On some local-global principles for linear algebraic groups over infinite algebraic extensions of global fields (Q2418965)

In this paper, the authors study to what extent known arithmetic results for (skew-)hermitian forms over global fields and cohomological Hasse principles for related algebraic groups can be extended to infinite algebraic extensions of such fields. The validity of an analogue of the strong Hasse-Minkowski principle for quadratic forms over an infinite algebraic extension \(k\) of a global field \(F\) was investigated by \textit{K. Koziol} and \textit{M. Kula} [Ann. Math. Sil. 12, 131--139 (1998; Zbl 0923.11064)], who showed that this principle holds for forms of rank at least 3. That is, a form over such a field \(k\) is isotropic if and only if it is isotropic over all localizations \(k(v)\) at places \(v\) of \(k\). In that result, the localization field \(k(v)\) is a subfield of the completion \(k_v\) of \(k\) at \(v\) defined as the direct limit of the completions in \(k_v\) of the finite extensions of \(F\) in \(k\). The main results of the present paper develop several aspects of the arithmetic theory of (skew-)hermitian forms over infinite algebraic extensions of local and global fields, and establish some local-global results for such forms, both with respect to the localization fields and with respect to the completions themselves. In particular, the authors prove the validity of the cohomological Hasse principle for \(H^1\) of semisimple simply connected algebraic groups defined over infinite algebraic extensions of global fields. As applications of this result, they derive, for example, analogues of the Hasse norm theorem and Albert-Brauer-Hasse-Noether theorem in this context. It is shown that this cohomological Hasse principle may fail for certain non-connected algebraic groups over infinite global fields; hence the connectedness of the group plays an essential role here. This paper is a sequel to a previous paper of these authors [Proc. Japan Acad. Ser. A Math. Sci. 90, No. 5, 73--78 (2014; Zbl 1300.11032)] and an expanded version of their paper [Proc. Japan Acad. Ser. A Math. Sci. 90, No. 8, 107--112 (2014; Zbl 1310.11043)].