{"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}
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.
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.