prove that if <i>G</i> has isotropic rank <span>\\(\\ge 1\\)</span> and <i>R</i> is a regular domain containing an infinite field <i>k</i>, then for any discrete Hodge algebra <span>\\(A=R[x_1,\\ldots ,x_n]/I\\)</span> over <i>R</i>, the map <span>\\(H^1_{\\mathrm {Nis}}(A,G)\\rightarrow H^1_{\\mathrm {Nis}}(R,G)\\)</span> induced by evaluation at <span>\\(x_1=\\cdots =x_n=0\\)</span>, is a bijection. If <i>k</i> has characteristic 0, then, moreover, the map <span>\\(H^1_{\\acute{\\mathrm{e}}\\mathrm {t}}(A,G)\\rightarrow H^1_{\\acute{\\mathrm{e}}\\mathrm {t}}(R,G)\\)</span> has trivial kernel. We also prove that if <i>k</i> is perfect, <i>G</i> is defined over <i>k</i>, the isotropic rank of <i>G</i> is <span>\\(\\ge 2\\)</span>, and <i>A</i> is square-free, then <span>\\(K_1^G(A)=K_1^G(R)\\)</span>, where <span>\\(K_1^G(R)=G(R)/E(R)\\)</span> is the corresponding non-stable <span>\\(K_1\\)</span>-functor, also called the Whitehead group of <i>G</i>. The corresponding statements for <span>\\(G={{\\mathrm{GL}}}_n\\)</span> were previously proved by Ton Vorst.</p>","PeriodicalId":636,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 2","pages":"509 - 524"},"PeriodicalIF":0.5000,"publicationDate":"2018-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0221-7","citationCount":"5","resultStr":"{\"title\":\"Isotropic reductive groups over discrete Hodge algebras\",\"authors\":\"Anastasia Stavrova\",\"doi\":\"10.1007/s40062-018-0221-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <i>G</i> be a reductive group over a commutative ring <i>R</i>. We say that <i>G</i> has isotropic rank <span>\\\\(\\\\ge n\\\\)</span>, if every normal semisimple reductive <i>R</i>-subgroup of <i>G</i> contains <span>\\\\(({{\\\\mathrm{{{\\\\mathbf {G}}}_m}}}_{,R})^n\\\\)</span>. We prove that if <i>G</i> has isotropic rank <span>\\\\(\\\\ge 1\\\\)</span> and <i>R</i> is a regular domain containing an infinite field <i>k</i>, then for any discrete Hodge algebra <span>\\\\(A=R[x_1,\\\\ldots ,x_n]/I\\\\)</span> over <i>R</i>, the map <span>\\\\(H^1_{\\\\mathrm {Nis}}(A,G)\\\\rightarrow H^1_{\\\\mathrm {Nis}}(R,G)\\\\)</span> induced by evaluation at <span>\\\\(x_1=\\\\cdots =x_n=0\\\\)</span>, is a bijection. If <i>k</i> has characteristic 0, then, moreover, the map <span>\\\\(H^1_{\\\\acute{\\\\mathrm{e}}\\\\mathrm {t}}(A,G)\\\\rightarrow H^1_{\\\\acute{\\\\mathrm{e}}\\\\mathrm {t}}(R,G)\\\\)</span> has trivial kernel. We also prove that if <i>k</i> is perfect, <i>G</i> is defined over <i>k</i>, the isotropic rank of <i>G</i> is <span>\\\\(\\\\ge 2\\\\)</span>, and <i>A</i> is square-free, then <span>\\\\(K_1^G(A)=K_1^G(R)\\\\)</span>, where <span>\\\\(K_1^G(R)=G(R)/E(R)\\\\)</span> is the corresponding non-stable <span>\\\\(K_1\\\\)</span>-functor, also called the Whitehead group of <i>G</i>. The corresponding statements for <span>\\\\(G={{\\\\mathrm{GL}}}_n\\\\)</span> were previously proved by Ton Vorst.</p>\",\"PeriodicalId\":636,\"journal\":{\"name\":\"Journal of Homotopy and Related Structures\",\"volume\":\"14 2\",\"pages\":\"509 - 524\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2018-11-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s40062-018-0221-7\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Homotopy and Related Structures\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40062-018-0221-7\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Homotopy and Related Structures","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40062-018-0221-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 5

摘要

设G是交换环r上的约化群,如果G的每一个正规半单约化r子群都含有\(({{\mathrm{{{\mathbf {G}}}_m}}}_{,R})^n\),则G具有各向同性秩\(\ge n\)。证明了如果G具有各向同性秩\(\ge 1\)且R是包含无限域k的正则域,则对于R上的任意离散Hodge代数\(A=R[x_1,\ldots ,x_n]/I\),在\(x_1=\cdots =x_n=0\)处求值所得到的映射\(H^1_{\mathrm {Nis}}(A,G)\rightarrow H^1_{\mathrm {Nis}}(R,G)\)是双射。如果k具有特征0,则映射\(H^1_{\acute{\mathrm{e}}\mathrm {t}}(A,G)\rightarrow H^1_{\acute{\mathrm{e}}\mathrm {t}}(R,G)\)具有平凡核。我们还证明了如果k是完美的,G在k上有定义,G的各向同性秩为\(\ge 2\), A是无平方的,则\(K_1^G(A)=K_1^G(R)\),其中\(K_1^G(R)=G(R)/E(R)\)是对应的不稳定的\(K_1\) -函子,也称为G的Whitehead群。\(G={{\mathrm{GL}}}_n\)的相应表述先前由Ton Vorst证明。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Isotropic reductive groups over discrete Hodge algebras

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank \(\ge n\), if every normal semisimple reductive R-subgroup of G contains \(({{\mathrm{{{\mathbf {G}}}_m}}}_{,R})^n\). We prove that if G has isotropic rank \(\ge 1\) and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra \(A=R[x_1,\ldots ,x_n]/I\) over R, the map \(H^1_{\mathrm {Nis}}(A,G)\rightarrow H^1_{\mathrm {Nis}}(R,G)\) induced by evaluation at \(x_1=\cdots =x_n=0\), is a bijection. If k has characteristic 0, then, moreover, the map \(H^1_{\acute{\mathrm{e}}\mathrm {t}}(A,G)\rightarrow H^1_{\acute{\mathrm{e}}\mathrm {t}}(R,G)\) has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is \(\ge 2\), and A is square-free, then \(K_1^G(A)=K_1^G(R)\), where \(K_1^G(R)=G(R)/E(R)\) is the corresponding non-stable \(K_1\)-functor, also called the Whitehead group of G. The corresponding statements for \(G={{\mathrm{GL}}}_n\) were previously proved by Ton Vorst.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Journal of Homotopy and Related Structures
Journal of Homotopy and Related Structures Mathematics-Geometry and Topology
自引率
0.00%
发文量
0
期刊介绍: Journal of Homotopy and Related Structures (JHRS) is a fully refereed international journal dealing with homotopy and related structures of mathematical and physical sciences. Journal of Homotopy and Related Structures is intended to publish papers on Homotopy in the broad sense and its related areas like Homological and homotopical algebra, K-theory, topology of manifolds, geometric and categorical structures, homology theories, topological groups and algebras, stable homotopy theory, group actions, algebraic varieties, category theory, cobordism theory, controlled topology, noncommutative geometry, motivic cohomology, differential topology, algebraic geometry.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信