nLab Grothendieck ring of varieties

Idea

Grothendieck group of varieties is the free Abelian group generated by isomorphism classes of quasiprojective varieties modulo all relations of the form [X][U]+[X\U][X]\sim [U] + [X\backslash U] where UU is open in XX. Product of varieties induces a multipication on this Abelian group, making it into a ring, the Grothendieck ring of varieties.

Literature

Chapter 2 of

  • Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag, Motivic integration, Progress in Mathematics 325 (2018) doi

  • Bjorn Poonen, The Grothendieck ring of varieties is not a domain, Math. Res. Lett. 9:4 (2002) 493-497 doi arXiv:math.AG/0204306

  • Michael Larsen, Valery A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3 (2003), no. 1, 85-95, 259 MR1996804 doi

We study the motivic Grothendieck group of algebraic varieties from the point of view of stable birational geometry. In particular, we obtain a counter-example to a conjecture of M. Kapranov on the rationality of motivic zeta-function.

An analogue in the setup of pretriangulated categories is introduced in

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