Homological and motivic invariants of torsors

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Tanania, Fabio (2020) Homological and motivic invariants of torsors. PhD thesis, University of Nottingham.

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Abstract

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.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Vishik, Alexander
Zhang, Kewei
Keywords: Homological invariants, Motivic invariants, Torsors, Algebraic geometry
Subjects: Q Science > QA Mathematics > QA440 Geometry
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 59725
Depositing User: Tanania, Fabio
Date Deposited: 14 Jul 2021 15:16
Last Modified: 14 Jul 2021 15:16
URI: https://eprints.nottingham.ac.uk/id/eprint/59725

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