Galois cohomology of the classical groups over imperfect fields
Serre's Conjecture II states that if \(G\) is a simply connected semi-simple linear algebraic group over a perfect field \(F\) of cohomological dimension at most \(2\), then the Galois cohomology set \(H^1(F,G)\) is trivial. Using Shapiro's Lemma, this problem can be reduced to the case where \(G\) is simply connected absolutely simple. This conjecture has been proved for groups of type \(^1A_n\) (Merkurjev and Suslin) and \(^2A_n\), \(B_n\), \(C_n\), \(D_n\) (except for trialitarian \(D_4\)), \(F_4\) and \(G_2\) (Bayer-Fluckiger and Parimala). Serre himself proposed a stronger version of the conjecture in the case of nonperfect fields, stating that \(H^1(F,G)\) is trivial for any simply connected absolutely simple \(G\) provided the \(p\)-separable dimension [in the sense of \textit{Ph.~Gille}, \(K\)-Theory 21, No. 1, 57--100 (2000; Zbl 0993.20031)] is at most \(2\) for all torsion primes \(p\) of the root system of \(G\). In the present paper, the authors prove this stronger conjecture for groups of type \(^2A_n\), \(B_n\), \(C_n\) and \(D_n\) (except trialitarian \(D_4\)). Similar to Bayer-Fluckiger and Parimala's approach, the idea is to reduce the question to certain classification results about quadratic and hermitian forms, involutions, and quadratic pairs which roughly say that these objects are classified by their ``classical'' invariants provided the separable dimension of the base field is at most \(2\). The crucial contribution in the present paper to these classification results is the characteristic \(2\) case. A further new aspect is the emphasis on involutions and quadratic pairs on central simple algebras rather than hermitian forms as in Bayer-Fluckiger and Parimala's work, thus avoiding the use of Morita equivalence and, as an additional bonus, making the proofs characteristic-free. In particular, the paper contains a detailed study, valid in all characteristics, of the hyperbolicity behavior of involutions and quadratic pairs under quadratic and odd-degree extensions.
scientific article; zbMATH DE number 5186628
scientific article