# The Grothendieck Ring of Varieties

• Antoine Chambert-Loir
• Johannes Nicaise
• Julien Sebag
Chapter
Part of the Progress in Mathematics book series (PM, volume 325)

## Abstract

In this chapter, we define the Grothendieck ring of varieties over an arbitrary base scheme. This is a ring of virtual varieties up to cut-and-paste relations; it takes a central place in the theory of motivic integration, because (after a suitable localization and/or completion) it serves as the ring where motivic integrals take their values. After the basic definitions in section 1, we define the notion of motivic measures, which are ring morphisms from the Grothendieck ring to other rings with a more explicit structure. Motivic measures are fundamental both for the understanding of Grothendieck ring itself and for extracting geometric information from its elements. Among the motivic measures, we develop in sections 3 and 5 the cohomological and motivic realizations. In sections 5 and 6, we study the main structure theorems for the Grothendieck ring over a field of characteristic zero: the theorems of Bittner and Larsen-Lunts. Bittner’s theorem gives a presentation of the Grothendieck ring in terms of smooth projective varieties and blow-up relations, which is quite useful to construct motivic measures. The theorem of Larsen and Lunts relates equalities in the Grothendieck ring to the notion of stable birational equivalence. In section 4 we discuss a process of dimensional completion for the Grothendieck ring of varieties.

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## Authors and Affiliations

• Antoine Chambert-Loir
• 1
• Johannes Nicaise
• 2
• Julien Sebag
• 3
1. 1.Université Paris Diderot, Sorbonne Paris CitéInstitut de Mathématiques de Jussieu-Paris Rive GaucheParisFrance
2. 2.Department of MathematicsImperial College LondonLondonUK
3. 3.IrmarUniversité de Rennes 1Rennes CedexFrance