On the motivic class of an algebraic group

TL;DR

This paper constructs examples of algebraic groups and their classifying stacks over certain fields that are stably rational but have nontrivial classes in the Grothendieck ring, challenging previous assumptions.

Contribution

It provides explicit examples of algebraic groups with stably rational classifying stacks whose classes differ from 1 in the Grothendieck ring, revealing new phenomena in motivic classes.

Findings

Existence of a torus with stably rational classifying stack but nontrivial Grothendieck class.

Existence of a finite étale group scheme with similar properties.

Counterexamples to expected relations between rationality and motivic classes.

Abstract

Let $F$ be a field of characteristic zero admitting a biquadratic field extension. We give an example of a torus $G$ over $F$ whose classifying stack $BG$ is stably rational and such that $\{BG\}\{G\}\neq 1$ in the Grothendieck ring of algebraic stacks over $F$. We also give an example of a finite \'etale group scheme $A$ over $F$ such that $BA$ is stably rational and $\{BA\}\neq 1$.