Motivic classes of some classifying stacks

Abstract

We prove that the class of the classifying stack $BPG{L}_n$ is the multiplicative inverse of the class of the projective linear group $PG{L}_n$ 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 $\{T\}=\{S\}\cdot \{PG{L}_n\}$ does not hold for all $PG{L}_n$-torsors $T\to S$, it holds for the universal $PG{L}_n$-torsors for said $n$.

25 pages. Final version

Cited by in corpus (5)

• The Binomial Theorem and motivic classes of universal quasi-split tori
• Arithmetic geometry of the moduli stack of Weierstrass fibrations over ${\mathbb{P}}^1$
• On the Noether Problem for torsion subgroups of tori
• On the motivic class of an algebraic group
• On the Mixed Tate property and the motivic class of the classifying stack of a finite group