TITLE: Etale local fibrations in the Grothendieck ring of varieties
QUESTION [7 upvotes]: Let $k$ be a field and $K_0(Var_k)$ the Grothendieck ring of varieties over $k$. This is the ring generated by isomorphism classes of varieties over $k$ with multiplication given by
$$
[X \times_k Y] = [X][Y]
$$
and the relation that $[X] = [X \setminus Z] + [Z]$ for any closed subvariety $Z \subset X$.
It is well known that if $\pi :X \to Y$ is a Zariski local fibration with fiber $F$, then $[X] = [F][Y]$. 
I assume this result is false if $\pi$ is only etale locally a fibration.  Is there a counterexample?

REPLY [10 votes]: To clarify what's happening, let us introduce the etale Grothendieck ring varieties $K^{et}(Var/k)$ by imposing the scissor congruence relation AND the relation $[X] = [F][Y]$ for every finite etale covering $X \to Y$ with (finite) fiber $F$.
Let us prove that at least when $k$ has characteristic zero, we have $K^{et}(Var/k)$ isomorphic to $\mathbb{Z}$ via the Euler characteristic with compact supports.
The proof goes in three steps. First we show by induction on dimension and using Noether Normalization that every element of $K^{et}(Var/k)$ is represented by a polynomial $P(\mathbf{L})$ in the Lefschetz class. The key observation is that any $n$-dimensional variety $X$ has an open subset $U$ which is a finite etale covering of an open subset of a projective space $U' \subset \mathbf  P^n$.
Then we show that $\mathbf{L} = 1$; this follows using the $2:1$ covering $\mathbf G_m \to \mathbf G_m$, $z \mapsto z^2$ as in Jim Bryan's comment. Thus every element in $K^{et}(Var/k)$ is represented by an integer. 
Finally, since the Euler characteristic respects etale coverings, it gives a well-defined homomorphism $K^{et}(Var/k) \to \mathbb Z$, and we just showed that this is an isomorphism!