On Galois cohomology of connected reductive groups and Kottwitz exact sequence
Let \(k\) be a global field and let \(G\) be a connected reductive algebraic group over \(k\). Write \(G'\) for the derived group of \(G\) and \(\hat{G}\) for the Langlands dual of \(G\). Let \(V\) denote the set of places of \(k\), let \(\mathbf{A}\) denote the \(k\)-adeles and put \(\mathbf{A}_s=\mathbf{A}\otimes_k k_s\), where \(k_s\) is a separable closure of \(k\). Define \(\mathscr{A}(G)= (\pi_0(Z(\hat{G})^{\operatorname{Gal}(k_s/k)}))^D\cong \mathrm{Pic}(G)^D\), where \((?)^D=\operatorname{Hom}_{\mathbb{Z}}(?,\mathbb{Q}/\mathbb{Z})\). The author generalizes several results of Kottwitz [\textit{R. E. Kottwitz}, Duke Math. J. 51, 611--650 (1984; Zbl 0576.22020); Math. Ann. 275, 365--399 (1986; Zbl 0577.10028)] concerning the Galois cohomology set \(\mathrm{H}^1(k,G)\) from the case of number fields to all global fields. The main result of the paper (Theorem~4.7) states that there exists an exact sequence \[ 1\to \operatorname{Sha}^1(G)\to \mathrm{H}^1(k,G )\to \bigoplus_{v\in V}\mathrm{H}^1(k_v,G) \to \mathscr{A}(G)\to \operatorname{Sha}^2(G/G')\to 1 \] which is functorial in \(G\); here, \(\mathrm{Sha}^i\) denotes the \(i\)-th Shafarevich-Tate group, or pointed set. In characteristic \(0\), the existence and exactness of the sequence formed by the left four terms were shown by Kottwitz [loc. cit., Prop. 2.6]. Further related results are proved along the way. Note first that for a \(k\)-torus \(T\), it is known that there is a natural isomorphism \(\mathrm{H}^1(k,T(\mathbf{A}_s)/T(k_s))\to \mathscr{A}(T)\) (Section~1.5). The author shows that this isomorphisms extends uniquely to a natural transformation \(\mathrm{H}^1(k,G(\mathbf{A}_s)/Z_G(k_s))\to \mathscr{A}(G)\), where \(Z_G\) is the center of \(G\). Moreover, the latter isomorphism fits into an exact sequence \( \mathrm{H}^1(k,G^{\mathrm{ad}})\to \mathrm{H}^1(k,G(\mathbf{A}_s)/Z(k_s)) \to \mathscr{A}(G)\), which is functorial in \(G\) (Theorem~3.14). It is further shown that \(\operatorname{Sha}^1(G)\cong \operatorname{Sha}^1(Z(\hat{G}))^D\) (Theorem~2.6). In characteristic \(0\), these results appear in [op.\ cit.]. The author applies the main result to give a new proof for a result of \textit{J. J. Sansuc} [J. Reine Angew. Math. 327, 12--80 (1981; Zbl 0468.14007), Cor.\ 8.14]. In an appendix joint with N.T.~Ngoan, they apply the main result to prove the Hasse principle for projective homogeneous spaces under \(G\).
scientific article; zbMATH DE number 7030204
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