INTEGRATION IN VALUED FIELDS

EHUD HRUSHOVSKI, DAVID KAZHDAN

Abstract. We develop a theory of integration over valued fields of residue characteristic zero. In particular we obtain new and base-field independent foundations for integration over local fields of large residue characteristic, extending results of Denef,Loeser, Cluckers. The method depends on an analysis of definable sets up to definable bijections. We obtain a precise description of the Grothendieck semigroup of such sets in terms of related groups over the residue field and value group. This yields new invariants of all definable bijections, as well as invariants of measure preserving bijections.

Contents

1. Introduction 3 1.1. The logical setting 3 1.2. Model theory 4 1.3. More general definable sets 4 1.4. Generalized algebraic varieties 5 1.5. Rational polyhedra over ordered Abelian groups 5 1.6. The Grothendieck semiring of definable sets 5 1.7. Integration of forms up to absolute value 7 1.8. Integrals over local fields: uniformity over ramified extensions 8 1.9. Bounded and unbounded sets 8 1.10. Finer volumes 9 1.11. Hopes 9 1.12. Course of the paper 9 2. First order theories 10 2.1. Basic examples of theories 10 2.2. Grothendieck rings 18 3. Some C-minimal geometry. 21 3.1. Basic geography of C-minimal structures 24 3.2. Generic types and orthogonality 26 3.3. Definable sets in group extensions 28 3.4. V-minimality 32 3.5. Definable completeness and functions on the value group 35 3.6. Transitive sets in dimension one 37 3.7. Resolution and finite generation 39 3.8. Dimensions 42 4. Descent to RV: objects 46 5. V-minimal geometry: Continuity and differentiation 51 5.1. Images of balls under definable functions. 51 5.2. Images of balls, II 52 5.3. Limits and continuity 55 5.4. Differentiation in VF 57 5.5. Differentiation and Jacobians in RV. 58 5.6. Comparing the derivatives 59 6. Lifting functions from RV to VF 60 7. Special bijections and RV-blowups 63 7.1. Special bijections 63 7.2. Special bijections in one variable and families of RV-valued functions. 64 7.3. Several variables 67 7.4. RV-blowups 68 7.5. The kernel of 𝕃 72 8. Definable sets over VF and RV: the main theorems 73 8.1. Definable subsets of varieties 74 8.2. Invariants of all definable maps 75 8.3. Definable volume forms: VF 76 8.4. Definable volume forms: RV 77 8.5. The kernel of 𝕃 in the measured case 78 8.6. Invariants of measure preserving maps, and some induced isomorphisms 82 9. The Grothendieck semirings of Γ 83 9.1. Finite sets 84 9.2. Euler characteristics of DOAG 86 9.3. Bounded sets: volume homomorphism 89 9.4. The measured case 90 10. The Grothendieck semirings of RV 91 10.1. Decomposition to Γ, RES. 91 10.2. Decomposition of μRV 95 11. Integration with an additive character 98 11.1. Definable distributions 101 11.2. Fourier transform 104 12. Expansions and rational points over Henselian fields 104 12.1. Expansions of the RV sort 105 12.2. Transitivity 107 12.3. Rational points over a Henselian subfield: constructible sets and morphisms 109 12.4. Quantifier elimination for Hensel fields 109 12.5. Rational points: definable sets and morphisms 111 12.6. Some specializations 111 12.7. Higher dimensional local fields 112 13. The Grothendieck group of algebraic varieties 112 References 115

1. Introduction

Since Weil’s Foundations, algebraic varieties have been understood independently of a particular base field; thus an algebraic group G exists prior to the abstract or topological groups of points G(F), taken over various fields F. For Hecke algebras, or other geometric objects whose definition requires integration, no comparable viewpoint exists. One uses the topology and measure theory of each local field separately; since a field F has measure zero from the point of view of any nontrivial finite extension, at the foundational level there is no direct connection between the objects obtained over different fields. The main thrust of this paper is the development of a theory of integration over valued fields, that is geometric in the sense of Weil. At present the theory covers local fields of residue characteristic zero or, in applications, large positive residue characteristic.

Our approach to integration continues a line traced by Kontsevich, Denef-Loeser, Loeser-Cluckers (cf. [7]). In integration over non-archimedean local fields there are two sources for the numerical values. The first is counting points of varieties over the residue field. Kontsevich explained that these numerical values can be replaced, with a gain of geometric information, by the isomorphism classes of the varieties themselves up to appropriate transformations, or more precisely by their classes in a a certain Grothendieck ring. This makes it possible to understand geometrically the changes in integrals upon unramified base change. In this aspect our approach is very similar. The main difference is a slight generalization of the the notion of variety over the residue field, that allows us to avoid what amounted to a choice of uniformizer in the previous theory.

The second source of the numbers is the piecewise linear geometry of the value group. We geometrize this ingredient too, obtaining a theory of integration taking values in an entirely geometric ring, a tensor product of a Grothendieck ring of generalized varieties over the residue field, and a Grothendieck ring of piecewise linear varieties over the value group.

Viewed in this way, the integral is an invariant of measure preserving definable bijections. We actually find all such invariants. In addition we consider and determine all possible invariants of definable bijections; we obtain in particular two Euler characteristics on definable sets, with values in the Grothendieck group of generalized varieties over the residue field.

At the level of foundations, until an additive character is introduced, we are able to work with Grothendieck semigroups rather than with classes in Grothendieck groups.

1.1. The logical setting. Let L be a valued field, with valuation ring 𝒪_{L}. ℳ denotes the maximal ideal. We let VF^{n}(L) = L^{n}. The notation VF^{n} is analogous to the symbol 𝔸^{n} of algebraic geometry, denoting affine n-space. Let RV^{m}(L) = L^{∗}/(1 + ℳ), Γ(L) = L^{∗}/𝒪_{L}^{∗}, k(L) = 𝒪_{L}/ℳ_{L}. Let rv : VF → RV and val : VF → Γ be the natural maps. The natural map RV → Γ is denoted val_{rv}. The exact sequence

0 → k^{∗} → RV → Γ → 0

shows that RV is, at first approximation, just a way to wrap together the residue field and value group.

We consider expressions of the form h(x) = 0 and valf(x) ≥ valg(x) where f, g, h ∈ L[X], X = (X_{1}, …, X_{n}). A semi-algebraic formula is a finite Boolean combination of such basic expressions. A semi-algebraic formula ϕ clearly defines a subset D(L) of VF^{n}(L). Moreover if f, g, h ∈ L_{0}[X], we obtain a functor L ↦ D(L) from valued field extensions of L_{0} to sets. We will later describe more general definable sets; but for the time being take a definable subset of VF^{n} to be a functor D = D_{ϕ} of this form.

An intrinsic description of definable subsets of RV^{m} is given in §2.1. In particular, definable of (k^{∗})^{m} coincide with constructible sets in the usual Zariski sense; while modulo (k^{∗})^{m}, a definable set is a piecewise linear subset of Γ^{m}. The structure of arbitrary definable subsets of RV^{m} is analyzed in §3.3.

The advantages of this approach are identical to the benefits in algebraic geometry of working with arbitrary algebraically closed fields, over arbitrary base fields. One can use Galois theory to describe rational points over subfields. Since function fields are treated on the same footing, one has a mechanism to inductively reduce higher dimensional geometry to questions in dimension one, and often in fact to dimension zero (as in algebraic geometry, statements about fields, applied to generic points, can imply birational statements about varieties.)

1.2. Model theory. Since topological tools are no longer available, it is necessary to define notions such as dimension in a different way. The basic framework comes from [15]; we recall and develop it further in §2 and §4. It is in many respects analogous to the o-minimal framework of [36], that has become well-accepted in real algebraic geometry.

In addition, whereas in geometry all varieties are made as it were of the same material, here a number of rather different types of objects co-exist, and the interaction between them must be clarified. In particular, the residue field and the value group are orthogonal in a sense that will be defined below; definable subsets of one can never be isomorphic to subsets of the other, unless both are finite. This orthogonality has an effect on definable subsets of VF^{n} in general; for example, closed disks behave very differently from open ones. Here we follow and further develop [16].

Note that the set of rational points of closed and open disks over discrete valuation rings, for instance, cannot be distinguished; as in rigid geometry, the geometric setting is required to make sense of the notions. Nevertheless they have immediate consequences to local fields. As an example, we define the notion of a definable distribution; these are defined as a function on the space of polydisks with certain properties. Making use of model theoretic properties of the space of polydisks, we show that any definable distribution agrees outside a proper subvariety with one obtained by integrating a function. This is valid over any valued field of sufficiently large residue characteristic. In particular for large p, the p-adic Fourier transform of a rational polynomial is a locally constant function away from an exceptional subvariety, in the usual sense ( Corollary 11.10.) The analogue for ℝ and ℂ was proved by Bernstein using D-modules. For an individual ℚ_{p}, the same result can be shown using Denef integration and a similar analysis of definable sets over ℚ_{p}. These results were obtained independently by Cluckers and Loeser, cf. [8].

1.3. More general definable sets. Throughout the paper we discuss not semi-algebraic sets, but definable subsets of a theory with the requisite geometric properties (called V-minimality.) This includes also the rigid analytic structures of [23]. The adjective “geometrically” can be take to mean here: in the sense of the V-minimal theory.

While we work geometrically throughout the paper, the isomorphisms we obtain are canonical and so specialize to rational points over substructures. Thus a posteriori our results apply to definable sets over any Hensel field of large residue characteristic. See §12.

For model theorists, this systematic use of of algebraically closed valued fields to apply to other Hensel fields is only beginning to be familar. As an illustration, see Proposition 12.9, where it is shown that after a little analysis of definable sets over algebraically closed valued fields, quantifier elimination for Henselian fields of residue characteristic zero becomes a consequence of Robinson’s earlier quantifier elimination in the algebraically closed case.

A third kind of generalization is an a-posteriori expansion of the language in the RV sort. Such an expansion involves loss of information in the integration theory, but is sometimes useful. For instance one may want to use the Denef-Pas language, splitting the exact sequence into a product of residue field and value group. Another example occurs in Theorem 12.5 where it is explained, given a valued field whose residue field is also a valued field, what happens when one integrates twice. To discuss this, the residue field is expanded so as to become itself a valued field.

1.4. Generalized algebraic varieties. We now describe the basic ingredients in more detail. Let L_{0} be a valued field with residue field k_{0} and value group A. For each point γ ∈ ℚ⊗A we have one-dimensional k-vector space

V_{γ} = {0} ∪ {x ∈ K : val(x) = γ}/(1 + ℳ)

As discussed above, V_{γ} should be viewed as a functor L ↦ V_{γ}(L) on valued field extensions L of L_{0}, giving a vector space over the residue field functor. If γ − γ′ ∈ A then V_{γ}, V′_{γ} are definably isomorphic, so one essentially has V_{γ} for γ ∈ (ℚ⊗A)/A.

Fix γ̄ = (γ_{1}, …, γ_{n}), and V_{i} = V_{γ_{i}}, Vγ̄ = Π_{i}V_{γ_{i}}. A γ̄- polynomial is an polynomial H(X) = ∑ a_{ν}X^{ν} with val_{p}(a_{ν}) + ∑_{i} ν(i)γ_{i} = 0 for each nonzero term a_{ν}X^{ν}. The coefficients a_{ν} are described in §5.5.1; for the purposes of the introduction, and of Theorem 1.3 below, it suffices to think of integer coefficients. Such a polynomial clearly defines a function H : Vγ̄ → k. In particular one has the set of zeroes Z(H). The generalized residue structure RES_{L_{0}} is the residue field, together with the collection of one-dimensional vector spaces V_{γ}(γ ∈ ℚ⊗A) over it, and the functions H : Vγ̄ → k associated to each γ̄-polynomial.

The intersection W of finitely many zero sets Z(H) is called a generalized algebraic variety over the residue field. Given a valued field extensions L of L_{0}, we have the set of points W (L) ⊆ Vγ̄(L). When L is a local field, W (L) is finite.

We will systematically use the Grothendieck group of generalized varieties residue field, rather than the usual Grothendieck group of varieties. They are fundamentally of a similar nature: base change to an algebraically closed value field makes them isomorphic. But the generalized residue field makes it possible to to see canonically objects that are only visible after base change in the usual approach. One application is Theorem 1.3 below.

K_{+}RES_{L_{0}}[n] denotes the Grothendieck group of generalized varieties of dimension ≤ n; in the paper we will omit L_{0} from the notation.

1.5. Rational polyhedra over ordered Abelian groups. Let A be an ordered Abelian group. A rational polyhedron Δ over A is given by an expression

Δ = {x : Mx ≥ b}

with x = (x_{1}, …, x_{n}), M a k × n - matrix with rational coefficients, and b ∈ A^{k}. We view this as a functor B ↦ Δ(B) on ordered Abelian group extensions B of A. This functor is already determined by its value at B = ℚ⊗A. In particular when A ≤ ℚ, Δ is an ordinary rational polyhedron.

K_{+}Γ_{A}[n] is the semigroup generated by such polyhedra, up to piecewise GL_{n}(ℤ)- transformations and A-translations; see §9. When A is fixed it is omitted from the notation.

In our applications, A will be the value group of a valued field L_{0}. If B is the value group of a valued field extension L, write Δ(L) for Δ(B).

