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In "The universal Euler characteristic for varieties of characteristic zero", Bittner shows that over a field $k$ of characteristic zero, the Grothendieck ring $K_{0}(Var_{k})$ of varieties is generated by symbols

$[X] \in K_{0}(Var_{k})$,

where $X$ is smooth and proper, subject only to the relations that if $Y \subseteq X$ is a closed immersion of smooth proper varieties and $E \subseteq Bl_{Y}(X)$ is the exceptional divisor of the associated blow-up, then

$[X] - [Y] = [Bl_{Y}(X)] - [E]$.

Question: Is Bittner's result known to hold after inverting $p$ if $k$ is of positive characteristic $p$? That is, do we have an analogous description of $K_{0}(Var_{k}) \otimes_{\mathbb{Z}} \mathbb{Z}[1/p]$?

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    $\begingroup$ Of course if resolution of singularities and weak factorisation hold, then the proof can just be carried over to positive characteristic. But even tensoring with $\mathbf Q$ does not seem to help, as far as I'm aware. Unlike hypercoverings in étale cohomology, you cannot break down an element of $K_0(\mathbf{Var}_k)\otimes\mathbf Q$ into smooth projective pieces using only alterations. (That is only to show that $K_0(\mathbf{Var}_k)$ is generated by smooth projective things; getting the Bittner relations is even harder as it needs weak factorisation instead of resolution of singularities.) $\endgroup$ Apr 23, 2023 at 21:23
  • $\begingroup$ @R.vanDobbendeBruyn Thank you! I was wondering if alterations are sufficient, hence the question. $\endgroup$ Apr 23, 2023 at 22:23
  • $\begingroup$ Possibly, one can use alterations to compute a certain quotient of $K_0(Var_k)[1/p]$ in this case. $\endgroup$ Apr 24, 2023 at 6:56

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