Galois cohomology of the classical groups over fields of cohomological dimension \(\leq 2\)
This carefully written paper concerns the following well-known conjecture of \textit{J.-P. Serre} [Cohomologie galoisienne (Springer 1964; Zbl 0128.26303) and (1994; Zbl 0812.12002), see also Sémin. Bourbaki, Vol. 1993/94, Exp. 783 (1995; Zbl 0837.12003)]: Conjecture II: Let \(k\) be a perfect field with \(\text{cd} (\Gamma_k) \leq 2\). Let \(G\) be a semisimple, simply connected linear algebraic group defined over \(k\). Then \(H^1 (k,G) = 0\). The main theorem is the following: ``The above conjecture holds for the classical groups of types \(A_n, B_n, C_n\) or \(D_n\)''. By Serre's recent work the same is true for exceptional groups of type \(G_2\) or \(F_4\) (the proof is repeated in \S 6 of the present paper). The cases \(E_6, E_7, E_8\) and trialitarian \(D_4\) are still open. By a classical result of Weil the four types of (almost simple) groups \(G\) correspond to \(SL_n(D)\), \(SU(h)\), \(U(h)\) and Spin \((h)\) where \(D\) is a division algebra over \(k\) and \(h\) is a hermitian form over \(D\) of unitary, symplectic or orthogonal type. In the first case the theorem is due to Merkurjev and Suslin (1985), and the other cases are settled in the present paper. Naturally a large part of the paper deals with hermitian forms \(h\) and conditions on \(h\) to be hyperbolic. A decisive role is played by the key exact sequence of \S 3 between the Witt groups of the various types of hermitian forms over central simple algebras with involution. The case \(\text{char} k = 2\), which is slightly exceptional, is treated in Appendix 1. But as for \(\text{char} k \neq 2\) the orthogonal case (type \(D_n)\) is the most difficult.
hermitian forms over central simple algebras
scientific article; zbMATH DE number 861962
scientific article