ag.algebraic geometry - 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., $mathcal Aotimesmathbb R$ 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
$Mbigotimes_{mathcal A}A$ characterised by $mathrm{Hom}(Mbigotimes_{mathcal A}A,B)=mathrm{Hom}_{mathcal A}(M,mathrm{Hom}(A,B))$ 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 $Mbigotimes_{mathcal A}A$ is isomorphic to
$Nbigotimes_{mathcal A}A$.

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 $A_1bigoplus A_2cong Abigoplus A$
as the left (resp. right) hand side is associated to $M_1bigoplus M_2$ (resp.
$mathcal A^2$). Therefore, to get an example it is enough to give an example of
an $mathcal A$ for which there exist $M_1notcong M_2$. 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 $(p-1)/24$.

• Next dg.differential geometry - Singular, holonomy-free connections on Riemannian surfaces?
• Previous ag.algebraic geometry - How to compute the ring of invariants of SO_3(k) acting on a polynomial ring