Galois Cohomology of Reductive Groups over Global Fields

<p> We give closed formulas for the abelian Galois cohomology groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript a b Superscript 1 Baseline left-parenthesis upper F comma upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ab</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^1_{\operatorname {ab}}(F,G)</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 H Subscript a b Superscript 2 Baseline left-parenthesis upper F comma upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ab</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^2_{\operatorname {ab}}(F,G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a connected reductive 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> over a global 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> in terms of the algebraic fundamental group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1 left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> π </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi _1(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> introduced earlier by one of us (M.B.). We further give closed formulas for the effects of restriction, corestriction, and localization, in terms of these formulas. Building on this, we give formulas, suitable for computer computations, for the first nonabelian Galois cohomology set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1 Baseline left-parenthesis upper F comma upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^1(F,G)</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> and for the second Galois cohomology group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H squared left-parenthesis upper F comma upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^2(F,T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of an <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> -torus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . </p> <p> As a preparation for the derivation of our formulas, we review the interpretation of Tate cohomology of a finite group in terms of the stable derived category of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z left-bracket normal upper Gamma right-bracket"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb Z}[\Gamma ]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -modules due to Buchweitz, and relate it to the explicit definition via cochains due to Kottwitz–Shelstad. We use this to construct the Tate–Nakayama isomorphisms for bounded complexes of tori over local and global fields