On the motivic class of an algebraic group

Abstract

We give an example of a torus G over a finitely generated field extension F of Q whose classifying stack BG is stably rational and such that {BG} = {G} −1 in the Grothendieck ring of algebraic stacks over F . We also give an example of a finite étale group scheme A such that BA is stably rational and {BA} = 1.

On the Mixed Tate property and the motivic class of the classifying stack of a finite group

“…E1×E2/k (G m ) be the k-torus of rank 3 used in the proof of [29,Theorem 1.5]. As in the first part of the proof of [29, Theorem 1.5] (which works as long as char k = 2), we have that B k G is stably rational, and moreover…”

“…As an application of Proposition 5.1, we generalize the results of [29] to the case where the characteristic of the base field is different from 2.…”

“…Here G acts on Γ via the natural surjection G → Γ. It follows from the Krull-Schmidt Theorem that K 0 (Rep G,F2 ) is freely generated on the set of isomorphism classes of indecomposable G-representations; see [29,Lemma 4.1]. This is in contradiction with (5.3), hence…”

On the Noether Problem for torsion subgroups of tori

“…If G is a connected group and k is algebraically closed, no counterexample to {BG}{G} = 1 is known; this is again in line with the Noether Problem, for which no negative answer is known among connected groups. In [22,Theorem 1.5], we exhibited the first connected counterexample T to the expected class formula, in the case when k is finitely generated over Q. More precisely, T := R Since {BT }{T } = 1, the previous proposition gives a more conceptual proof of [22,Theorem 1.6].…”

“…In [22,Theorem 1.5], we exhibited the first connected counterexample T to the expected class formula, in the case when k is finitely generated over Q. More precisely, T := R Since {BT }{T } = 1, the previous proposition gives a more conceptual proof of [22,Theorem 1.6].…”