1.6. The Grothendieck semiring of definable sets. Fix a base field L_{0}. The word “definable” will mean: T_{L_{0}}-definable, with T a fixed V-minimal theory. To have an example in mind one can read “semi-algebraic over L_{0}” in place of “definable”.

Let VF[n] be the category of definable subsets X of n-dimensional algebraic varieties over L_{0}; a morphism X → X′ is a definable bijection X → X′ (see Definition 3.65 for equivalent definitions.) K_{+}VF[n] denotes the Grothendieck semi-group, i.e. the set of isomorphism classes of VF[n] with the disjoint sum operation. [X] denotes the class of X in the Grothendieck semi-group.

We explain how an isomorphism class of VF[n] is determined precisely by isomorphism classes of generalized algebraic varieties, and rational polyhedra, whose dimensions add up to n.

If X ⊆ RES^{m} and f : X → RES^{n} is a finite-to-one map, let

𝕃(X, f) = VF^{n} ×_{rv,f} X = {(v_{1}, …, v_{n}, x) : v_{i} ∈ VF, x ∈ X, rv(v_{i}) = f_{i}(x)}

The VF[n]-isomorphism class [𝕃(X, f)] does not depend on f, and is also denoted [𝕃X].

When S is a smooth scheme over 𝒪, X a definable subset of S(k), π : S(𝒪) → S(k) the natural reduction map, we have [𝕃X] = [π^{−1}X],

We let RES[n] be the category of pairs (X, f) as above; a morphism (X, f) → (X′, f′) is just a definable bijection X → X′. Let K_{+}RES[∗] be the direct sum of the Grothendieck semigroups K_{+}RES[n].

On the other hand, we have already defined K_{+}Γ[n]. Let K_{+}Γ[∗] be the direct sum of the K_{+}Γ[n]. An element of K_{+}Γ[n] is represented by a definable X ⊆ Γ[n]. Let 𝕃X = val^{−1}(X), 𝕃[X] = [𝕃X].

It is shown in Proposition 10.2 that the Grothendieck semiring of RV is the tensor product K_{+}RES[∗]⊗K_{+}Γ[∗] over the semiring K_{+}Γ^{fin} of classes of finite subsets of Γ; see §9.

Note that 𝕃([1]_{1}) = 𝕃([1]_{0}) + 𝕃([(0, ∞)]_{1}), where [1]_{1} ∈ K_{+}RES[1], [1]_{0} ∈ K_{+}RES[0] are the classes of the singleton set 1, and [(0, ∞)]_{1} is the class in K_{+}Γ[1] of the semi-infinite segment (0, ∞). Indeed 𝕃([1]_{1}) is the unit open ball around 1, 𝕃([1]_{0}) is the point {1}, while 𝕃([(0, ∞)]_{1}) is the unit open ball around 0, isomorphic by a shift to the unit open ball around 1. This is the one relation that cannot be understood in terms of the Grothendieck semi-ring of RV; it will be seen to correspond to the analytic summation of geoemtric series in the Denef theory. Let I_{sp} be the congruence on the ring K_{+}RES[∗]⊗K_{+}Γ[∗] generated by: [1]_{1} ∼ [1]_{0} + [(0, ∞)]_{1}.

The following theorem summarizes the relation between definable sets in VF and in RV; it follows from Theorem 8.4 together with Proposition 10.2 in the text.

Theorem 1.1. 𝕃 induces an surjective homomorphism of filtered semirings

K_{+}RES[∗]⊗K_{+}Γ[∗] → K_{+}(VF)

The kernel is precisely the congruence I_{sp}.

The inverse isomorphism K_{+}(VF) → K_{+}RES[∗]⊗K_{+}Γ[∗]/I_{sp} can be viewed as a kind of Euler characteristic, respecting products and disjoint sums, and functorial in various other ways.

The values of this Euler characteristc are themselves geometric objects, both on the algebraicgeometry side (RES) and the combinatorial-analytic side (Γ). This is valuable for some purposes; in particular it becomes clear that the isomorphism is compatible with taking rational points over Henselian subfields (cf. Proposition 12.6)

For other applications, however, it would be useful to obtain more manageable numerical invariants; for this purpose one needs to analyze the structure of K_{+}(Γ[∗]). We do not fully do this here, but using a number of homomorphisms on K_{+}(Γ[∗]) we obtain a number of invariants. In particular using the ℤ-valued Euler characteristics on K(Γ[∗]) (cf. §9 and [26], [20]), we obtain two homomorphisms on K_{+}VF[n] essentially to K(RES[n]). The reason there are two rather than one has to do with Poincaré duality, see Theorem 10.5.

For instance, when F is a field of characteristic 0 we obtain an invariant of rigid analytic varieties over F ((t)), with values in the Grothendieck ring K(Var_{F}) of algebraic varieties over F; and another in K(Var_{F})[[𝔸_{1}]^{−1}] ( Proposition 10.8). It is instructive to compare this with the invariant of [25], with values in K(RES[n])/[G_{m}]. ^{1} Since any two closed balls are isomorphic, via additive translation and multiplicative contractions, all closed balls must have the same invariant. Working with a discrete value group tends to force [G_{m}] = 0, since it appears that a closed ball B_{0} of valuation radius 0 equals G_{m} times a closed ball B_{1} of valuation radius 1. Since our technology is based on divisible value groups, the “equation” [B_{0}] = [B_{1}][G_{m}] is replaced for us by [B_{0}] = [B_{0}^{o}][G_{m}], where B_{0}^{o} is the open ball of valuation radius 0. Though B_{1} and B_{0}^{0} have the same F ((t))-rational points, they are geometrically distinct (cf. Lemma 3.46) and so no collapse takes place.

See also subsections 12.6.1 and 12.6.2 for two previously known cases.

By such Euler characteristic methods we can prove a statement purely concerning algebraic varieties, partially answering a question of Gromov and Kontsevich ([13], p. 121.). In particular, two elliptic curves with isomorphic complements in projective space were previously known to be isogenous, by zeta function methods; we show that they are isomorphic. This also follows from [22]; the method there requires strong forms of resolution of singularities. See Theorem 13.1.

1.7. Integration of forms up to absolute value. Over local fields, data for integration consists of a triple (X, V, ω), with X a definable subset of a smooth variety V, and ω volume form on V. We are interested an integral of the form ∫_{X} |ω|, so that multiplication of ω by a function with norm one does not count as a change, nor does removing a subvariety of V of smaller dimension. Using an equivalent description of VF[n], where the objects come with a distinguished finite-to-one map into affine space, we can represent an integrand as a pair (X, ω) with X ∈ Ob VF[n] and ω a function on X into Γ. Isomorphisms are essential bijections, preserving the form up to a function of norm 1. See Definition 8.10 for a precise definition of this category, the category μ_{Γ}VF[n].

Integration is intended to be an invariant of isomorphisms in this category. Thus we can find the integral if we determine all invariants. We do this in complete analogy with Theorem 1.1.

For n ≥ 0 let Γ[n] be the category whose objects are finite unions of rational polyhedra over the group A of definable points of Γ. A morphism f : X → Y of Γ[n] is a bijection such that for some partition X = ∪_{i=1}^{k}X_{i} into rational polyhedra, f|X_{i} is given by an element of GL_{n}(ℤ)⋉A^{n}. Let μΓ[n] be the category of pairs (X, ω), with X an object of Γ[n], and ω : X → Γ a piecewise affine map. A morphism f : (X, ω) → (X′, ω′) is a morphism f : X → X′ of Γ[n] such that ∑_{i=1}^{l} x_{i} + ω(x) = ∑_{i=1}^{l} x′_{i} + ω′(x′) whenever (x′_{1}, …, x′_{n}) = f(x_{1}, …, x_{n}). Given (X, ω) ∈ Ob μΓ[n], define 𝕃X as above, and adjoint the pullback of ω to obtain an object of μ_{Γ}VF[n]. This gives a homomorphism K_{+}μΓ[n] → K_{+}μ_{Γ}VF[n]. Theorem 1.2. 𝕃 induces an surjective homomorphism of filtered semirings

K_{+}RES[∗]⊗_{ℕ}K_{+}μΓ[∗] → K_{+}(μ_{Γ}VF)[∗]

The kernel is generated by the relations p⊗1 = 1⊗[(val_{rv}(p), ∞)] and 1⊗a = val_{rv}^{−1}(a)⊗1.

In the statement of the theorem, p ranges over definable points of RES (actually one value suffices), and a ranges over definable points of Γ.

This can also be written as (K_{+}RES[∗]⊗_{K_{+}(μΓ^{fin})}K_{+}μΓ[∗])/μI_{sp} ≃ K_{+}(μ_{Γ}VF)[∗], where K_{+}(μΓ^{fin}) is the subsemiringof subsets of μΓ with finite support, and μI_{sp} is a semiring congruence defined similarly to I_{sp}. The base of the tensor leads to the identification of a point of Γ with with a coset of k^{∗} in RES, while μI_{sp} identifiese a point of RES with an infinite interval of Γ. The inverse isomorphism can be viewed as an integral.

1The setting is somewhat different: Loeser-Sebag can handle positive characteristic too, but assume smoothness.

We introduce neither additive nor multiplicative inverses in K_{+}RES[∗] formally, so that the target of integration is completely geometric.

We proceed to give an application of the first part of the theorem (the surjectivity) in terms of ordinary p-adic integration.

1.8. Integrals over local fields: uniformity over ramified extensions. Let L be a local field, finite extension of ℚ_{p} or 𝔽_{p}((t)). We normalize the Haar measure μ in such a way that the maximal ideal has measure 1, the norm by |a| = μ{x : |x| < |a|}. Let RES_{L} be the generalized residue field, and Γ_{L} be the value group. We assume ℚ_{p} or 𝔽_{p}((t)) have value group ℤ, and identify Γ_{L} with a subgroup of ℚ.

Given c = (c_{1}, …, c_{k}) ∈ L^{k} and s = (s_{1}, …, s_{k}) ∈ ℝ^{k} with s_{i} ≥ 1, let |c|^{s} = Π_{i=1}^{k}|c_{i}|^{s_{i}}.

Let λ be a multiplicative character ℝ^{n} → ℝ^{∗}. Define:

ev_{λ}(Δ(B)) = ∑_{b∈Δ(B)} λ(b)

provided this sum is absolutely convergent. Given linear functions h_{0}, …, h_{k} on ℝ^{n} and s_{1}, …, s_{k} ∈ ℝ, let ev_{h,s,Q} = ev_{λ} where λ(x) = Q^{h_{0}(x)+∑ s_{i}h_{i}(x)} Theorem 1.3. Fix n, d, k ∈ ℕ. Let p be a large prime compared to n, d, k, and let f ∈ ℚ_{p}[X_{1}, …, X_{n}]^{k} have degrees ≤ d. Then there exist finitely many generalized varieties X_{i} over RES(ℚ_{p}), rational polyhedra Δ_{i}, γ(i) ∈ ℚ^{≥0}, n_{i} ∈ ℕ, and linear functions h_{0}^{i}, …, h_{k}^{i} with rational coefficients, such that for any finite extension L of ℚ_{p} with residue field GF (q) and val(L^{∗}) = (1/r)ℤ, val(p) = 1, and any s ∈ ℝ_{≥1}^{k},

∫_{𝒪_{L}^{n}} |f|^{s} = ∑_{i} q^{rγ(i)}(q − 1)^{n_{i}}|X_{i}(L)|ev_{h^{i},s,q^{r}}(Δ_{i}(L))

Note:

1)Δ_{i}(L) depends only on the ramification degree r of L over ℚ_{p}.

2) The formula is a sum of non-negative terms.

3) ev_{h,s,q^{r}}(Δ_{i}((1/r)ℤ)) can be written in closed form as a rational function of q^{rs}. This follows from Denef, who shows it for more general sets Δ_{i} definable in Pressburger arithmetic; such analytic summation is an essential component of his integration theory. Since it plays no role in our approach we leave the statement in geometric form.

4) The generalized varieties X_{i} and polyhedra Δ_{i} are simple functions of the coefficients f. Here we wish to emphasize not this, but the uniformity of the expression over ramified extensions of ℚ_{p}.

The proof follows Proposition 10.10 (it uses only the easy surjectivity in this proposition, and Proposition 4.5.)

1.9. Bounded and unbounded sets. The isomorphism of semirings of Theorem 1.2 obviously induces an isomorphism of rings. However introducing additive inverses loses information on the Γ side; the class of the interval [0, 1) becomes 0, since [0, ∞) and [1, ∞) are isomorphic. The classical remedy is to cut down to bounded sets before groupifying. This presents no difficulty, since the isomorphism respects boundedness.

In higher dimensional local fields, stronger notions of boundedness may be useful, such as those introduced by Fesenko. Since these questions are not entangled with the theory of integration, and can be handled a posteriori, we will deal with them in a sequel.

Here we mention only that even if one insists on integrating all definable integrands, with no boundedness condition, into a ring, some but not all information is lost. This is due to the existence of Euler characteristics on Γ, and thus again to the fact that we work geometrically, with divisible groups, even if the base field has a discrete group. We will see ( Lemma 9.12) that K_{+}(μΓ[n]) can be identified with the group of definable functions Γ → K_{+}(Γ[n]). Applying an appropriate Euler characteristic reduces to the group of piecewise constant functions on Γ into ℤ. Recombining with RES we obtain a consistent definition of an integral on unbounded integrands, compatible with measure preserving maps, sums and products, with values in K(RES)[A]/[𝔸_{1}]_{1}K(RES)[A], where A is the group of definable points of Γ, and [𝔸_{1}]_{1} is the class of the affine line. See Theorem 10.11.

