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Cohomological Invariants of Simple Linear Algebraic Groups Arising via the Killing Form

  • Author / Creator
    Bishop, Andrew E.
  • Let $G$ be a linear algebraic group defined over a ground field $k$, and let $\mu$ be a $\Gal(k^\sep/k)$-module.A \tb{cohomological invariant} is a morphism $a:H^1(-,G)\to H^n(-,\mu)$ of two functors from the category of field extensions over $k$ to the category of setswhere $H^1(-,G)$ is the functor of isomorphism classes of $G$-torsors and $H^n(-,\mu)$ isthe functor of abelian Galois cohomology groups with coefficients in $\mu.$The objective of this thesis is to investigate the existence of nontrivial cohomological invariantsarising via the Killing form in several settings, with the primary targetbeing split groups of type $E8.$ We note that for such groups not much is known.The only known invariant is due to M. Rost and it lives in dimension 3.To deal with the type $E8$ we first study its subgroup of type $D8$.In Chapter VI we give results regarding the existence of cohomological invariantsfor groups of type $Dn$, not necessary simply connected or adjoint. After that we passto type $E8$. Our main result establishes the existence ofa nontrivial cohomological invariant in degree 6 for the subfunctor of $H^1(-,E8)$ consisting of torsors spitting over a quadratic extension of the base field. It is worth mentioning that alltorsors in the kernel of the Rost invariant have this property,so that our result will complement the recent result of N. Semenov who constructeda cohomological invariant for the kernel of the Rost invariant for $E_8$ in degree 5.

  • Subjects / Keywords
  • Graduation date
    Spring 2019
  • Type of Item
    Thesis
  • Degree
    Master of Science
  • DOI
    https://doi.org/10.7939/r3-hpd3-jh84
  • License
    Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.