{"title":"Automorphism groups of affine varieties consisting of algebraic elements","authors":"Alexander Perepechko, Andriy Regeta","doi":"10.1090/proc/16759","DOIUrl":null,"url":null,"abstract":"<p>Given an affine algebraic variety <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\"application/x-tex\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we prove that if the neutral component <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper A normal u normal t Superscript ring Baseline left-parenthesis upper X right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"normal\">A</mml:mi> <mml:mi mathvariant=\"normal\">u</mml:mi> <mml:mi mathvariant=\"normal\">t</mml:mi> </mml:mrow> <mml:mo>∘<!-- ∘ --></mml:mo> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathrm {Aut}^\\circ (X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a closed connected nested ind-subgroup <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H subset-of upper G\"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">H\\subset G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g element-of upper G\"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">g\\in G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> some positive power of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g\"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding=\"application/x-tex\">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> belongs to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=\"application/x-tex\">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G equals upper H\"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">G=H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"138 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16759","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Given an affine algebraic variety XX, we prove that if the neutral component Aut∘(X)\mathrm {Aut}^\circ (X) of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group GG contains a closed connected nested ind-subgroup H⊂GH\subset G, and for any g∈Gg\in G some positive power of gg belongs to HH, then G=HG=H.
给定仿射代数簇 X X，我们证明如果自变群的中性分量 A u t ∘ ( X ) \mathrm {Aut}^\circ (X) 由代数元组成，那么它是嵌套的，即是代数子群的直接极限。这改进了我们之前的结果（见 Perepechko 和 Regeta [Transform. Groups 28 (2023), pp.）为了证明这一点，我们得到以下事实。如果一个连通的吲哚群 G 包含一个封闭的连通嵌套吲哚子群 H ⊂ G H\subset G ，并且对于任意 g ∈ G g\in G，g 的某个正幂次属于 H H ，那么 G = H G=H 。
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This journal is devoted to shorter research articles (not to exceed 15 printed pages) in all areas of pure and applied mathematics. To be published in the Proceedings, a paper must be correct, new, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Longer papers may be submitted to the Transactions of the American Mathematical Society. Published pages are the same size as those generated in the style files provided for AMS-LaTeX.
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