1.10. Finer volumes. We also consider a finer category of definable sets with RV-volume forms. This means that a volume form ω is identified with gω only when g − 1 ∈ ℳ; val(g) = 0 does not suffice. We obtain an integral whose values themselves are definable sets with volume forms; in particular including algebraic varieties with volume forms over the residue field. Theorem 1.4. 𝕃 induces an surjective homomorphism of graded semirings

K_{+}μRV[∗] → K_{+}(μVF)[∗]

The kernel is precisely the congruence μI_{sp}.

μRV is the category of definable subsets of μRV^{∗} enriched with volume forms; see Definition 8.13. Again, an isomorphism is induced in the opposite direction, that can be viewed as a motivic integral

∫ : K_{+}(μVF)[∗] → K_{+}μRV[∗]/μI_{sp}

This allows an iteration of the integration theory, either with an integral of the same nature if the residue field is a valued field, or with a different kind of integral if for instance the residue field is ℝ.

1.11. Hopes. We mention three. Until now, a deep obstacle existed to extending Denef’s theory to positive characteristic; namely the theory was based on quantifier elimination for Hensel fields of residue characteristic 0, or for finitely ramified extensions of ℚ_{p}, and it is known that no similar quantifier elimination is possible for 𝔽_{p}((t)), if any is. On the other hand Robinson’s quantifier elimination is perfectly valid in positive characteristic. This raises hopes of progress in this direction, though other obstacles remain.

It is natural to think that the theory can be applied to higher dimensional local fields; we will consider this in a sequel.

Another important target is asymptotic integration over ℝ. Nonstandard extensions of ℝ admit natural valued field structures. This is the basis of Robinson’s nonstandard analysis. These valued fields have divisible value groups, and so previous theories of definable integration do not apply. The theory of this paper applies however, and we expect that it will yield connections between p-adic integration and asymptotics of real integrals.

1.12. Course of the paper. After recalling some basic model theory in §2, we proceed in §3 to V-minimal theories.

In §4 we show that any definable subset of VF^{n} admits a constructible bijection with some 𝕃(X, f). In fact only a very limited class bijections is needed; a typical one has the form: (x_{1}, x_{2}) ↦ (x_{1}, x_{2} + f(x_{1}, x_{2})), so it is clearly measure preserving. The proof is simple and brief, and uses only a little of the preceding material. We note here that for many applications this statement is already sufficient; in particular it suffices to give the surjectivity in Theorems 1.1 and 1.2, and hence the application Theorem 1.3.

In §5 we return to the geometry of V-minimal structures, developing a theory of differentiation. We show the compatibility between differentiation in RV and in VF. This is needed for Theorem 1.4. Differentiation in VF involves much finer scales than in RV; in effect RV can only see distances measured by valuation 0, while the derivative in VF involves distances of arbitrarily large valuation. The proof uses a continuity argument on scales. It fails in positive characteristic, in its present form.

§6 is devoted to showing that 𝕃 yields a well-defined map K_{+}(RV) → K_{+}(VF); in other words not only objects, but also isomorphism can be lifted.

Sections §7 and §8 investigate the kernel of 𝕃 in Theorem 1.1. This is the most technical part of the paper, and we have not been able to give a proof as functorial as we would have liked. See Question 7.9.

In §9 we study the piecewise linear Grothendieck group; see the introduction to this section. §10 decomposes the Grothendieck group of RV into the components RES and Γ, used

througout this introduction.

§11 introduces an additive character, and hence the Fourier transform. The isomorphism of volumes given by Theorem 1.4 suffices for this extension; it is not necessay to redo the theory from scratch, but merely to follow through the functoriality.

§12 contains the extension to definable sets over Hensel fields mentioned above, and §13 the application to the Grothendieck group of varieties.

Thanks to Aviv Tatarsky and Moshe Kaminsky, and to Lou Van den Dries, Clifton Ealy, and Jana Maříková, for useful comments and corrections.

2. First order theories

The bulk of this paper uses no deep results from logic, beyond Robinson’s quantifier elimination the theory of algebraically closed valued fields ([32]). However, it is imbued with a model theoretic viewpoint. We will not explain the most basic notions of logic: language, theory, model. Let us just mention that a language consists of basic relation and function symbol, and formulas are built out of these, using symbols for Boolean operations and quantifiers; cf. e.g.

[11] or [19], or the first section of [9]); but we attempt in this section to bridge the gap between these and the model-theoretic language used in the paper.

A language L consists of a family of “sorts” S_{i}, a collection of variables ranging over each sort, a set of relation symbols R_{j}, each intended to denote a subset of a finite product of sorts, and a set of function symbols F_{k} intended to denote functions from a given finite product of sorts to a given sort. From these, and the logical symbols &, ¬, ∀, ∃ one forms formulas. A sentence is a formula with no free variables (cf. [11].) A theory T is a set of sentences of L. A theory is called complete if for every sentence ϕ of L, either ϕ or its negation ¬ϕ is in T.

A universe M for the language L consists, by definition, of a set S(M) for each sort S of L. An L-structure consists of such a universe, together with an interpretation of each relation and function symbol of L. One can define the truth value of a sentence in a structure M; more generally, if ϕ(x_{1}, …, x_{n}) is a formula, with x_{i} a variable of sort S_{i}, then one defines the interpretation ϕ(M) of ϕ in M, as the set of all d ∈ S_{1}(M) × … × S_{n}(M) of which ϕ is true. If every sentence in T is true in M, one says that M is a model of T (M ⊧ T). The fundamental theorem here is a consequence of Gödel’s completeness theorem called the compactness theorem: a theory T has a model if every finite subset of T has a model.

The language L_{rings} of rings, for example, has one sort, three function symbols +, ⋯, −, two constants 0, 1; any ring is an L_{rings}-structure; one can obviously write down a theory T_{fields} in this language whose models are precisely the fields.

2.1. Basic examples of theories. We will work with a number of theories associated with valued fields.

(1) ACF , the theory of algebraically closed fields. The language is the language of rings {+, ·, −, 0, 1}, mentioned earlier. The theory states that the model is a field, and for each n, that every monic polynomial of degree n has a root. For instance for n = 2:

(∀u_{1})(∀u_{0})(∃x)(x^{2} + u_{1}x + u_{0} = 0)

ACF(0) includes in addition the sentences: 1 + 1 ≠ 0, 1 + 1 + 1 ≠ 0, . . .. This theory is complete (Tarski-Chevalley.) It will arise as the theory of the residue field of our valued fields.

(2) Divisible ordered Abelian groups (DOAG). The language consists of a single sort, a binary relation symbol <, a binary function symbol +, a unary function symbol − and a constant symbol 0. The theory states that a model is an ordered Abelian group. In addition axioms asserting divisibility by n for each n, for instance: (∀x)(∃y)(y +y = x).

This is the theory of the value group of a model of ACVF.

(3) The RV sort. (Extension of (2) by (1)). The language has one official sort, denoted RV, and includes: Abelian group operations ·, / on RV, a unary predicate k^{∗} for a subgroup, and an operation + : k^{2} → k, where k is k^{∗} augmented by a constant 0. Finally, there is a partial ordering; the theory states that k^{∗} is the equivalence class of 1; that ≤ is a total ordering on k^{∗}-cosets, making RV/k^{∗} =: Γ a divisible ordered Abelian group, and that (k, +, ·) is an algebraically closed field. (We thus have an exact sequence 0 → k^{∗} → RV → Γ → 0, but we treat Γ as an imaginary sort.) This theory TRV is complete too.

We will sometimes view RV as an autonomous structure; but it will arise from an algebraically closed valued field, as in (5) below.

(4) Let M ⊧ TRV, and let A be a subgroup of Γ(M). Within TRV_{A} we see an interpretation of ACF, namely the algebraically closed field k. In addition for each a ∈ A we have a 1-dimensional k-space, the fiber of RV lying over Γ augmented by 0. Collectively the field k with this collection of vector spaces will be denoted RES.

(5) ACVF, the theory of algebraically closed valued fields. According to Robinson, the completions, denoted ACVF(q, p), are obtained by specifying the characteristic q and residue characteristic p. We will be concerned with ACVF(0, 0) in this paper. However since any sentence of ACVF(0, 0) lies in ACVF(0, p) for almost all primes p, the results will a posteriori apply also to valued fields of characteristic zero and large residue characteristic.

We will take ACVF(0, 0) to have two sorts, VF and RV = VF^{∗}/(1+ℳ). The language includes the language of rings (1) on the VF sort, the language (3) on the RV sort, and a function symbol rv for a function VF^{∗} → RV. Denote rv^{−1}(RV^{≥0}) = 𝒪, rv^{−1}(0) = ℳ.

The theory states that VF is a valued field, with valuation ring 𝒪 and maximal ideal ℳ; that rv : VF^{∗} → RV is a surjective group homomorphism, and the restriction to 𝒪 (augmented by 0 ↦ 0) is a surjective ring homomorphism.

The structure ACVF_{A} induces on Γ is of a uniquely divisible Abelian group, with constants for the elements of Γ(A). Thus every definable subset of Γ is a finite union of points and open intervals (possibly infinite.)

(6) Rigid analytic expansions (Lipshitz) The theory ACVF^{R} of algebraically closed valued fields expanded by a family R of analytic functions. See [23] and [24]. Our theory of definable sets will be carried out axiomatically, and are thus also valid for these rigid analytic expansions.

A definable set D is not really a set, but a functor from the category of models of T to the category of sets of the form M ↦ ϕ(M), where ϕ is a formula of L. Model theorists do not really distinguish between the definable set D and the formula ϕ defining it; we will usually refer to definable sets rather than to formulas. If R ⊆ D ×D′ and for any model M ⊧ T, R(M) is the graph of a function D(M) → D(M′), we say R is a definable function of T. Similarly we say D is finite if D(M) is finite for any M ⊧ T, etc. It follows from the compactness theorem that if D is finite, then for some integer m we have |D(M)| ≤ m for any M ⊧ T. We sometimes write S^{∗} to denote S^{n} for some unspecified n.

By a map between L-structures A, B we mean a family f = (f_{S}) indexed by the sorts of L, with f_{S} : S(A) → S(B); one extends f to products of sorts by setting f((x_{1}, …, x_{n})) = (f(x_{1}), …, f(x_{n})). f is an embedding of structures if f^{−1}R(B) = R(A) for any atomic formula R of L. Taking R to be the equality relation, this includes in particular the statement that each f_{S} is injective.

On occasion we will use ∞-definable sets. An ∞-definable set is a functor of the form M ↦ ∩𝒟, where 𝒟 is a given collection of definable sets. In a complete theory a definable set is determined by the value it has at a single model, this is of course false for ∞-definable sets.

We write: a ∈ D to mean: a ∈ D(M) for some M ⊧ T. It is customary, since Shelah, to choose a single universal domain 𝕌 embedding all “small” models, and let a ∈ D mean: a ∈ D(𝕌); we will not require this interpretation, but the reader is welcome to take it.

We will sometimes consider imaginary sorts. If D is a definable set, and E a definable equivalence relation on D, then D/E may be considered to be an imaginary sort; as a definable set it is just the functor M ↦ D(M)/E(M). A definable subset of a product Π_{i=1}^{n}D_{i}/E_{i} of imaginary sorts (and ordinary sorts) is taken to be a subset whose preimage in Π_{i=1}^{n}D_{i} is definable; the notion of a definable function is thus also defined. In this way the imaginary sorts can be treated on the same footing as the others. The set of all elements of all imaginary sorts of a structure M is denoted M^{eq}. It is easy to construct a theory T^{eq} in a language L^{eq} whose category of models is (essentially) {M^{eq} : M ⊧ T }. See [34], [30], §16d.

Given a definable set D ⊆ S × X, where S, X are definable sets, and given s ∈ S, let D(s) = {x ∈ X : (s, x) ∈ D}. Thus D is viewed as a family of definable subsets of X, namely {D(s) : s ∈ S}. If s ≠ s′ implies D(s) ≠ D(s′), we say that the parameters are canonical, or that s is a code for D(s). In particular, if E is a definable equivalence relation, the imaginary elements a/E can be considered as codes for the classes of E.

T is said to eliminate imaginaries if every imaginary sort admits a definable injection into a product of some of the sorts of L. For instance, the theory of algebraically closed fields eliminates imaginaries. See [31] for an excellent exposition of these issues. We note that T admits elimination of imaginaries iff for any family D ⊆ S ×X there exists a family D′ ⊆ S′ ×X such that for any t ∈ S there exists a unique t′ ∈ S′ with D(t) = D′(t′).

(Recall that t ∈ S means: t ∈ S(M) for some M ⊧ T. The uniqueness of t′ implies in this case that one can choose t′ ∈ S′(M) too.) In this case we also say that t′ is called a canonical parameter or code for D(t).

