Motivic classes of some classifying stacks
Daniel Bergh

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 is the multiplicative inverse of the class of the projective linear group in the Grothendieck ring of stacks for and under mild conditions on the base field . In particular, although it is known that the multiplicativity relation does not hold for all -torsors , it holds for the universal -torsors for said .
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