TITLE: Should the Grothendieck ring of varieties be K_0 of numerical motives?
QUESTION [6 upvotes]: Assuming the Standard Conjectures, should the Grothendieck ring of varieties be the $K_0$ of the abelian category of numerical motives?

REPLY [8 votes]: 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.