Example 2.1. let b be a nondegenerate closed ball in a model the theory ACVF of algebraically closed valued fields. Then b = {x : val(x − c) ≥ val(c − c′)} for some elements c ≠ c′ of the field . b is coded by b̄ = (c, c′)/E, where (c, c′)E(d, d′) iff val(c − c′) = val(d − d′) ≤ val(c − d). However we often fail to distinguish notationally between b and b̄, and in particular we write A(b) = A(b̄).

The only imaginary sorts that will really be essential for us are the sorts 𝔅 of closed and open balls. The closed balls around 0 can be identified with their radius, hence the valuation group Γ(M) = VF^{∗}(M)/𝒪^{∗}(M) of a valued field M is embedded as part of 𝔅.

Notation Let 𝔅 = 𝔅^{o} ∪ 𝔅^{cl}, the sorts of open and closed sub-balls of VF. Let Γ^{+} = {γ ∈ Γ : γ ≥ 0}.

𝔅^{cl} = ⋃^{•}_{γ∈Γ}𝔅_{γ}^{cl}, 𝔅_{γ}^{cl} = VF/γ𝒪

𝔅^{o} = ⋃^{•}_{γ∈Γ}𝔅_{γ}^{o}, 𝔅_{γ}^{o} = VF/γℳ

Here γℳ = {x ∈ VF : val(x) > γ}, γ𝒪 = {x ∈ RES : val(x) ≥ γ}. The elements of 𝔅_{γ}^{cl} , fB_{γ}^{o} will be referred to as closed and open balls of valuative radius γ; though this valuative definition of radius means that bigger balls have smaller radius. The word “distance” will be used similarly.

By a thin annulus we will mean: a closed ball of valuative radius γ, with an open ball of valuative radius γ removed.

Fix a model M of T. A substructure A of M (written: A ≤ M) consists of a subset A_{S} of S(M), for each sort S of L, closed under all definable functions of T. For example, the substructures of models of T_{fields} are the integral domains.

In general, the definable closure of a set A_{0} ⊂ M is the smallest substructure containing A_{0}; it is denoted dcl(A_{0}) or < A_{0} >. An element of < A_{0} > can be written as g(a_{1}, …, a_{n}) with a_{i} ∈ A_{0} and g a definable function; i.e., it is an element satisfying a formula ϕ(x, a_{1}, …, a_{n}) of L_{A_{0}} in one variable that has exactly one solution in M. If A is a substructure, dcl(A ∪ {c}) is also denoted A(c). These notions apply equally when A, c contain elements of the imaginary sorts. If B is contained in sorts S_{1}, …, S_{n}, then dcl(B) is said to be an S_{1}, …, S_{n}-generated substructure. In the special case of valued fields, where one of the sorts VF is the “main” valued field sort, a VF-generated structure will be said to be field-generated, or sometimes just “a field”.

For any definable set D, we let D(A) be the set of points of D(M) with coordinates in A. If S = D/E is an imaginary sort, S(A) is the set of a ∈ S whose preimage is defined over A. We have D(A)/E(A) ⊆ S(A). D(A)/E(A) is of course closed under definable functions S^{m} → S that lift to definable functions D^{m} → D, but it is not necessarily closed under arbitrary definable functions, i.e functions whose graph is the image of a definable subset of D^{m} × D. For example x ↦ (1/n)x is a definable function on the value group of a model of ACVF, but if A ≤ M ⊧ ACVF, Γ(A) need not be divisible.

When A ≤ M, B ≤ N with M, N ⊧ T, a function f : A → B is called a (partial) elementary embedding (A, M) → (B, N) if for any definable set D of L, f^{−1}D(B) = D(A). In particular, when A = M, B = N, one say that M is an elementary submodel of N.

By a constructible set over A, we mean the functor L ↦ ϕ(L) on models M ⊧ T_{A}, where ϕ = ϕ(x_{1}, …, x_{n}, a_{1}, …, a_{m}) is a quantifier-free formula with parameters from A.

We say that T admits quantifier-elimination if every definable set coincides with a constructible set. It follows in this case that for any A, any A-definable set is A-constructible. When T admits quantifier elimination,f : A → B is a partial elementary embedding iff it is an embedding of structures.

The theories (1-5) of §2.1 admit quantifier elimination in their natural algebraic languages (theorems of Tarski-Chevalley and Robinson; cf. [16].) The sixth admits quantifier elimination in a language that needs to be formulated with more care, see [23].

In all of this paper except for §12.1, §12.3, we will use only structural properties of definable sets, and not explicit formulas. In this situation quantifier elimination can be assumed softly, by merely increasing the language by definition so that all definable sets become equivalent to quantifier-free ones. The above distinctions will only directly come into play in the two sections §12.1, §12.3.

If A ≤ M ⊧ T, L_{A} is the language L expanded by a constant c_{a} for each element a of A, so that an L_{A}-structure is the same as an L-structure M together with a function A_{S} → S(M), for each sort S. T_{A} is the set of L_{A} sentences true in M when the constant symbol c_{a} is interpreted as a; the models of T_{A} are models M of T, together with an isomorphic embedding of A as a substructure of M. In particular, M with the inclusion of A in M is an L_{A}-structure denoted M_{A}. For any subset A_{0} ⊆ M, we write T_{A_{0}} for T_{<A_{0}>}, where < A_{0} > is the substructure generated by A_{0}.

A definable set of T_{A} will also be referred to as A-definable; similarlly for other notions such as those defined just below.

A parametrically definable set of T is by definition a T_{A} - definable set for some A.

An almost definable set is the union of classes of a definable equivalence relation with finitely many classes. An element e is called algebraic (resp. definable) if the singleton set {e} is almost definable (resp. definable). When T is a complete theory, the set of algebraic (definable) elements of a model M of T forms a substructure that does not depend on M, up to (a unique) isomorphism.

Let A_{0} ⊆ M ⊧ T; the set of e ∈ M almost definable over A_{0} is called the algebraic closure of A_{0}, acl(A_{0}). If A_{0} is contained in sorts S_{1}, …, S_{n}, any substructure of acl(A_{0}) containing dcl(A_{0}) is said to be almost S_{1}, …, S_{n}-generated. Example 2.2. If a definable set D carries a definable linear ordering, then every algebraic element of D is definable. This is because the least element of a finite definable set F is clearly definable; the rest are contained in a smaller finite definable subset of D, so are definable by induction.

If in addition D has elimination of imaginaries, and Y is almost definable and definable with parameters from D, then Y is definable. Indeed using elimination of imaginaries in D, the set Y can be defined using canonical parameters. These are algebraic elements of D, hence definable.

Two definable functions f : X → Y, f′ : X → Y′ will be called isogenous if for all x ∈ X, acl(f(x)) = acl(f′(x)).

2.1.1. Compactness. Compactness often allows to replace arguments in relative dimension one over a definable set, by arguments in dimension one over a different base structure. Here is an example: Lemma 2.3. Let f_{i} : X_{i} → Y be definable maps between definable sets of T (i = 1, 2.) Assume that for any M ⊧ T and b ∈ Y (M), X_{1}(b) := f_{1}^{−1}(b) is T_{b}-definably isomorphic to X_{2}(b) = f_{2}^{−1}(b). Then X_{1}, X_{2} are definably isomorphic.

Proof. Let ℱ be the family of pairs (U, h), where U is a definable subset of Y, and h : f_{1}^{−1}U → f_{2}^{−1}U is a definable bijection. Claim For any b ∈ Y (M), M ⊧ T, there exists (U, h) ∈ ℱ with b ∈ U.

Proof. Let b ∈ Y (M). There exists a T_{b}-definable bijection X_{1}(b) → X_{2}(b). This bijection can be written: x ↦ g(x, b), where g is a definable function. Let U = {y ∈ Y : (x ↦ g(x, y)) is a bijection X_{1}(y) → X_{2}(y)}. Then (U, g(x, f_{1}(x))) ∈ ℱ, and b ∈ U. □

Now by compactness, there exist a finite number of definable subsets U_{1}, …, U_{k} of Y, with Y = ∪_{i}U_{i}, and with (U_{i}, h_{i}) ∈ ℱ for some h_{i}. Let U′_{i} = U_{i} \ (U_{1} ∪ … ∪ U_{i−1}), and define h = ∪_{i}h_{i}|U′_{i}. Then h : X_{1} → X_{2} is the required bijection. □

Here is another example of the use of compactness: Example 2.4. If D is a definable set, and for any a, b ∈ D, a ∈ acl(b), then D is finite. More generally, if a ∈ acl(b) for any b ∈ D, then a ∈ acl(∅)

Proof. We prove the first statement, the second being similar. For any model M, pick a ∈ M; then D(M) ⊆ acl(a). For b ∈ acl(a), let ϕ_{b} be the formula x ≠ b&D(x). So the set of formulas T h(M)_{M} ∪ {ϕ_{b}} has no common solution. By compactness, some finite subset already has no solution; this is only possible if D(M) is finite. □

2.1.2. Transitivity, orthogonality. A definable set D is transitive if it has no proper, nonempty definable subsets. (The usual word is “atomic”. One also says that D generates a complete type.) It is (finitely) primitive if it admits no nontrivial definable equivalence relation (with finitely many classes). Remark 2.5. Let A be a VF-generated substructure of a model of ACVF. When A is VFgenerated, we will see that an ACVF_{A}-definable ball b is never transitive in ACVF_{A}; indeed it always contains an A-definable finite set. But b is always ACVF_{A(b)}-definable, and quite often it is transitive; cf. Lemma 3.8.

Two definable sets D, D′ are said to be orthogonal if any definable subset of D^{m} × D^{l} is a finite union of rectangles E × F, E ⊆ D^{m}, F ⊆ D^{l}. In this case, the rectangles E, F can be taken to be almost definable. If the rectangles can actually be taken definable, we say D, D′ are strongly orthogonal.

2.1.3. Types. Let S be a product of sorts, and let M ⊧ T, a ∈ S(M). We write tp(a) = tp(a; M) (the type of a) for the set of definable sets D with a ∈ D; when p = tp(a) we write a ⊧ p. A complete type is the type of some element in some model. If q = tp(a), we say a is a realization of q. The set 𝒯p_{S} of complete types belonging to S can be topologized: a basic open set is the set of types including a given definable set D. The compactness theorem of model theory implies that this is a compact topological space: if {D_{i}} is any collection of definable sets with nonempty finite intersections, the compactness theorem asserts the existence of M ⊧ T with ∩_{i}D_{i}(M) ≠ ∅.

The compactness theorem is often used by way of a construction called saturated models; cf. [9]. These are models where all types over “small” sets are realized. They enjoy excellent Galois-theoretic properties: in particular if M is saturated, then dcl(A_{0}) = F ixAut(M/A_{0}) for any finite A_{0} ⊆ M. If D is acl(A_{0})-definable, then there exists an A_{0}-definable D′ which is a finite union of Aut(M/A_{0})-conjugates of D.

A type p can also be identified with the functor P from models of T (under elementary embeddings) into sets; P (M) = {a ∈ M : a ⊧ p}. As with definable sets, we speak as if P is simply a set. Unlike definable sets, the value of P (M) at a single model does not determine P (it could be empty; but it does determine P if M is sufficiently saturated.)

Any definable map f : S → S′ induces a map f_{∗} : 𝒯p_{S} → 𝒯p_{S′}; as another consequence of the compactness theorem, f_{∗} is continuous. We also have a restriction map from types of T_{A} to types of T, tp_{T (A)}(a) ↦ tp_{T} (a).

If L ⊆ L′ and T ⊆ T′, we say that T′ is an expansion of T. In this case any T′-type p′ restricts to a T -type p. If p′ is the unique type of T′ extending p, we say that p implies p′.

The simplest kind of expansion is an expansion by constants, i.e. a theory T_{A} (where A ≤ M ⊧ T.) If c ∈ M^{n}, or more generally if c ∈ M^{eq}, the type of c for M_{A} is denoted tp(c/A). It is rare for tp(c) to imply tp(c/A), but significant when it happens.

An instance of this is strong orthogonality: it is easy to see that strong orthogonality of two definable sets D, D′ is equivalent to the following condition:

(*) If A′ is generated by elements of D′, then any type of elements of D generates a complete type over A′.

The asymmetry in (*) is therefore only apparent.

Similarly, we have:

Lemma 2.6. Let D, D′ be definable sets. Then (1) ⟺ (2), (3) ⟺ (4).

(1) Every definable function f : D → D′ is piecewise constant, i.e. there exists a partition D = ∪_{i=1}^{n}D_{i} of D into definable sets, with f constant on D_{i}.

(2) If d ∈ D, d′ ∈ D′, d′ ∈ dcl(d) then d′ ∈ dcl(∅).

(3) If f : E → D is a definable finite-to-one map, and g : E → D′ is definable, then g(E) is finite.

(4) If d ∈ D, d′ ∈ D′, d′ ∈ acl(d) then d′ ∈ acl(∅).

Proof. Let us show that (3) implies (4). Let M ⊧ T, d ∈ D(M), d′ ∈ D′(M), d′ ∈ acl(d). Then d′ lies in some finite T_{d}-definable set D′(d) ⊆ D′. Since T_{d} is obtained from T by adding a constant symbol for d, there exists a formula ϕ(x, y) of the language of T and some m such that M ⊧ ϕ(d, d′) and M ⊧ (∃^{≤m}z)ϕ(d, z). Let X_{0} = {(x : (∃^{≤m}y)ϕ(x, y)}, E = {(x, y) : x ∈ X_{0}, ϕ(x, y)}, f(x, y) = x, g(x, y) = y. Then by (3) g(E) is finite, but d′ ∈ g(E), so d′ ∈ acl(∅).

