Doctor of Philosophy
Jardine, John F.
Let X be a simplicial object in a small Grothendieck site C, and let G be a sheaf of groups on C. We define a notion of G-torsor over X, generalizing a definition of Gillet, and prove that there is a bijection between the set of isomorphism classes of G-torsors over X, and the set of maps in the homotopy category of simplicial presheaves on C, with respect to the local weak equivalences, from X to BG. We prove basic results about the resulting non-abelian cohomology invariant, including an exact sequence associated to a central extension of sheaves of groups, as well as a characterization of the second sheaf cohomology group of BG with coefficients in a sheaf of abelian groups A, in terms of central extensions of G by A. It is well-known that, if k is a perfect field, the motivic cohomology of the classifying space BGL_n of the general linear group is a polynomial algebra over the motivic cohomology of k; we give a proof that takes advantage of this theory of torsors over simplicial schemes. Finally, using the work of Vistoli, we prove that, working over the complex numbers, the map from the Chow groups of the etale classifying space of the projective linear group PGL_p to the Chow groups of the Nisnevich classifying space of PGL_p is injective, when p is an odd prime.
Summary for Lay Audience
Algebraic groups are important objects of study in Algebraic Geometry. An important example for this thesis is the projective linear group PGL_n, which is the algebraic group that encodes the automorphisms, or symmetries, of projective space of dimension n-1. There are various ways to construct "classifying spaces" for algebraic groups; if one can calculate the cohomology of the classifying space of an algebraic group, then one has found universal characteristic classes for the algebraic structures whose automorphism groups are given by that group. This thesis presents results concerning the cohomology of the classifying spaces of algebraic groups. We begin in Chapter 2 by proving some basic results in a very general setting, where the classifying space is constructed using simplicial methods from Algebraic Topology. This simplicial construction has a long history, and the results of this part of the thesis are connected to work of Jardine, and earlier work of Giraud and Breen. The general linear group plays a central role in the theory of algebraic groups; in Chapter 3, we study the motivic cohomology of its classifying space. The results of this chapter are not new, but this presentation has not appeared in print before. And, the ideas involved illustrate ways in which the results of Chapter 2 can be applied. Finally, in Chapter 4, we prove a theorem about the relationship between the motivic cohomology of two different notions of classifying space for the projective linear group PGL_p, where p is an odd prime. These two notions of classifying space correspond to the Nisnevich and etale topologies, respectively. To the best of my knowledge, the motivic cohomology of the Nisnevich classifying space has not before been studied in the literature, except in the rare cases in which it coincides with that of the etale classifying space.
Rolle, Alexander S., "Torsors over Simplicial Schemes" (2019). Electronic Thesis and Dissertation Repository. 6312.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.