Abstract
In this article, we survey recent results of Eva Bayer-Fluckiger and the author on the Galois cohomology of classical groups over fields of virtual cohomological dimension 2. Number fields are examples of such fields. We begin by describing a well-known classification theorem for quadratic forms over number fields in terms of the so-called classical invariants (§ 2). We explain in § 3 how this classification leads to Hasse principle for principal homogeneous spaces for Spin q over number fields. In § 4 and § 7, we state the conjecture of Serre concerning the triviality of principal homogeneous spaces under semi-simple, simply connected, linear algebraic groups over perfect fields of cohomological dimension 2 and its real analogue due to Colliot-Thélène and Scheiderer in the form of a Hasse principle, if the field has virtual cohomological dimension ≤ 2. As in the case of Spin q over number fields, a main step in the proof of these conjectures is a classification theorem of hermitian forms over involutorial division algebras defined over fields of virtual cohomological dimension ≤ 2, which is described in § 6 and § 7.
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© 1999 Hindustan Book Agency (India) and Indian National Science Academy
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Parimala, R. (1999). Galois Cohomology of Classical Groups. In: Passi, I.B.S. (eds) Algebra. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-80250-94-6_7
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DOI: https://doi.org/10.1007/978-93-80250-94-6_7
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