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 $[X]\sim [U] + [X\backslash U]$ where $U$ is open in $X$. 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

• A. I. Bondal, M. Larsen, V. A. Lunts, Grothendieck ring of pretriangulated categories, Intern. Math. Res. Notices 29 (2004) 1461–1495 doi

  math.AG/0401009

category: algebraic geometry