On the Mixed Tate property and the motivic class of the classifying stack of a finite group (2003.10683v1)

Abstract: Let $G$ be a finite group, and let ${B_{\mathbb{C}}G}$ the class of its classifying stack $B_{\mathbb{C}}G$ in Ekedahl's Grothendieck ring of algebraic $\mathbb{C}$-stacks $K_0(\operatorname{Stacks}{\mathbb{C}})$. We show that if $B{\mathbb{C}}G$ has the mixed Tate property, the invariants $H^i({B_{\mathbb{C}}G})$ defined by T. Ekedahl are zero for all $i\neq 0$. We also extend Ekedahl's construction of these invariants to fields of positive characteristic.