Next, (4) implies (3): let f, E, g be as in (3) , and suppose g(E) is infinite. In particular, for any finite F ⊆ acl(∅) there exists d′ ∈ g(E) \ F. Thus the family consisting of g(E) and the complement of all finite definable sets has nonempty intersections of finite subfamilies, so by the compactness theorem, in some M ⊧ T there exists d′ ∈ g(E) \ acl(∅).

Let d ∈ E(M) be such that d′ = g(d). Then d′ ∈ acl(f(d)), but f(d) ∈ D, contradicting (4). Thus (4) implies (3).

The equivalence of (1),(2) is similar. □

Example 2.7. Let P be a complete type, and f a definable function. Then f(P) is a complete type P′. If f is injective on P then there exist definable D ⊇ P, D′ ⊇ P′ such that f restricts to a bijection of D with D′.

Proof. For any definable D′, f^{−1}D′ is definable, so P ⊆ f^{−1}D′ or P ∩f^{−1}D′ = ∅. Thus P′ ⊆ D′ or P′ ∩ D′ = ∅. So P′ is complete.

Let {D_{i}} be the family of definable sets containing P. Let R_{i} = {(x, y) ∈ D_{i}^{2} : x ≠ y, f(x) = f(y)}. Then ∩_{i}R_{i} = ∅. Since the family of {D_{i}} is closed under finite intersections, it follows from the compactness theorem that for some i, R_{i} = ∅. Let D = D_{i}, D′ = f(D). □

2.1.4. Naming almost definable sets. As special case of an expansion by constants, we can move from a complete theory T to the theory T_{A}, where A = acl(∅) is the set of all algebraic elements of a model M of T, including imaginaries. The effect is a theory where each class of any definable equivalence relation E with finitely many classes is definable. Since T is complete, the isomorphism type of acl(∅) in a model M does not depend on the choice of model; so the theory T_{A} is determined. A definable set in this theory corresponds to an almost definable set in T.

When D is a constructible set, T |D denotes the theory induced on D: If T eliminates quantifiers, the language is just the restriction to D of the relations and functions of L. If the language is countable, the countable models of D_{A} are of the form D(M), where M is a countable model of T_{A}.)

2.1.5. Stable embeddedness. A definable subset D of any product of sorts (possibly imaginary) is called stably embedded (in T) if for any A, any T_{A}-definable subset of D^{m} is T_{B}-definable for some B ⊂ D. For example, the set of open balls is not stably embedded in ACVF, since the set of open balls containing a point a ∈ K cannot in general be defined using a finite number of balls. Lemma 2.8. Let D be a family of sorts of L; let T |D be the theory induced on the sorts D; If D is stably embedded and T |D admits elimination of imaginaries, then for any definable P and definable S ⊂ P × D^{m}, viewed as a P -indexed family of subsets S(a) ⊆ D^{m}, a ∈ P, we have a definable function f : P → D^{n}, with f(a) a canonical parameter for S(a).

Proof. By stable embeddedness there exists a family S′ ⊂ P′ × D^{m} yielding the same family, i.e. {S(a) : a ∈ P } = {S′(a′) : a′ ∈ P′}, and with P′ ⊆ D^{n}; using elimination of imaginaries we can take S′ to be a canonical family; now a define f(a) to be the unique a′ ∈ P′ with S(a) = S′(a′). □

Corollary 2.9. If D is stably embedded and admits elimination of imaginaries, then for any substructure A,

(1) (T_{A})|D = (T |D)_{A∩D}

(2) for a ∈ A, tp(a/A ∩ D) implies tp(a/D).

□

Examples of definable sets of ACVF satisfying the hypotheses include the residue field k, or the value group Γ, as well as RV ∪ Γ. The stable embeddedness in this case is an immediate consequence of quantifier elimination; cf. Lemma 3.30.

If M is saturated and D is stably embedded in T, then we have an exact sequence

1 → Aut(M/D(M)) → Aut(M) → Aut(D(M)) → 1

where Aut(M/D(M)) is the group of automorphisms of M fixing D(M) pointwise, and Aut(D(M)) is the group of permutations of D(M) preserving all definable relations. Moreover Aut(M/D(M)) is has a good Galois theory; in particular elements with a finite orbit are almost definable over some finite subset of D. This and some other characterizations can be found in the appendix to [5].

2.1.6. Generic types. Let T be a complete theory with quantifier-elimination. Let 𝒞 be the category of substructures of models of T, with L-embeddings, and let 𝒮 be the category of pairs (A, p) with A ∈ Ob 𝒞 and p a type over A. We define Mor ((A, p), (B, q)) = {f ∈ Mor_{𝒞} (A, B) : f^{∗}(q) = p}.

By a generic type we will mean a function p on Ob 𝒞, denoted A ↦ (p|A), such that A ↦ (A, p|A) is a functor 𝒞 → 𝒮. One example, when T is the theory of algebraically closed fields, is provided by any absolutely irreducible variety V: given a field F, let p|F be the type of an F - generic point of V, i.e. the type of a point of V (L) avoiding U(L) for every proper F -subvariety U of V, where L is some extension field of F. Other examples will be given below, beginning with Example 3.3.

Lemma 2.10. Let p be a generic type of T, and let M ⊧ T, a, b ∈ M. Let c ⊧ p|M.

(1) If a ∉ dcl(∅), then a ∉ dcl(c).

(2) If a ∉ acl(∅), then a ∉ acl(c).

(3) If a ∉ acl(b), then a ∉ acl(b, c).

Proof. (1) Since a ∉ dcl(∅), there exists a′ ≠ a with tp(a) = tp(a′). Let c′ ⊧ p| < {a, a′} >. Since tp(a) = tp(a′), there exists an isomorphism < a >→< a′ >; by functoriality of p, tp(a, c) = tp(a′, c). If a ∈ dcl(c), then a is the unique realization of tp(a/c), so a = a′; a contradiction.

(2) If a ∈ acl(c), then for some n there are at most n realizations of tp(a/c). Since a ∉ acl(∅), there exist distinct realizations a_{0}, …, a_{n} of tp(a). Proceed as in (1) to get a contradiction.

(3) By (2) for T_{<b>}. □

2.2. Grothendieck rings. .

We define the Grothendieck group and associated objects of a theory T; cf. [10]. Def(T) is the category of definable sets and functions. Let 𝒞 be a subcategory of Def(T). We assume Mor(X, Y) is a sheaf on X: if X_{1} = X_{2} ∪ X_{3} are subobjects of X, and f_{i} ∈ Mor(X_{i}, Y) with f_{1}|(X_{2} ∩ X_{3}) = f_{2}|(X_{2} ∩ X_{3}), then there exists f ∈ Mor(X_{1}, Y) with f|X_{i} = f_{i}. Thus the disjoint union of two constructible sets in Ob𝒞 is also the category theoretic disjoint sum.

If only the objects are given, we will assume Mor 𝒞 is the collection of all definable bijections between them.

The Grothendieck semigroup K_{+}(𝒞) is defined to be the semigroup generated by the isomorphism classes [X] of elements X ∈ Ob 𝒞, subject to the relation:

[X] + [Y] = [X ∪ Y] + [X ∩ Y]

In most cases, 𝒞 has disjoint unions; then the elements of K_{+}(𝒞) are precisely the isomorphism classes of 𝒞.

If 𝒞 has Cartesian products, we have a semiring structure given by:

[X][Y] = [X × Y]

In all cases we will consider when products are present, the symmetry isomophism X × Y → Y × X will be in the category, as well as the associativity morphisms, so that K_{+}(𝒞) is a commutative semiring.

(The assumption on Cartesian products is taken to include the presence of an object {p} = X^{0} such that the bijections X → {p} × X, x ↦ (p, x), and X → X × {p}, x ↦ (x, p), are in Mor_{𝒞} for all X ∈ Ob_{𝒞}. All such p gives the same element 1 = [{p}] ∈ K(𝒞), which serves as the identity element of the semiring.)

Let K(𝒞) be the Grothendieck group, the formal groupification of K_{+}(𝒞). When 𝒞 has products, K(𝒞) is a commutative ring.

We will often have dimension filtrations on out categories, and hence on the semi-ring.

By an semi-ring ideal we mean a congruence relation, i.e. an equivalence relation on the semiring R that is a sub-semi-ring of R ×R. To show that an equivalence relation E is a congruence on a commutative semi-ring R, it suffices to check that if (a, b) ∈ E then (a + c, b + c) ∈ E and (ac, bc) ∈ E.

Remark When T is incomplete, let S be the (compact, totally disconnected) space of completions of T. Then {K(t) : t ∈ S} are the fibers of a sheaf of rings over S. K(T) can be identified with the ring of continuous sections of this sheaf. In this sense, Grothendieck rings reduce to the case of complete theories.

This last remark is significant even when T is complete: if one adds a constant symbol c to the language, T becomes incomplete, and so the Grothendieck ring of T in L(c) is the Boolean power of K(T_{a}), where T_{a} ranges over all L(c)-completions of T. Say c is a constant for an element of a sort S. Then an L(c)-definable subset of a sort S′ corresponds to an L-definable subset of S × S′. This allows an inductive analysis of the Grothendieck ring of a structure, given good information about definable sets in one variable (cf. Lemma 2.3).

2.2.1. Groups of functions into ℛ. Let 𝒞(T) be a subcategory of the category of definable sets and bijections, defined systematically for T and for expansions by constants T. Let ℛ(T) = K_{+}(𝒞(T)) be the Grothendieck semigroup of 𝒞(T). When V is a definable set, we let 𝒞_{V}, ℛ_{V} denote the corresponding objects over V; the objects of 𝒞_{V} are definable sets X ⊆ (V × W) such that for any a ∈ V, X_{a} ∈ 𝒞_{a}, and similarly the morphisms. In practice, ℛ will be the Grothendieck semgroup of all definable sets and definable isomorphisms satisfying some definable conditions, such as a boundedness condition on the objects, or a “measure preservation” condition on the definable bijections.

To formalize the notion of of “definable function into ℛ” we will need to look at classes X_{a} of parametrically definable sets. The class of X_{a} makes sense only in the Grothendieck groups associated with T_{a}, not T. Moreover equality of such classes, say of X_{a} and of X_{b}, begins to make sense only in Grothendieck groups of T_{(a,b)}. Expressions like

[X] =_{a,b} [Y]

will therefore mean: X, Y are both definable in T_{a,b}, [X], [Y] denote their classes in the Grothendieck group of T_{a,b}, and these classes are equal.

If V is a definable set, we define the semigroup of definable functions V → ℛ, denoted F n(V, ℛ). An element of F n(V, ℛ) is represented by a definable X ∈ 𝒞_{V}, viewed as the function: a ↦ [X_{a}], where [X_{a}] is a class in ℛ_{a}. X, X′ represent the same function if for all a, [X_{a}], [X′_{a}] are the same element of ℛ_{a}. Note that despite the name, the elements of F n(V, ℛ) should actually be viewed as sections V → Π_{a∈V} ℛ_{a}.

Addition is given by disjoint union in the image (i.e. disjoint union over X.)

Usually ℛ has a natural grading by dimension; in this case F n(V, ℛ) inherits the grading. Assume V is a definable group and ℛ = K_{+}(T) is the Grothendieck semiring of all definable

sets and functions of T, there is a natural convolution product on F n(V, ℝ). If h_{i}(a) = [H_{i}(a)], H_{i} ⊂ V × B_{i}, the convolution h_{1} ∗ h_{2} is represented by

H = {(a_{1} + a_{2}, (a_{1}, a_{2}, y_{1}, y_{2})) : (a_{i}, y_{i}) ∈ H_{i}} ⊆ V × (V^{2} × B_{1} × B_{2})

so that h_{1} ∗ h_{2}(a) = H(a) = {(a_{1}, a_{2}, y_{1}, y_{2}) : (a_{i}, y_{i}) ∈ H_{i}, a_{1} + a_{2} = a}.

2.2.2. Grothendieck groups of orthogonal sets.

Lemma 2.11. Let T be a theory with two strongly orthogonal definable sets D_{1}, D_{2}, D_{12} = D_{1}× D_{2}. Let K_{+}D_{i}[n] be the Grothendieck semigroup of definable subsets of D_{i}^{n}. Then K_{+}D_{12}[n] ≃ K_{+}D_{1}[n]⊗K_{+}D_{2}[n]

Proof. This reduces to n = 1. Given definable sets X_{i} ⊆ D_{i}^{n}, it is clear that the class of X_{1} ×X_{2} in K_{+}D_{12}[n] depends only on the classes of X_{i} on D_{i}[n]. Define [X_{1}]⊗[X_{2}] = [X_{1} × X_{2}]. This is clearly ℤ-bilinear, and so extends to a homomorphism η : K_{+}D_{1}[1] × K_{+}D_{2}[1] → K_{+}D_{12}[1]. By strong orthogonality, η is surjective.

