Intégration motivique sur les schémas formels

Abstract

Nous généralisons la théorie de l'intégration motivique au cadre des schémas formels. Nous définissons etétudions l'anneau booléen des ensembles mesurables, la mesure motivique, l'intégrale motivique et nous démontrons un théorème de changement de variables pour cette intégrale. Abstract (Motivic Integration on Formal Schemes).-We generalize the theory of motivic integration on formal schemes. In particular, we define and study the boolean ring of mesurable subsets, the motivic measure, the motivic integral and we prove a theorem of change of variables for this integral.

A trace formula for varieties over a discretely valued field

“…Of course, one has to show that this value only depends on X and not on the choice of a weak Néron model. If X is smooth and quasi-compact, this was proven in [32] using the theory of motivic integration on formal R-schemes [43], and the general case can be deduced from this result.…”

A Short Survey on the Integral Identity Conjecture and Theories of Motivic Integration

“…If, in addition, X is smooth and of relative dimension d, then by [29,Lemma 3.4.2], the morphism π n : Gr(X) → Gr n (X n ) is surjective, and the canonical projection Gr n+m (X n+m ) → Gr n (X n ) is a locally trivial fibration for the Zariski topology with fiber A dm k . We refer to [29,Section 4.2] for the definition of piecewise trivial fibration mentioned in the following Proposition 3.2 (Sebag [29], Lemma 4.3.25). Let X be a flat separated, quasi-compact, topologically of finite type formal R-scheme of relative dimension d. There is an integer c ≥ 1 such that, for e ∈ Z and n ∈ N with n ≥ ce, the projection…”

“…By [29], the image π n (Gr(X)) of Gr(X) in Gr n (X n ) is a constructible subset of Gr n (X n ). If, in addition, X is smooth and of relative dimension d, then by [29,Lemma 3.4.2], the morphism π n : Gr(X) → Gr n (X n ) is surjective, and the canonical projection Gr n+m (X n+m ) → Gr n (X n ) is a locally trivial fibration for the Zariski topology with fiber A dm k . We refer to [29,Section 4.2] for the definition of piecewise trivial fibration mentioned in the following Proposition 3.2 (Sebag [29], Lemma 4.3.25).…”

“…The Greenberg functor. The main reference for this paragraph is [12]; we may see also [29] and [22].…”

“…Because everything behind the integral identity conjecture is motivic integration in the sense of Sebag, Loeser and Nicaise for formal schemes and rigid varieties (cf. [29], [22], [24] and [28], etc. ), we shall express in what follows some important points of this theory.…”