Torsors over Simplicial Schemes
Abstract
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.