Motivic classes of some classifying stacks

TL;DR

This paper demonstrates that for certain cases, the class of the classifying stack of PGL_n is the multiplicative inverse of the class of PGL_n in the Grothendieck ring of stacks, revealing new algebraic relations.

Contribution

It establishes the inverse relation between the class of the classifying stack and PGL_n in the Grothendieck ring for n=2 and 3 under mild conditions, a novel result in algebraic geometry.

Findings

The class of B PGL_n is the inverse of PGL_n in the Grothendieck ring for n=2,3.

The multiplicativity relation holds for universal PGL_n-torsors in these cases.

Provides new insights into the structure of classifying stacks and their classes in algebraic geometry.

Abstract

We prove that the class of the classifying stack $B PGL_n$ is the multiplicative inverse of the class of the projective linear group $PGL_n$ in the Grothendieck ring of stacks for $n = 2$ and $n = 3$ under mild conditions on the base field $k$. In particular, although it is known that the multiplicativity relation $\{T\} = \{S\}\cdot\{PGL_n\}$ does not hold for all $PGL_n$-torsors $T \to S$, it holds for the universal $PGL_n$-torsors for said $n$.