Sur un invariant cohomologique de degré 3 pour les involutions unitaires
(2013)
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- Authors
- Supervisors
- Tignol, Jean-Pierre
- Abstract
- (en) Through a thorough study of cohomological invariants for central simple algebras with involution, this project pursues three specifics objectives. First of all, it aims to get on with relative invariants for unitary involutions as well as orthogonal involutions. Firstly, with regard to central simple algebra with unitary involution with trivial discriminant algebra, one should to complete the isomorphism criterion for degree 6 algebras and to demonstrate the decomposability criterion for degree 8 and 12 algebras. Moreover, concerning othogonal involutions, the objective is to continue studying Arason invariant (induced by Rost invariant) for algebras with orthogonal involutions and the analysis of the discriminant of symplectic involutions. This would allow to reply to a Garibaldi conjecture on orthogonal involutions on degree 16 algebras in $I^4$. Besides this in-depth analysis of relative invariants, this project intends to build a so-called absolute invariant in two cases, namely with regard to orthogonal involutions on central simple algebras of degree 8m as well as unitary involutions, and this by using trace forms. This would allow to create a exhaustive system of cohomological invariants for involutions in caracteristic different from 2. Finally, the last but not least objective of this project is to implement these cohomological invariants for algebras with involution into the caracteristic two case. They are with values in the cohomological sets modified by Kato. However, as the most part of ours invariants have a non trivial 2-torsion component, they cannot be applied without any modification in characteristic 2. The creation of such cohomological invariants set would allow, for example, to detect the R-triviality of the classical adjoint groups in characteristic 2.
- Affiliations
Citations
Masquelein, A. (2013). Sur un invariant cohomologique de degré 3 pour les involutions unitaires. https://hdl.handle.net/2078.5/206841