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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.

14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14A10 Varieties and morphisms
14G35 Modular and Shimura varieties
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