To prove injectivity, note that any element of K_{+}D_{1}[n]⊗K_{+}D_{2}[n] can be written ∑[X_{1}^{i}]⊗[X_{2}^{i}], with X_{1}^{1}, …, X_{1}^{k} pairwise disjoint. To see this, begin with some expression ∑[X_{1}^{i}]⊗[X_{2}^{i}]; use the relation [X′ ∪^{.} X′′]⊗[Y] = [X′]⊗[Y] + [X′′]⊗[Y] to replace the X_{1}^{i} by the atoms of the Boolean algebra they generate, so that the new X_{i}^{i} are equal or disjoint; finally use the relation [X′⊗Y′] + [X′⊗Y′′] = [X′]⊗[Y′ ∪^{.} Y′′] to amalgamate the terms with equal first coordinate.

Hence it suffices to show that if [∪_{i}X_{1}^{i} ×X_{2}^{i}] = [∪_{j}(Y_{1}^{i} ×Y_{2}^{i}], with the X_{1}^{i} and the Y_{1}^{i} pairwise disjoint, then ∑[X_{1}^{i}] × [X_{2}^{i}] = ∑[Y_{1}^{i}] × [Y_{2}^{i}]. Let F : ∪_{i}X_{1}^{i} × X_{2}^{i} → ∪_{j}Y_{1}^{i} × Y_{2}^{i} be a definable bijection. By strong orthogonality, the graph of F is a disjoint union of rectangles. Since F is a bijection, it is easy to see that each of these rectangles has the form f_{1}^{k} × f_{2}^{k}, where for ν = 1, 2, f_{ν}^{k} : X_{ν}(k) → Y_{ν}(k) is a bijection from a subset of ∪_{i}X_{ν}^{i} to a subset of ∪_{j}Y_{ν}^{j}. The rest follows by an easy combinatorial argument; we omit the details, since a somewhat more complicated case will be needed and proved later, see Proposition 10.2. □

2.2.3. Integration by parts. The following will be used only in §9, to study the Grothendieck semiring of the valuation group. Definition 2.12. Let us say that Y ∈ Ob 𝒞 is treated as discrete if for any X ∈ Ob 𝒞 and any definable F ⊂ X × Y such that T ⊧ F is the graph of a function, the projection map F → X is an invertible element of Mor_{𝒞} (F, X).

To explain the terminology, suppose each X ∈ Ob_{𝒞} is endowed with a measure μ_{X}, and 𝒞 is the category of measure preserving maps. If μ_{Y} is the counting measure, and μ_{X×Y} is the product measure, then for any function f : X → Y, x ↦ (x, f(x)) is measure preserving.

We will assume 𝒞 is closed under products.

If Y_{1}, Y_{2} are treated by 𝒞 as discrete, so is Y_{1} × Y_{2}: if F ⊂ X × (Y_{1} × Y_{2}) is the graph of a function X → (Y_{1} × Y_{2}), then the projection to F_{1} ⊂ X × Y_{1} is the graph of a function, hence the projection F_{1} → X is in 𝒞; now F ⊂ (F_{1} × Y_{2}) is the graph of a function, and so F → F_{1} is invertibly represented too; thus so is the composition.) In particular if Y is discretely treated, any bijection U → U′ between subsets of Y^{n} is represented in 𝒞.

If ℛ is a Grothendieck group or semigroup, we write [X] =_{R} [Y] to mean that X, Y have the same class in ℛ. Lemma 2.13. Let f, f′ ⊂ X × L be objects of 𝒞 such that [f(c)] =_{K(𝒞_{c})} [f′(c)] for any c in X.

Then [f] =_{K(𝒞)} [f′]. Similarly for K_{+}.

Proof. By assumption, there exists g(c) such that f(c) + g(c), f′(c) + g(c) are 𝒞_{c}-isomorphic. By compactness (cf. end of proof of Lemma 2.3) this must be uniform (piecewise in L, and hence by glueing globally): there exists a definable g ⊂ Z × L and a definable isomorphism f + g ≃ f′ + g, inducing the isomorphisms of each fiber. By definition of 𝒞_{c}, and since 𝒞 is closed under finite glueing, f + g, f′ + g are in Ob 𝒞 and the isomorphism between them is in Mor 𝒞. □

Let L be an object of 𝒞, treated as discrete in 𝒞, and assume given a definable partial ordering on L.

Notation 2.14. Let f ⊂ X × L. For y ∈ L, let f(y) = {x : (x, y) ∈ f}. Denote: ∑_{γ<β} f(γ) = [{(x, y) : x ∈ f(y), y < γ}]

Notation 2.15. Let ϕ : L → K(X) be a constructible function, represented by f ⊂ X × L; so that ϕ(y) = [f(y)], f(y) = {x : (x, y) ∈ f}. Denote: ∑_{γ<β} ϕ(γ) = [{(x, y) : x ∈ f(y), y < γ}]

Note by Lemma 2.13 that this is well-defined.

Below, we write fg for the pointwise product of two functions in K(𝒞); [fg(y)] = [f(y)×g(y)].

Lemma 2.16 (Integration by parts). Let Γ be an object of 𝒞, treated as discrete in 𝒞, and assume given a definable partial ordering of Γ. Let f ⊂ X × Γ, F (β) = ∑_{γ<β} f(γ), g ⊂ Y × Γ, G(β) = ∑_{γ≤β} g(γ).

Then

F G(β) = ∑_{γ<β} fG(γ) + ∑_{γ≤β} F g(γ)

Proof. Clearly

F G(β) = Σ_{γ<β,γ′≤β}f(γ)g(γ′)

We split this into two sets: γ < γ′, and γ′ ≤ γ. Now

Σ_{γ<γ′≤β}f(γ)g(γ′) = Σ_{γ′≤β}F (γ′)g(γ′)
Σ_{γ′≤γ<β}f(γ)g(γ′) = Σ_{g<β}f(γ)G(γ′)

The lemma follows. □

This is particularly useful when L is treated as discrete in 𝒞, since then, if the sets f(γ) are disjoint, [f] = [∪_{γ}f_{γ}]. Another version, with G(β) = ∑_{γ<β} g(γ):

F G(β) = ∑_{γ<β}(fG + gF + fg)(γ)

3. Some C-minimal geometry.

We will isolate the main properties of the theory ACVF, and work with an arbitrary theory T satisfying these properties. This includes the rigid analytic expansions ACVF^{R} of [23].

The right general notion, C-minimality, has been introduced and studied in [15]. They obtain many of the results of the present section. Largely for expository reasons, we will describe a slightly less general version; it is essentially minimaility with respect to an ultrametric structure in the sense of [30]. We will use notation suggestive of the case of valued fields; thus denoting the main sort by VF, and a binary function by val(x − y). Some additional assumptions will be made explicit later on.

Let T be a theory in a language L, extending a theory T in a language L. T is said to be T-minimal if for any M ⊧ T, any L_{M}-formula in one variable is T_{M}-equivalent to an L_{M} formula.

More generally, if D is a definable subset of T (i.e. a formula of L), we say that D is Tminimal if if every for any M ⊧ T, any T_{M}-definable subset of D is T_{M}-equivalent to one defined by an L_{M} formula.

Strong minimality. Let L = ∅. The only atomic formulas of L are thus equalities x = y of two variables. T the theory of infinite sets. T-minimality is known as strong minimality; see [1], [28]. A theory T is strongly minimal iff for any M ⊧ T, any T_{M}-definable subset of M is finite or cofinite. For us the primary example of a strongly minimal theory is ACF, the theory of algebraically closed fields.

Let M ⊧ T. If D is strongly minimal, and X a definable subset of D^{∗}, we define the Ddimension of X to be the least n such that X admits a T_{M}-definable map into D^{n} with finite fibers. In the situation we will work in, there will be more than one definable strongly minimal set up to isomorphism, and even up to definable isogeny; in particular there will be the various sets of RES_{M}. However, between any of these, there exists an M-definable isogeny; so the k-dimension agrees with the D-dimension for any of them. We will call it the RES-dimension. It agrees with Morley rank, a notion defined in greater generality, that we will not otherwise need here.

O-minimality. L = {<}, T = DLO the theory of dense linear orders without endpoints. (Cf. [9]) DLO minimality is known as O-minimality, and can also be stated thus: any T_{M}-definable subset of M is a finite union of points and intervals. This also forms the basis of an extensive theory; see [36].

Let D be O-minimal. Then the O-minimal dimension of a definable set X ⊆ D^{∗} is the least n such that X admits a T_{M}-definable map into D^{n} with bounded finite fibers.

The Steinitz exchange principle states that if a ∈ acl(B ∪ {b}) but a ∉ acl(B), then b ∈ acl(B ∪ {a}).

This holds for both strongly minimal and O-minimal structures, cf. [36].

For us the relevant O-minimal theory is DOAG itself. We will occasionally use stronger facts valid for this theory. Quantifier elimination for DOAG implies that Lemma 3.1. (1) any parameterically definable function f of one variable is piecewise

affine; there exists a finite partition of the universe into intervals and points, such that on each interval I in the partition, f(x) = αx + c for some rational α and some definable c.

(2) DOAG admits elimination of imaginaries.

Proof. (1) Follows from quantifier elimination for DOAG.

(2) It follows from (1) that any function definable with parameters in DOAG has a canonical code, consisting of the endpoints of the intervals of the coarsest such partition, together with a specification of the rationals α and the constants c. But from this it follows on general grounds that every definable set is coded (cf. [16] 3.2.2). So DOAG admits elimination of imaginaries. □

C-minimality. Let T = T_{um} be the theory of ultrametric spaces, or equivalently chains of equivalence relations. (cf. [30].)

In more detail, L has two sorts, VF and Γ_{∞}. The relations on Γ_{∞} are a constant ∞ and a binary relation <. In addition L has a function symbol VF^{2} → Γ_{∞}, written val(x − y). T states:

(1) Γ_{∞} is a dense linear ordering with no least element, but with a greatest element ∞.

(2) val(x − y) = ∞ iff x = y.

(3) val(x − y) ≥ α is an equivalence relation; the classes are called closed α-balls. Hence so is the relation val(x − y) > α; whose classes are called open α-balls.

(4) Let Γ = Γ_{∞}\{∞}. For α ∈ Γ, every closed α-ball contains infinitely many open α-balls. A T_{um}-minimal theory will be said to be C-minimal. The notion considered in [15] is a little

more general, but for theories T_{um} they coincide. Since we will be interested in fields, this level of generality will suffice.

A theory T extending ACV F is C-minimal iff for any M ⊧ T, every T_{M}-definable subset of VF(M) is a Boolean combination of open balls, closed balls and points. If T is C-minimal, A ≤ M ⊧ T, and b is an A-definable ball, or an infinite intersection, let p_{A}^{b} be the collection of A-definable sets not contained in a finite union of proper sub-balls of b. Then by C-minimality, p_{A}^{b} is a complete type over A.

Let T be C-minimal. Then in T, Γ is O-minimal; and for any closed α-ball C, the set of open α-sub-balls of C is strongly minimal. Denote it C/(1 + ℳ). (These facts are immediate from the definition.)

Assume T is C-minimal with a distinguished point 0. We define: val(x) = val(x − 0); ℳ = {x : val(x) > 0}. Let 𝔅_{cl} be the family of all closed balls, including points. Among them

are 𝔅_{cl}^{c}_{α}(0) = {x : val(x) ≥ α}. Let RV = ⋃^{•}_{γ∈Γ}B_{γ}^{c}(0)/(1+ℳ), and let rv : VF\{0} → RV and val_{rv} : RV → Γ be the natural map. By an rv-ball we mean an open ball of the form rv^{−1}(c).

The T -definable fibers of val_{rv} are referred to, collectively, as RES_{T}. Later we will fix a theory T, and write RES for RES_{T}; we will also write RES_{A} for RES_{T_{A}}. The unqualified notion “definable”, as well as many derived notions, will implicity refer to T.

A certain notion of genericity plays an essential role in these theories. Example 3.2. Let T be a strongly minimal theory. For any A ≤ M ⊧ T, any A-definable set is finite or has finite complement. Therefore, the collection of co-finite sets forms a complete type. A realization of this type is called a generic element of M, over A. Example 3.3. Let T be an O-minimal theory. For any A ≤ M ⊧ T, any A-definable set contains, or is disjoint from, an infinite interval (b, ∞) for some b ∈ M. The set of A-definable sets containing such an interval is thus a complete type, the generic type of large elements of Γ. Similarly, the set of A-definable sets containing an interval (0, a) with 0 < a is the generic type of small positive elements. More generally, given a subset S ⊆ A S′ = {b ∈ A : (∀s ∈ S)(s < b)}; then the definable sets x > a(a ∈ S), x < b(b ∈ S′) generate a complete type over A, called the type of elements just bigger than S . Definition 3.4. Let T be C-minimal. Let b be a T_{A} -definable ball, or an infinite intersection of balls. The generic type p_{b} of b is defined by: p_{b}|A′ = p_{A′}^{b}, for any A ≤ A′ ≤ M ⊧ T.

The completeness follows from C-minimality, since for any A′-definable subset S of b, either S is contained in a finite union of proper sub-balls of b, or else the complement b\S is contained in such a finite union.

A realization of p_{b}|A′ is said to be a generic point of b over A′. An A′- definable set is said to be b-generic if it contains a generic point of b over A′.

See §3.2 for some generalities about generic types. For our purposes it will suffice to consider generic types in one VF variable. For more information see [16] §2.5. Remark 3.5. If A = acl(A) then any type of a field element tp(c/A) coincides with p_{b}|A, where b is the intersection of all A-definable balls containing c.

