Timeline for Do we expect abelian varieties (and “Artin motives”) to generate the Grothendieck ring of varieties over a finite field?
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| Apr 9, 2018 at 17:50 | comment | added | jmc | 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? | |
| Apr 9, 2018 at 16:41 | comment | added | SashaP | 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 | |
| Apr 9, 2018 at 9:16 | history | edited | jmc | CC BY-SA 3.0 |
Typo
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| Apr 9, 2018 at 9:05 | history | asked | jmc | CC BY-SA 3.0 |