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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

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    $\begingroup$ Wouldn't this imply that resolution of singularities or weak factorization fails over finite fields? Assuming these theorems, Larsen and Lunts give a ring homomorphism to the group ring of the monoid of abelian varieties $Alb:K(Var)\to Z[Av]$ which sends a smooth proper variety to the class of its Albanese variety. It implies that the map $Z[Av]\to K(Var)$ is injective. The positive answer to your question would imply that this homomorphism is also surjective(at least over $\bar{\mathbb{F}_q}$ where we can disregard classes of field extensions) which cannot be the case since $Alb$ has a kernel $\endgroup$
    – SashaP
    Commented Apr 9, 2018 at 16:41
  • $\begingroup$ SashaP, thanks for your comment! It convinces me that we should not expect a positive answer to my question. Would you mind turning your comment into an answer? $\endgroup$
    – jmc
    Commented Apr 9, 2018 at 17:50

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