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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Notes
- 1.
(…) let L(k) be the “K group” defined by schemes of finite type over k together with relations coming from decomposition into pieces (…)
- 2.
- 3.
It is not necessary to recall here the precise definition of an excellent scheme and we refer to (ÉGA IV4, §7) and to Raynaud and Laszlo ( 2014 ) for an introduction. In fact, the following examples will suffice for our purposes: fields are excellent, as well as Z and, more generally, Dedekind rings whose field of fractions has characteristic zero; complete noetherian local rings are excellent; moreover, localizations of schemes of finite type over a (quasi-)excellent scheme are (quasi-)excellent.
- 4.
Recall that we only consider Q-Hodge structures.
- 5.
Also called “constant tordus”
- 6.
We thank B. Le Stum for having communicated us this proof.
- 7.
The English word motive has two meanings, either a factor inducing to act in a certain way or, especially in art, a distinctive feature, a structural principle, a pattern. Grothendieck referred to the latter meaning.
- 8.
“ […] one obtains a natural homomorphism
$$\displaystyle{\mathrm{L}(k) \rightarrow \mathrm{ M}(k),}$$which is actually a homomorphism of rings […]. The general question that can be asked is to know what one can say on this homomorphism, is it far from being bijective? […]”
- 9.
This proof is due to J. Kollár and E. Szabó, see the appendix of Reichstein and Youssin (2000).
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Chambert-Loir, A., Nicaise, J., Sebag, J. (2018). The Grothendieck Ring of Varieties. In: Motivic Integration. Progress in Mathematics, vol 325. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-7887-8_2
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