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

### Federico Scavia

Vol. 16 (2022), No. 10, 2265–2287
##### Abstract

Let $G$ be a finite group, and let $\left\{{B}_{ℂ}G\right\}$ the class of its classifying stack ${B}_{ℂ}G$ in Ekedahl’s Grothendieck ring of algebraic $ℂ$-stacks ${K}_{0}\left({\mathrm{Stacks}}_{ℂ}\right)$. We show that if ${B}_{ℂ}G$ has the mixed Tate property, the invariants ${H}^{i}\left(\left\{{B}_{ℂ}G\right\}\right)$ defined by Ekedahl are zero for all $i\ne 0$. We also extend Ekedahl’s construction of these invariants to fields of positive characteristic.

##### Keywords
mixed Tate, Grothendieck ring, classifying stack, algebraic group
##### Mathematical Subject Classification
Primary: 14A20, 14C15
Secondary: 14J10