A neat construction of Bjorn Poonen shows that the Grothendieck ring of varieties (over a field of char. 0) is not a domain: <http://arxiv.org/abs/math/0204306> 


Is the Grothendieck ring of varieties reduced? (My guess: the answer is yes, the proof is easy enough that several people have observed this without writing it up anywhere. But I don't know how to show it.) 



Is the Grothendieck ring of varieties reduced?
Qing Liu's example probably works, only we don't know if an abelian variety in
positive characteric is determined by its class in
$K\_0(\mathrm{Var}\_k)$. However, we do know that in characteristic zero (this is
what Bjorn Poonen uses in his examples) and the non-cancellation is a purely
arithmetic phenomenon and hence can be realised in characteristic zero.


Hence, we let $\mathcal A$ be a maximal order in a definite (i.e., $\mathcal
A\otimes\mathbb R$ is non-split) quaternion algebra over $\mathbb Q$. There is
an abelian variety $A$ over some field $k$ of characteristic zero with $\mathcal
A=\mathrm{End}(A)$ (Bjorn works hard to get his example defined over $\mathbb
Q$, here I make no such claim). For any (right) f.g. projective (i.e., torsion
free) $\mathcal A$-module $M$ we may define an abelian variety
$M\bigotimes\_{\mathcal A}A$ characterised by $\mathrm{Hom}(M\bigotimes\_{\mathcal
 A}A,B)=\mathrm{Hom}\_{\mathcal A}(M,\mathrm{Hom}(A,B))$ for all abelian
varieties (concretely it is constructed by realising $M$ is the kernel of an
idempotent of some $\mathcal A^n$ and then taking the kernel of the same
idempotent acting on $A^n$). In any case we see that $M$ and $N$ are isomorphic
precisely when $M\bigotimes\_{\mathcal A}A$ is isomorphic to
$N\bigotimes\_{\mathcal A}A$. 


Now (all the arithmetic results used below can be found in for instance Irving
Reiner: Maximal orders, Academic Press, London-New York), the class group of
$\mathcal A$ is equal to the ray class group of $\mathbb Q$ with respect to the
infinite prime, i.e., the group of fractional ideals of $\mathbb Q$ modulo
ideals with a strictly positive generators. As that is all ideals we find that
the class group is trivial. Furthermore, we have the Eichler stability theorem
which says that projective modules of rank $\geq2$ are determined by their rank and
image in the class group and hence are determined by their rank (the rank
condition comes in in that $\mathrm{M}\_k(\mathcal A)$ is a central simple
algebra which is indefinite at the infinite prime). In particular if $M\_1$ and
$M\_2$ are two rank $1$ modules over $\mathcal A$ and $A\_1$ and $A\_2$ are the
corresponding abelian varieties we get that $A\_1\bigoplus A\_2\cong A\bigoplus A$
as the left (resp. right) hand side is associated to $M\_1\bigoplus M\_2$ (resp.
$\mathcal A^2$). Therefore, to get an example it is enough to give an example of
an $\mathcal A$ for which there exist $M\_1\not\cong M\_2$. The number (or more
easily the mass) of isomorphism classes of ideals can be computed using mass
formulas and tends to infinity with the discriminant of $\mathcal A$. It is
interesting to note that when the discriminant is a prime $p$ we can go
backwards using supersingular elliptic curves: The mass is equal to the mass of
supersingular elliptic curves in characteristic $p$ and the latter mass can be
computed geometrically to be equal to $(p-1)/24$.


