## On the motivic class of an algebraic group.(English)Zbl 1454.14116

Summary: 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 $$\{BG\}\neq\{G\}^{-1}$$ in the Grothendieck ring of algebraic stacks over $$F$$. We also give an example of a finite étale group scheme $$A$$ over $$F$$ such that $$B\!A$$ is stably rational and $$\{B\!A\}\neq 1$$.

### MSC:

 14L15 Group schemes 14D23 Stacks and moduli problems 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry
Full Text:

### References:

 [1] 10.1016/j.aim.2006.11.003 · Zbl 1138.14014 [2] 10.1017/CBO9780511623622 [3] 10.1112/jlms/jdv059 · Zbl 1375.14020 [4] 10.1112/S0010437X03000617 · Zbl 1086.14016 [5] 10.1307/mmj/1457101817 · Zbl 1375.14021 [6] 10.1016/j.jalgebra.2007.03.048 · Zbl 1141.14028 [7] 10.1007/s00222-007-0079-5 · Zbl 1136.14035 [8] 10.1017/CBO9780511607219 [9] 10.1093/qmath/ham019 · Zbl 1131.14005 [10] ; Kunyavskiĭ, Investigations in number theory, 90 (1987) [11] 10.1016/j.jpaa.2015.08.019 · Zbl 1332.13007 [12] 10.7146/math.scand.a-25693 · Zbl 1401.14051 [13] 10.1093/imrn/rnx208 · Zbl 1444.14030 [14] ; Serre, Anneaux de Chow et applications. Séminaire Claude Chevalley, 3 (1958) [15] 10.1112/blms.12072 · Zbl 1386.14064 [16] ; Voskresenskiĭ, Algebraic groups and their birational invariants. Transl. Math. Monogr., 179 (1998) · Zbl 0974.14034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.