Motivic classes of some classifying stacks. (English) Zbl 1375.14020
Summary: We prove that the class of the classifying stack \(\mathrm{BPGL}_n\) is the multiplicative inverse of the class of the projective linear group \(\mathrm{PGL}_n\) in the Grothendieck ring of stacks \(\mathrm{K}_0(\mathrm{Stack}_k)\) for \(n = 2\) and \(n = 3\) under mild conditions on the base field \(k\). In particular, although it is known that the multiplicativity relation \(\{ T \} = \{ S \}\cdot \{ \mathrm{PGL}_n \}\) does not hold for all \(\mathrm{PGL}_n\)-torsors \(T \to S\), it holds for the universal \(\mathrm{PGL}_n\)-torsors for said \(n\).
MSC:
| 14C15 | (Equivariant) Chow groups and rings; motives |
| 14A20 | Generalizations (algebraic spaces, stacks) |
| 14C25 | Algebraic cycles |
| 14L30 | Group actions on varieties or schemes (quotients) |
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