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• W3003293136 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected noetherian commutative ring, and let <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> be a simply connected reductive group over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of isotropic rank <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=greater-than-or-equal-to 2> <mml:semantics> <mml:mrow> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>ge 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The elementary subgroup <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E left-parenthesis upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G left-parenthesis upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>G(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the subgroup generated by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper U Subscript upper P Sub Superscript plus Baseline left-parenthesis upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>P</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>U_{P^+}(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper U Subscript upper P Sub Superscript minus Baseline left-parenthesis upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>P</mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:msup> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>U_{P^-}(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper U Subscript upper P Sub Superscript plus-or-minus> <mml:semantics> <mml:msub> <mml:mi>U</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>P</mml:mi> <mml:mo>±<!-- ± --></mml:mo> </mml:msup> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>U_{P^pm }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are the unipotent radicals of two opposite parabolic subgroups <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P Superscript plus-or-minus> <mml:semantics> <mml:msup> <mml:mi>P</mml:mi> <mml:mo>±<!-- ± --></mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>P^pm</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <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>. Assume that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 element-of upper R Superscript times> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>×<!-- × --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>2in R^times</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <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> is of type <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper B Subscript n Baseline comma upper C Subscript n Baseline comma upper F 4 comma upper G 2> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>B_n,C_n,F_4,G_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3 element-of upper R Superscript times> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>×<!-- × --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>3in R^times</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <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> is of type <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G 2> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>G_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that the congruence kernel of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E left-parenthesis upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, defined as the kernel of the natural homomorphism <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=ModifyingAbove upper E left-parenthesis upper R right-parenthesis With caret right-arrow ModifyingAbove upper E left-parenthesis upper R right-parenthesis With bar> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mover> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo accent=false>¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding=application/x-tex>widehat {E(R)}to overline {E(R)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> between the profinite completion of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E left-parenthesis upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the congruence completion of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E left-parenthesis upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to congruence subgroups of finite index, is central in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=ModifyingAbove upper E left-parenthesis upper R right-parenthesis With caret> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>^<!-- ^ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding=application/x-tex>widehat {E(R)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the course of the proof, we construct Steinberg groups associated to isotropic reducive groups and show that they are central extensions of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E left-parenthesis upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a local ring." @default.
• W3003293136 created "2020-02-07" @default.
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• W3003293136 date "2020-03-31" @default.
• W3003293136 modified "2023-09-26" @default.
• W3003293136 title "On the congruence kernel of isotropic groups over rings" @default.
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