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• W3204361402 startingPage "57" @default.
• W3204361402 abstract "We define and compute the asymptotic Brauer <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimension of a field <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding=application/x-tex>F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, denoted <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A upper B r d Subscript p Baseline left-parenthesis upper F right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mi>r</mml:mi> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>ABrd_p(F)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in cases where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding=application/x-tex>F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a rational function field or Laurent series field. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A upper B r d Subscript p Baseline left-parenthesis upper F right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mi>r</mml:mi> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>ABrd_p(F)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined like the Brauer <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimension except it considers finite sets of Brauer classes instead of single classes. Our main result shows that for fields <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F 0 left-parenthesis alpha 1 comma ellipsis comma alpha Subscript n Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>α<!-- α --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>α<!-- α --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>F_0(alpha _1,dots ,alpha _n)</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 F 0 left-parenthesis left-parenthesis alpha 1 right-parenthesis right-parenthesis ellipsis left-parenthesis left-parenthesis alpha Subscript n Baseline right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mspace width=negativethinmathspace /> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>α<!-- α --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=negativethinmathspace /> <mml:mo stretchy=false>)</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mspace width=negativethinmathspace /> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>α<!-- α --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=negativethinmathspace /> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>F_0 (!( alpha _1)!) dots (!(alpha _n)!)</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 F 0> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>F_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a perfect field of characteristic <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than-or-equal-to 2> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the asymptotic Brauer <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimension is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also show that it is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n minus 1> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F equals upper F 0 left-parenthesis left-parenthesis alpha 1 right-parenthesis right-parenthesis ellipsis left-parenthesis left-parenthesis alpha Subscript n Baseline right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mspace width=negativethinmathspace /> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>α<!-- α --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=negativethinmathspace /> <mml:mo stretchy=false>)</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mspace width=negativethinmathspace /> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>α<!-- α --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=negativethinmathspace /> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>F=F_0 (!( alpha _1)!) dots (!(alpha _n)!)</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 F 0> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>F_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is algebraically closed of characteristic not <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We conclude the paper with examples of pairs of cyclic algebras of odd prime degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a field <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding=application/x-tex>F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper B r d Subscript p Baseline left-parenthesis upper F right-parenthesis equals 2> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Brd</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {Brd}_p(F)=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that share no maximal subfields despite their tensor product being non-division." @default.
• W3204361402 created "2021-10-11" @default.
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• W3204361402 date "2023-01-01" @default.
• W3204361402 modified "2023-09-26" @default.
• W3204361402 title "Asymptotic Brauer 𝑝-dimension" @default.
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• W3204361402 doi "https://doi.org/10.1090/conm/785/15775" @default.
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