Central extensions and the classifying spaces of projective linear groups

TL;DR

This paper establishes a connection between sheaf cohomology and central extensions of presheaves of groupoids, and explores the motivic cohomology and Chow rings of classifying spaces of projective linear groups.

Contribution

It proves a bijection between sheaf cohomology groups and central extensions, and investigates the motivic cohomology and Chow rings of classifying spaces of PGL groups.

Findings

Sheaf cohomology $H^2(BG, A)$ corresponds to central extensions of $G$ by $A$.

Chow ring of $PGL_{p}$ injects into the motivic cohomology of $BPGL_{p}$ in characteristic zero.

Results apply to the motivic cohomology of Nisnevich classifying spaces over fields.

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 $H^2 (BG, A)$ 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 $BG$, when $G$ is a presheaf of groups on the smooth Nisnevich site over a field, and particularly when $G = PGL_{n}$. Finally, we show that, when $p$ is an odd prime, the Chow ring of the classifying space of $PGL_{p}$ injects into the motivic cohomology of the Nisnevich classifying space $BPGL_{p}$, over any field of characteristic zero containing a primitive $p^{th}$ root of unity.