Mathematical Research Letters

Volume 9 (2002)

Number 4

The Grothendieck ring of varieties is not a domain

Pages: 493 – 497



Bjorn Poonen (University of California at Berkeley)


If $k$ is a field, the ring $K_0(\mathcal{V}_k)$ is defined as the free abelian group generated by the isomorphism classes of geometrically reduced $k$-varieties modulo the set of relations of the form $[X-Y] = [X] - [Y]$ whenever $Y$ is a closed subvariety of $X$. The multiplication is defined using the product operation on varieties. We prove that if the characteristic of $k$ is zero, then $K_0(\mathcal{V}_k)$ is not a domain.

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