An invariant Ƴ of quasiprojective K-varieties X with values in a commutative ring Λ is motivic
if Ƴ (X)= Ƴ (Y)+ Ƴ (X\Y) for Y closed in X, and Ƴ (X× Y)= Ƴ (X) Ƴ (Y). Examples include Euler
characteristics χ and virtual Poincaré and Hodge polynomials. We first define a unique
extension Ƴ′ of Ƴ to finite type Artin K-stacks, which is motivic and satisfies Ƴ′([X/G])= Ƴ
(X)/Ƴ (G) when X is a K-variety, G a special K-group acting on X, and [X/G] is the quotient
stack. This only works if Ƴ (G) is invertible in Λ for all special K-groups G, which excludes Ƴ …