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Assuming the Standard Conjectures, should the Grothendieck ring of varieties be the $K_0$ of the abelian category of numerical motives?

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No. If $E$ and $E'$ are distinct isogenous elliptic curves, say over a field of characteristic zero, then $[E]-[E'] \neq 0$ in the Grothendieck ring of varieties (as explained e.g. in Bjorn Poonen's paper on zero divisors in the Grothendieck ring of varieties), but $E$ and $E'$ define isomorphic motives.

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  • $\begingroup$ Only with rational coefficients are the motives of $E$ and $E'$ isomorphic. $\endgroup$ – ACL Mar 16 '15 at 15:42
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As explained by Dan Petersen, the answer is no. There are additional issues though. For example, multiplication by the class $\mathbf L$ of the affine line $\mathbf A^1_k$ corresponds to Tate twist on the motivic level, hence is “motivically regular”. However, it has been proved recently by Lev Borisov that $\mathbf L$ is a zero-divisor in the Grothendieck group of varieties.

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