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Let $K_0(Var_k)$ be the grothendieck group of the category of $k$-varieties, and call its elements virtual motives. $\mathbb{L}:=[\mathbb{A}^1_k]$ is called the Lefschetz motive. I think that if a virtual motive is divisible by arbitrarily high powers of $\mathbb{L}$, then it must be $0$. Is this true? If so, is there a proof of this?

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This is an open problem. It's probably quite hard.

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  • $\begingroup$ May I please have a reference for this? $\endgroup$ – user110215 Jul 1 '17 at 14:37
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    $\begingroup$ One reference is Daniel Litt, "Symmetric powers do not stabilize", Remark 3. I do not know who first asked this question. $\endgroup$ – Dan Petersen Jul 1 '17 at 15:00
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    $\begingroup$ This is an important question for the motivic integration theory; so you may look at surveys on this subject. $\endgroup$ – Mikhail Bondarko Jul 1 '17 at 15:44

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