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• W2963844460 abstract "Let <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> be a smooth scheme, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Z> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding=application/x-tex>Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a smooth closed subscheme, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper U> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=application/x-tex>U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the open complement. Given any localizing and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper A Superscript 1> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-homotopy invariant of dg categories <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding=application/x-tex>E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we construct an associated Gysin triangle relating the value of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding=application/x-tex>E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at the dg categories of perfect complexes of <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>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Z> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding=application/x-tex>Z</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> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=application/x-tex>U</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the particular case where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding=application/x-tex>E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is homotopy <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory, this Gysin triangle yields a new proof of Quillen’s localization theorem, which avoids the use of devissage. As a first application, we prove that the value of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding=application/x-tex>E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at a smooth scheme belongs to the smallest (thick) triangulated subcategory generated by the values of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding=application/x-tex>E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at the smooth projective schemes. As a second application, we compute the additive invariants of relative cellular spaces in terms of the bases of the corresponding cells. Finally, as a third application, we construct explicit bridges relating motivic homotopy theory and mixed motives on the one side with noncommutative mixed motives on the other side. This leads to a comparison between different motivic Gysin triangles as well as to an étale descent result concerning noncommutative mixed motives with rational coefficients." @default.
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• W2963844460 date "2017-08-15" @default.
• W2963844460 modified "2023-09-27" @default.
• W2963844460 title "The Gysin triangle via localization and 𝐴¹-homotopy invariance" @default.
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