This is intended to include the case of closed balls of valuative radius ∞, i.e. points; these are the algebraic types x = c. Note also the degenerate case that c is not in any A-definable ball; then b = VF and tp(c/A) is the generic type of VF over A.

Not every generic 1-type is of the form p_{b} for a ball b as above. For instance, let b be an open ball, c ∈ b; then the generic type p_{b}((x − c)^{−1}) is not of this form.)

For V-minimal theories (defined below) it can be shown that every generic 1-type is of the form p_{b} or p_{b}((x − c)^{−1}).

Let T be a C-minimal theory. Let b be a definable ball, or an infinite intersection of definable balls. We say that b is centered if it contains a proper definable finite union of balls. If b is open, or a properly infinite intersection of balls, we have:

(*) If b contains a proper finite union of balls, then it contains a definable closed ball (the smallest closed ball containing the finite set.)

For C-minimal fields of residue characteristic 0, (*) is true of closed balls: the set of maximal open sub-balls of b forms an affine space over the residue field k, where the center of mass of a finite set is well-defined.

Clearly b is centered over acl(A) if and only if it is centered over A. The term “centered” will be justified to some extent by the assertion of Lemma 3.39, that when A is generated by elements of VF ∪ RV ∪ Γ, any A-definable closed ball contains an A-definable point, and thus a centered ball has a definable “center”. Lemma 3.6. b is centered over A iff b is not transitive over A.

This is immediate from the definition, and from C-minimality, since any proper definable subset would have to be a Boolean combination of balls.

An often useful corollary of C-minimality: Lemma 3.7. Let T be C-minimal, X a definable subset of VF, and Y a definable set of disjoint balls. Then for all but finitely many b ∈ Y, either b ⊆ X or b ∩ X = ∅.

Proof. X is a finite Boolean combination of balls, so it suffices to prove this when X is a ball; then X is contained in at most one ball b ∈ Y; for any other b ∈ Y, either b ⊆ X or b ∩ X = ∅. □

Lemma 3.8. Let (b_{t} : t ∈ Q) be a definable family of pairwise disjoint balls. Then for any nonalgebraic t ∈ Q, b_{t} is transitive over < t > Proof. Consider a definable R′ ⊆ Q × VF with R′(t) ⊆ b_{t}. Let Y = ∪_{t∈Q}R′(t). Then Y is a definable subset of VF, hence a finite combination of a finite set H of balls. The b_{t} are pairwise disjoint, so at most finitely many can contain an element of H, and thus no nonalgebraic b_{t} contains an element of H. Thus each ball in H is disjoint from, or contains, any given b_{t}. It follows that Y is disjoint from, or contains, any given b_{t}. Thus b_{t} ∩ Y cannot be a non-empty proper subset of b_{t}. □ Internalizing finite sets. The following lemma will be generalized later to finite sets of balls. It is of such fundamental importance in this paper that we include it separately in its simplest form. The failure of this lemma in residue characteristic p > 0 is the main reason for the failure of the entire theory to generalize, in its present form. Recall the definition of RV, §2.1.

Lemma 3.9. Let T be a C-minimal theory of fields of residue characteristic 0 (possibly with additional structure), A ≤ M ⊧ T. Let F be a finite T_{A}-definable subset of VF^{n}. Then there exists F′ ⊆ RV^{m}, and a T_{A}-definable bijection h : F → F′.

Proof. First consider F = {c_{1}, …, c_{n}} ⊆ VF. Let c = (∑_{i=1}^{n} c_{i})/n be the average; then F is T_{A}-definably isomorphic to {c_{1} − c, …, c_{n} − c}. Thus we may assume the average is 0. If there is no nontrivial A-definable equivalence relation on F, then val(x − y) = α is constant on x ≠ y ∈ F. In this case rv is injective on F and one can take h = rv. Otherwise, let E be a nontrivial A-definable equivalence relation on F. By an E-symmetric polynomial, we mean a polynomial H(x_{1}, …, x_{n}) with coefficients in A, invariant under the symmetric group on each E-class. For any such H, H(F) is a T_{A}-definable set with < n elements. There exists H such that H(F) has more than one element. By induction, there exists an injective A-definable function h_{0} : H(F) → RV^{m}. Let h_{1} = h_{0} ∘ H. For d ∈ h_{0}(H(F)), and d′ = h_{0}^{−1}d, let F_{d} = H^{−1}h_{0}^{−1}(d) = H^{−1}(d′). By induction again, there exists an A(d) = A(d′)- definable injective function g_{d} : F_{d} → RV^{m′}. (We can take the same m′ for all d.) Define h(x) = (h_{1}(x), g_{h_{1}(x)}(x)). Then clearly h is A-definable and injective.

The case F ⊆ VF^{n} follows using a similar induction, or by finding a linear projection with ℚ-coefficients VF^{n} → VF which is injective on F. □

3.1. Basic geography of C-minimal structures. Let T be a C-minimal theory. We begin with a rough study of the existence and non-existence of definable maps between various regions of the structure: k, Γ, RV, VF and VF/𝒪.

We will occasionally refer to stable definable sets.

A definable set D of a theory T is called stable if there is no model M ⊧ T and M-definable relation R ⊆ D^{2} and infinite subset J ⊆ M(D) such that R ∩ J^{2} is a linear ordering. This is a model-theoretic finiteness condition, greatly generalizing finite Morley rank, and in turn strong minimality (cf. [28].)

It is shown in [16], that a definable subset of ACVF_{A}^{eq} is stable if and only if it has finite Morley rank, if an only if it admits no parametrically definable map onto an interval of Γ; and this is if and only it embeds, definably over acl(A), into a finite dimensional k-vector space. These vector spaces have the general form Λ /ℳΛ, with Λ ≤ VF^{n} a lattice. Within the sorts we are using here, the relevant ones are the finite products of vector spaces of RES. More generally in a C-minimal structure with sorts VF, RV, all stable sets are definably embeddable (with parameters) into RES. We will however make no use of these facts, beyond justifying the terminology. Thus “X is a stable definable set” can simply be read as “ there exists a definable bijection between X and a subset of RES^{∗}”.

The first fact is the unrelatedness of k and Γ.

Lemma 3.10. Let Y be a stable definable set. Then Y, Γ are strongly orthogonal. In particular, any definable map from Y to Γ has finite image.

Proof. We prove the second statement first: let M ⊧ T. Let f : Y → Γ be an M-definable map. Then f(Y) is stable, and linearly ordered by <_{Γ}; hence by definition of stability it is finite.

Let γ = (γ_{1}, …, γ_{m}) ∈ Γ. We have to show that for a Y -generated structure A, tp(γ) implies tp(γ/A). It suffices to show that for any a_{,} …, a_{n} ∈ A, tp(γ_{i}/ < γ_{1}, …, γ_{i−1} >) implies tp(γ_{i}/ < γ_{1}, …, γ_{i−1}, a_{1}, …, a_{n} >), for each i. By passing to T_{<γ_{1},...,γ_{i−1}>} we may assume m = 1, γ ∈ Γ. Similarly we may assume n = 1; let a = a_{1} ∈ Y. To show that tp(γ) implies tp(γ/a), it suffices to show that any T_{a}-definable subset of Γ is definable. By O-minimality, any T_{a}-definable subset of Γ is a finite union of intervals, so (in view of the linear ordering) it suffices to show this for intervals (c_{1}, c_{2}). But if the interval is T_{a}-definable then so are the endpoints, so c_{i} = c_{i}(a) is a value of a definable map Y → Γ. But such maps have finite images, so c_{i} lies in a finite definable set. Using the linear ordering, we see that c_{i} itself is definable, and hence so is the interval. □

Lemma 3.11. There are no definable sections of val_{rv} : RV → Γ over an infinite subset of Γ. In fact if Y ⊂ RV^{n} is definable and val_{rv} is finite-to-one on Y, then Y is finite.

Proof. Looking at the fibers of the projection of Y to RV^{n−1}, and using induction, we reduce the lemma to the case n = 1. In this case, by Lemma 3.7, every definable set is a Boolean combination of pullbacks by val_{rv} of subsets of Γ, and finite sets. □

Lemma 3.12. Let M ⊧ T and let Y ⊂ 𝔅_{cl}^{n} be an infinite definable set. Then there exists a surjective M-definable map of Y to a proper interval in Γ.

Proof. Since Γ is O-minimal, any infinite M-definable subset contains a proper interval. Thus it suffices to find an M-definable map of Y onto an infinite subset of Γ.

If the projection of Y to 𝔅_{cl}^{n−1} as well as every fiber of this projection are finite, then Y is finite. Otherwise, replacing Y by one of the fibers or by the projection, we reduce inductively to the case n = 1.

Let v(y) ∈ Γ be the valuative radius of the ball y. Then v(Y) is an M-definable subset of Γ. If it is infinite, we are done; otherwise we may assume all elements of Y have the same valuative radius γ.

Let W = ∪Y. By C-minimality, W is a Boolean combination of balls b_{i} (open, of valuative radius < γ, or closed, of valuative radii δ_{i} ≤ γ.) If W contains some W′ = b_{i} \ (b_{j_{1}} ∪ ⋯ ∪ b_{j_{l}}), where b_{j_{i}} is a proper sub-ball of b_{i}, and δ_{i} < γ, pick a point c in W′; then for any δ with γ > δ > δ_{i} there exists c′ ∈ W′ with val(c − c′) = δ. It follows that the balls b_{γ}(c), b_{γ}(c′) of radius γ around c, c′ are both in Y; but infinitely many such δ exist; fixing c, we obtain a map β_{γ}(c′) ↦ val(c − c′) into an infinite subset of Γ.

Otherwise, W can only be a finite set of balls of valuative radius γ. So Y is finite. □

Corollary 3.13. 𝔅_{cl}^{n} contains no stable definable set. In particular VF contains no strongly minimal set. □

By contrast,

Lemma 3.14. Any infinite definable subset of RV^{n} contains a strongly minimal M-definable subset.

Proof. By Lemma 3.11, the inverse image of some point in Γ^{n} must be infinite. □

Lemma 3.15. Let M ⊧ T. Let Y ⊆ 𝔅_{cl} be a definable set. Let rad(y) be the valuative radius of the ball y. Then either rad : Y → Γ is finite-to-one, or else there exists an M- definable map of an M-definable Y′ ⊆ Y onto a strongly minimal set.

Proof. If rad is not finite-to-one, then Y contains an infinite set Y′ of balls of the same radius α. Then ∪Y′ contains a closed ball b of valuative radius β < α. The set S of open sub-balls b′ of b of valuative radius β forms a strongly minimal set; the map sending y ∈ Y′ to the unique b′ ∈ S containing y is surjective. □

The following lemma regarding VF/𝒪 will be needed for integration with an additive character (§11).

Lemma 3.16. Let Y be a stable definable set, Z ⊂ VF × Y a definable set such that for y ∈ Y, Z(y) = {x : (x, y) ∈ Z} is additively ℳ invariant. Then for all but finitely many 𝒪-cosets C, Z ∩ (C × Y) is a rectangle C × Y′.

Proof. For y ∈ Y, Z(y) is a T_{y}-definable subset of VF, hence a Boolean combination of a finite < y >-definable set of balls b_{1}(y), …, b_{k}(y). Let B_{i}(y) be the smallest closed ball containing b_{i}(y). According to Lemma 3.13, since the set of closed balls occuring as B_{i}(y) for some y is stable, it is finite:

{B_{i}(y) : y ∈ Y } = {B_{1}, …, B_{l}}

All the B_{i} are 𝒪-invariant. Let R be the set of 𝒪-cosets C that are equal to some B_{i}.

If B_{i}(y) has valuative radius < 0 (i.e. it is bigger than an 𝒪-coset), then so is b_{i}(y), so the characteristic function of such a b_{i}(y) is constant on any closed 𝒪-coset C. If C ∉ R, then it is disjoint from any B_{i} of valuative radius equal to (or greater than) 0, so the characteristic functions of the corresponding b_{i}(y) are also constant on it. Thus with finitely many exceptional C, any such characteristic function is constant on C, and the claim follows. □

3.2. Generic types and orthogonality. Two generic types p, q are said to be orthogonal if for any base A′, if c ⊧ p|A′, d ⊧ q|A′, then p generates a complete type over A(d), equivalently q generates a complete type over A(c). We will see that generics of different kinds are orthogonal (cf. Lemma 3.19). This orthogonality of types is weaker than the orthogonality of definable sets mentioned in the introduction, and in the present case is only an indirect consequence of the orthogonality between the residue field and value group; these types do not have orthogonal definable neighborhoods.

If γ ∈ Γ and rk_{ℚ}(Γ(C(a))/Γ(C)) = n, we say that tp(γ/C) has Γ-dimension n.

Lemma 3.17. Let p_{Γ} be a T_{A}- type of elements of Γ^{n} of Γ-dimension n. Let P = val^{−1}(p_{Γ}). Then

(1) val^{−1}(p_{Γ}) is a complete type over A. In other words, for any A-definable set X, either val^{−1}(p_{Γ}) ⊆ X or val^{−1}(p_{Γ}) ∩ X = ∅.

