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The Grothendieck ring of varieties is not a domain. (English) Zbl 1054.14505
Summary: 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.

##### MSC:
 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14A10 Varieties and morphisms 14G35 Modular and Shimura varieties
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