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

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 $K_0({\mathrm{Stacks}}_{\mathbb{C}})$. 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 $i\ne 0$. We also extend Ekedahl's construction of these invariants to fields of positive characteristic.

References in corpus (8)

• The Grothendieck group of algebraic stacks
• The motive of a classifying space
• A geometric invariant of a finite group
• Approximating classifying spaces by smooth projective varieties
• The Ekedahl Invariants for finite groups
• Motivic classes of some classifying stacks
• On Chow-pure cohomology and Euler characteristics for motives and varieties, and their relation to unramified cohomology and Brauer groups
• On the motivic class of an algebraic group