{"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":null,"pages":null},"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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