The Tate conjecture implies that the category of motives over a finite field is generated (as tensor category) by the motives of abelian varieties and Artin motives. See [1] for details.

Let $K(\textrm{Var}_{\mathbb{F}_q})$ be the Grothendieck ring of varieties over the finite field $\mathbb{F}_q$. By definition, this is the ring generated by isomorphism classes of varieties over $\mathbb{F}_q$ modulo the scissor relations: $[U] + [Z] = [X]$ for every closed embedding $Z \hookrightarrow X$ with open complement $U = X - Z$. (One may be pretty liberal with the definition of “variety” in this context: let us take it to mean a scheme of finite type over $\mathbb{F}_q$.)

> **Q:** Do we expect $K(\textrm{Var}_{\mathbb{F}_q})$ to be generated (as ring) by the isomorphism classes of abelian varieties and finite field extensions? If so, what is know in this direction?

[1]: Milne, J. “Motives over finite fields”. Available at: http://jmilne.org/math/articles/1994aP.pdf