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• W2146560763 abstract "Let <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> be 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>-local profinite <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>-Galois extension of an <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E Subscript normal infinity> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>E_infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-ring spectrum <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (in the sense of Rognes). We show that <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> may be regarded as producing a discrete <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>-spectrum. Also, we prove that if <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 a profaithful <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>-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes’s Galois correspondence extends to the profinite setting. We show that the function spectrum <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F Subscript upper A Baseline left-parenthesis left-parenthesis upper E Superscript h upper H Baseline right-parenthesis Subscript k Baseline comma left-parenthesis upper E Superscript h upper K Baseline right-parenthesis Subscript k Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>E</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>h</mml:mi> <mml:mi>H</mml:mi> </mml:mrow> </mml:msup> <mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>E</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>h</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> </mml:msup> <mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>F_A((E^{hH})_k, (E^{hK})_k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is equivalent to the localized homotopy fixed point spectrum <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis left-parenthesis upper E left-bracket left-bracket upper G slash upper H right-bracket right-bracket right-parenthesis Superscript h upper K Baseline right-parenthesis Subscript k> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy=false>[</mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mi>G</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo stretchy=false>]</mml:mo> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>h</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> </mml:msup> <mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>((E[[G/H]])^{hK})_k</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 H> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=application/x-tex>H</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 K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are closed 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>. Applications to Morava <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>-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action in terms of the derived functor of fixed points." @default.
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• W2146560763 date "2010-04-14" @default.
• W2146560763 modified "2023-10-13" @default.
• W2146560763 title "The homotopy fixed point spectra of profinite Galois extensions" @default.
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