Galois cohomology in degree three and homogeneous varieties
[See also the author's article, C. R. Acad. Sci., Paris, Sér. I 321, 891-896 (1995; Zbl 0871.14038).] Let \(G\) be a semisimple linear algebraic group over a field \(k\), \(V\) a generalized flag variety under \(G\). The main result of this paper is the existence of finite extensions \(k_i/k\), \(1\leq i\leq m\), of elements \(\alpha_i\in \text{Br}(k_i)\) and a natural exact sequence \[ 0 \to H^1(V,{\mathcal K}_2) \to \bigoplus_{i=1}^m k_i^* @>\varphi>> \text{Ker} (H^3(k, \mathbb{Q}/\mathbb{Z}(2))\to H^3(k(V), \mathbb{Q}/ \mathbb{Z}(2)) @>\psi>> CH^2(V)_{\text{tors}} \to 0. \] Here \(\varphi\) is the map \(N_{k_i/k} (.\bigcup_{\alpha_i})\) (the determination of \(\text{Ker }\varphi\) is due to Merkurjev (1996)). This generalizes a previous theorem of the author [Proc. Symp. Pure Math. 58, Part 2, 369-401 (1995; Zbl 0837.14006)] on products of Severi-Brauer varieties. Many applications are given: connection with \textit{I. A. Panin}'s result on \(K_*V\) [\(K\)-Theory 8, 541-585 (1994; Zbl 0854.19002)], examples in low-dimensional cases, examples arising from orthogonal groups, explicitation of the map \(\psi\) for special groups \(G\), construction of negligible classes in the cohomology of a central extension of \(\mathbb{F}_p\)-vector spaces \dots{}.
scientific article; zbMATH DE number 1236095