| Summary: | Many interesting objects in algebraic geometry arise as torsors of linear algebraic groups over a field. Some notable examples are provided by vector bundles, quadratic forms, Hermitian forms, octonion algebras, Severi-Brauer varieties and many others. The main aim of this thesis is to investigate torsors from a motivic homotopic perspective, by using Nisnevich classifying spaces and their characteristic classes. In order to do so, we will need a Gysin long exact sequence induced by fibrations with motivically invertible reduced fiber. The leading example is provided by the work of Smirnov and Vishik where they introduce subtle Stiefel-Whitney classes, by computing the motivic cohomology of the Nisnevich classifying spaces of orthogonal groups, with the purpose of studying quadratic forms.
In this work, we will mainly deal with spin groups and unitary groups. In particular, we will give descriptions of the motivic cohomology rings of their Nisnevich classifying spaces. These will provide us with subtle characteristic classes for Spin-torsors and for Hermitian forms. As a result, we will obtain information about the kernel invariant of quadratic forms belonging to I^3 on the one hand, and of quadratic forms divisible by a one-fold Pfister form on the other. Moreover, in order to approach the case of Severi-Brauer varieties, we will develop a Serre spectral sequence induced by fibrations with motivically cellular fiber. This could be a successful approach to compute the motivic cohomology of the Nisnevich classifying spaces of projective general linear groups.
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