Cohomological invariants of central simple algebras with involution Jean-Pierre Tignol To Parimala, with gratitude and admiration This text is based on two lectures delivered in January 2009 at the conference “Quadratic forms, linear algebraic groups and Galois cohomology” held at the Uni- versity of Hyderabad. The purpose was to survey the cohomological invariants that have been defined for various types of involutions on central simple algebras on the model of quadratic form invariants. I seized the occasion to make explicit some of the classification or structure results that may be expected from future invariants, and to compile a fairly extensive list of references. As this list makes clear, Pari- mala’s contributions to the subject are all-pervasive. 1 Introduction: classification of quadratic forms . 58 2 From quadratic forms to involutions . 60 3 Orthogonal involutions . 62 3.1 The split case . 63 3.2 The case of index 2 . 64 3.3 Discriminant . 65 3.4 Clifford algebras . 67 3.5 Higher invariants . 70 4 Unitary involutions . 73 4.1 The (quasi-) split case . 74 4.2 The discriminant algebra . 77 4.3 Higher invariants . 79 5 Symplectic involutions . 81 5.1 The case of index 2 . 81 5.2 Invariant of degree 2 . 83 5.3 The discriminant . 84 References . 89 Throughout these notes, F denotes a field of characteristic different from 2 and Fs a separable closure of F. We identify µ2 := 1 with Z/2Z. For any integer {± } Jean-Pierre Tignol Departement´ de mathematique,´ Universite´ catholique de Louvain, Chemin du cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium; e-mail:
[email protected] 57 58 Jean-Pierre Tignol n 0, we let Hn(F) be the Galois cohomology group ≥ n n H (F) := H (Gal(Fs/F),Z/2Z).