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• W1570557922 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an algebraically closed field. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Lamda> <mml:semantics> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:annotation encoding=application/x-tex>Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a noetherian commutative ring annihilated by an integer invertible in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script l> <mml:semantics> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:annotation encoding=application/x-tex>ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime number different from the characteristic of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that if <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> is a separated algebraic space of finite type over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> endowed with an action of a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=k> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=application/x-tex>k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebraic 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>, the equivariant étale cohomology algebra <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript asterisk Baseline left-parenthesis left-bracket upper X slash upper G right-bracket comma normal upper Lamda right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mi>X</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>H^*([X/G],Lambda )</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=left-bracket upper X slash upper G right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mi>X</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[X/G]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the quotient stack 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> by <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 finitely generated over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Lamda> <mml:semantics> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:annotation encoding=application/x-tex>Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, for coefficients <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K element-of upper D Subscript c Superscript plus Baseline left-parenthesis left-bracket upper X slash upper G right-bracket comma double-struck upper F Subscript script l Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>c</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mi>X</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>F</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ℓ<!-- ℓ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>K in D^+_c([X/G],mathbb {F}_{ell })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> endowed with a commutative multiplicative structure, we establish a structure theorem for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript asterisk Baseline left-parenthesis left-bracket upper X slash upper G right-bracket comma upper K right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mi>X</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>H^*([X/G],K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, involving fixed points of elementary abelian <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script l> <mml:semantics> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:annotation encoding=application/x-tex>ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroups 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>, which is similar to Quillen’s theorem in the case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K equals double-struck upper F Subscript script l> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>F</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ℓ<!-- ℓ --></mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>K = mathbb {F}_{ell }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. One key ingredient in our proof of the structure theorem is an analysis of specialization of points of the quotient stack. We also discuss variants and generalizations for certain Artin stacks." @default.
• W1570557922 created "2016-06-24" @default.
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• W1570557922 date "2016-02-10" @default.
• W1570557922 modified "2023-10-18" @default.
• W1570557922 title "Quotient stacks and equivariant étale cohomology algebras: Quillen’s theory revisited" @default.
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• W1570557922 doi "https://doi.org/10.1090/jag/674" @default.
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