Central extensions and the classifying spaces of projective linear groups

Alexander Rolle

paper

Central extensions and the classifying spaces of projective linear groups

arXiv:1810.07850

Abstract

If $G$ is a presheaf of groupoids on a small site, and $A$ is a sheaf of abelian groups, we prove that the sheaf cohomology group $H^{2} (B G, A)$ is in bijection with a set of central extensions of $G$ by $A$ . We use this result to study the motivic cohomology of the Nisnevich classifying space $B G$ , when $G$ is a presheaf of groups on the smooth Nisnevich site over a field, and particularly when $G = P G L_{n}$ . Finally, we show that, when $p$ is an odd prime, the Chow ring of the classifying space of $P G L_{p}$ injects into the motivic cohomology of the Nisnevich classifying space $B P G L_{p}$ , over any field of characteristic zero containing a primitive $p^{t h}$ root of unity.

15 pages. The final result is stated in greater generality than in version 1. References have also been added, and the exposition improved