(2) If D is a stable A-definable set and d_{1}, …, d_{n} ∈ D, then P implies a complete type over A(d_{1}, …, d_{n}).

(3) If c ∈ P then D(A(c)) = D(A).

(4) P is complete over A.

Proof. (1) reduces inductively to the case n = 1. Since val^{−1}(p_{Γ}) is a disjoint union of open balls, (1) for n = 1 follows from Lemma 3.7: an A-definable set X cannot intersect nontrivially each of an infinite family of open balls. Therefore either X is disjoint from almost all,or X contains almost all open balls val^{−1}(c), c ⊧ p_{Γ}; in the former case the complement of X contains val^{−1}(p_{Γ}), and in the latter X contains val^{−1}(p_{Γ}) since p_{Γ} is complete.

(2) By strong orthogonality, p_{Γ} generates a complete type q′ over A(d), of Γ-dimension n. By (1) over A(d), val^{−1}(p_{Γ}) is complete over A(d). But if c ∈ P then val(c) ⊧ p_{Γ} so

c ∈ val^{−1}(p_{Γ}). Thus P implies a complete type over A(d).

(3) follows from (2): if d ∈ D(A(c)) then there exists a formula ϕ such that ⊧ ϕ(d, c) and such that ϕ(x, c) has a unique solution. By (2) ϕ is a consequence of P (c) ∪ tp(d/A), and hence by compactness of a formula ϕ_{1}(x)&ϕ_{2}(c) where ϕ_{2} ∈ tp(d/A). So already ϕ_{1}(x) has the unique solution d, and thus d ∈ D(A).

(4) is immediate from (1). □

Lemma 3.18. Let q be a T_{A}-type of elements of RES_{A}^{n} of RES- dimension n. Let Q = rv^{−1}(q). Then Q is complete over A. Moreover, if γ_{1}, …, γ_{m} ∈ Γ then Q implies a complete type over A(γ_{1}, …, γ_{m}).

Proof. Again the lemma reduces inductively to the case n = 1, and for n = 1 follows from Lemma 3.7, since val^{−1}(q) is a union of disjoint annuli; the “moreover” also follows from orthogonality as in the proof of Lemma 3.17 (2). □

Lemma 3.19 ( [16] §2.5). (1) If b is an open ball, or a properly infinite intersection of balls, and b′ a closed ball, then p_{b}, p_{b′} are orthogonal.

(2) Any b-definable map to k is constant on b away from a proper sub-ball of b.

Proof. We recall the proof from [16] §2.5: The statement becomes stronger if the base set is enlarged. Thus we may assume that b and b′ are centered; by translating we may assume both are centered at 0, and by a multiplicative renormalization that b′ is the unit closed ball. So

(*) c ⊧ p_{b′}|A iff c ∈ 𝒪 and res(c) ∉ acl(A).

On the other hand let p_{Γ} be the type of elements of Γ that are just bigger than the valuative radius of b (cf. Example 3.3). Then d ⊧ p_{b}|A iff val(d) ⊧ p_{Γ}, i.e. p_{b} is now the type P described in Lemma 3.17. By Lemma 3.17, if c′ ∈ P then k(A(c′)) = k(A). It follows that if c ⊧ p_{b′}|A then res(c) ∉ acl(k(A(c′)). By (*) c ⊧ p_{b′}|A(c′).

For the second statement, let g be a definable map b → k; by Lemma 3.17 (3), g is constant on the generic type of b; by compactness, g is constant on b away from some proper sub-ball of b. □

Lemma 3.20. Let a = (a_{1}, …, a_{n}) ∈ RV^{n}, and assume a_{i} ∉ acl(A(a_{1}, …, a_{i−1})) for 1 ≤ i ≤ n. Then the formula D(x) = ⋀_{i=1}^{n} rv(x_{i}) = a_{i} generates a complete type over A(a), and indeed over any RV ∪ Γ-generated structure A′′ over A.

In particular, if q = tp(a/A), any A-definable function f : rv^{−1}(q) → RV ∪ Γ factors through rv(x) = (rv(x_{1}), …, rv(x_{n})).

Proof. This reduces inductively to the case n = 1. If we replace A by a bigger set M (such that a_{i} ∉ acl(A(a_{1}, …, a_{i−1})) for 1 ≤ i ≤ n), the assertion becomes stronger; so we may assume A = M ⊧ T. Let rv(c) = rv(c′) = a. Either val(c) = val(c′) ∉ M, or else val(c) = val(c′) = val(d) for some d ∈ M, and res(c/d) = res(c′/d) ∉ M; in either case, by Lemma 3.17 or Lemma 3.18, we have tp(c/M) = tp(c′/M). So tp(c, rv(c)/M) = tp(c′, rv(c′)/M), i.e.tp(c/M(a)) = tp(c′/M(a)). This proves completeness over A(a).

Let A′ be a structure generated over A by finitely many elements of Γ. Then A′(a) = A(γ_{1}, …, γ_{k}, a), where γ_{i} ∈ Γ, and γ_{i} ∉ A(γ_{1}, …, γ_{i−1}, val(a)). It follows that rv(a) ∉ A(γ_{1}, …, γ_{k}), so D(x) generates a complete type over A(γ_{1}, …, γ_{k})(a) = A′(a).

Let A′′ be generated over A′(a) by elements of stable A-definable sets. Since D(x) is the (unique, and therefore) generic type of an open ball over A′(a), by Lemma 3.17, it generates a complete type over A′′.

Now if A′′ = A(γ_{1}, …, γ_{k}, r_{1}, …, r_{n}, d) where γ_{j} ∈ Γ, r_{i} ∈ RV and d lies in a stable set over A, let A′ = A(γ_{1}, …, γ_{k}, val_{rv}(r_{1}), …, val_{rv}(r_{n})); then A′/A is Γ-generated, and A′′/A is generated by elements of stable sets (including val_{rv}^{−1}(r_{i}).) Thus the above applies.

The last statement follows by applying the first part of the lemma over A′′ = A(f(c)): the formula f(x) = f(c) must follow from the formula D(x), since D(x) generates a complete type over A′′. □

3.3. Definable sets in group extensions. We will analyze the structure of RV in a slightly more abstract setting. In the following lemmas we assume R is a ring, and 0 → A → B → C → 0 is a definable exact sequence of R-modules in T. This means that A, B, C are definable sets, and that one is also given definable maps +_{A} : A^{2} → A, f_{A}^{r} : A → A for each r ∈ R, and similarly for B, C; and definable maps ι : A → B, ϑ : B → C, such that in any M ⊧ T, A(M), B(M), C(M) are R-modules under the corresponding functions, and 0 → A(M) →_{ι} B(M) →_{ϑ} C(M) → 0 is an exact sequence of homomorphisms of R-modules.

Lemma 3.21. Consider a theory with a sequence 0 → A → B →_{ϑ} C → 0 of definable Rmodules and homomorphisms (carrying additional structure.) Assume:

(1) A, C are stably embedded and orthogonal.

(2) Every almost definable subgroup of A^{n} is defined by finitely many R-linear equations.

(3) (“No definable quasi-sections”.) If P is a definable subset of B^{n} whose projection to

C^{n} is finite-to-one, then P is finite. Then every almost definable subset Z of B^{n} is a finite union of sets of the form

{b : ϑ(b) ∈ W, Nb ∈ Y }

where N ∈ B_{n,k}(R) is an n × k matrix, Y is an almost definable subset of a single coset of A^{k}, W is an almost definable subset of C^{n}.

Note:

(1) To verify (3), it suffices to check it for n = 1 but for parametrically definable P.

(2) If C is definably linearly ordered, and Z is definable, then Y, W may be taken definable.

Proof. Using a base change as in §2.1.4, we may assume almost definable sets are definable. Replacing B by B^{n} and R by M_{n}(R), we may assume n = 1. Let Z be a definable subset of B. Given X ⊂ A, let [X] denote the class of X up to translation; so [X] = [X′] if X = X′ + a for some a ∈ A. Now a definable subset U of a coset b + A of A has the form b + X, X ⊂ A; the class [X] is well-defined, and we will denote [U] = [X]. We obtain a map

c ↦ [Z ∩ ϑ^{−1}(c)]

In more detail: for any b ∈ (ϑ^{−1}(c) ∩ Z), we have (ϑ^{−1}(c) ∩ Z) − b ⊆ A, and so by stable embeddedness of A we can write (ϑ^{−1}(c) ∩ Z) − b = X(a) for some a ∈ A^{m}. The tuple a is not well-defined; but the class of a in the definable equivalence relation:

x ∼ x′ ⟺ (∃t ∈ A)(t + X(x)) = X(x′)

is obviously a function of c alone. By the orthogonality assumption, this map is piecewise constant. So we may assume it is constant, and fix C_{0} with [Z ∩ ϑ^{−1}(c)] = [C_{0}]. Let S be the stabilizer S = {a ∈ A : a + C_{0} = C_{0}}. Then for a ∈ S, a + (Z ∩ ϑ^{−1}(c)) = (Z ∩ ϑ^{−1}(c))} for any c ∈ C, so that also S = {a ∈ A : a + Z = Z}, and S is definable.

Now Z ∩ ϑ^{−1}(c) = C_{0} + f(c) for some f(c) ∈ ϑ^{−1}(c); f(c) + S is well-defined.

By assumption (2), S = Ker(r_{1}) ∩ … ∩ Ker(r_{m}) for some r_{i} ∈ R. Let I = {r_{1}, …, r_{m}}. For r ∈ I, f_{r}(c) := rf(c) is a well-defined element of B, and for all c ∈ ϑ(Z), r(Z ∩ ϑ^{−1}(c)) = rC_{0} + f_{r}(c).

We have ϑf_{r}(c) = rc. If d ∈ Ker(r : C → C), then f_{r}(d+c) = rc also, so f_{r}(d+c)−f_{r}(c) ∈ A. By orthogonality, for fixed r, f_{r}(d + c) − f_{r}(c) takes finitely many values as c, d vary in C. In other words, {rf(c) : c ∈ ϑ(Z) is a quasi-section above rϑ(Z). By (3), rϑ(Z) is finite, for each r ∈ I. Let N = (r_{1}, …, r_{m}), Y′ = NZ. Then ϑ(Y′) is finite. It follows that Y′ is contained in a finite union of cosets of A, so C, Y′ are orthogonal.

Thus {(ϑ(z), Nz) : z ∈ Z} is a finite union of rectangles; upon dividing Z further, we may assume this set is a rectangle W × Y. Now if ϑ(b) ∈ W and Nb ∈ Y then for some z ∈ Z, ϑ(b) = ϑ(z) and Nb = Nz; it follows that b − z ∈ A and b − z ∈ S; so b ∈ S + Z = Z. Thus Z is of the required form. □

Corollary 3.22. T be a complete theory in a language L satisfying the assumptions of Lemma 3.21. Let L ⊆ L′, T ⊆ T′, and assume (1)-(3) persist to T′. If T, T′ induce the same structure on A and on C, up to constants they induce the same structure on B, i.e. every T -definable subset of B^{∗} is parameterically T′-definable.

Proof. Apply Lemma 3.21 to T′, and note that every definable set in the normal form obtained there is already parametrically definable in T. □

We will explicitly use imaginaries in RV only rarely; but our ability to work with RV, using Γ as an auxiliary, is partly explained by:

Corollary 3.23. Let 0 → A → B →_{ϑ} C → 0 be as in Lemma 3.21, and assume C carries a definable linear ordering. Let V̄ be the disjoint union of the definable cosets of A in B, with structure induced from T. Let e be an imaginary element of B. Then < e >=< (a′, c′) > for some pair (a′, c′) consisting of an imaginary of V̄ and an imaginary of C. Thus if V̄ , C eliminate imaginaries, so does B ∪ C ∪ V̄.

Proof. Let e be an imaginary element of B; let E_{0} be the set of A, V̄ -imaginaries that are algebraic over e.

By Lemma 3.21, applied to a definable set with code e in the theory T_{E_{0}}, there exist almost definable subsets of V̄ , C^{n} from which e can be defined. These are coded by imaginaries permitted in the definition of E_{0}. Thus e is E_{0}-definable. So e = g(d) for some definable function g and some tuple d from E_{0}. Let us now show that e is equi-definable with a finite set, i.e. an imaginary of the form (f_{1}, …, f_{n})/Sym(n). Let W be the set of elements with the same type as d over e; W is finite by definition of E_{0}, and is e-definable. But e = g(w) for any element w ∈ W, so e is definable from {W }.

It remains to see that every finite set of elements of E_{0} is coded by imaginaries of A and C and elements of B. Since C is linearly ordered, it suffices to consider finite sets whose image in C^{m} consists of one point. These are subsets of some definable coset of A^{m}, so again by elimination of imaginaries there they are coded. □

Corollary 3.24. The structure induced on RV ∪ Γ from ACVF eliminates imaginaries.

Proof. Γ_{E_{0}} eliminates imaginaries, and so does ACF (cf. [30]). Note that V̄ is essentially a family of 1-dimensional k-vector spaces, closed under tensor products and roots and duals. Hence by [18], V̄_{E_{0}} eliminates imaginaries too. Our only application of this lemma will be in a situation when parameters can be freely added; in this case, it suffices to quote elimination of imaginaries in ACF. □

Corollary 3.25. Let T be a theory as in Lemma 3.21, with R = ℤ, and C a linearly ordered group. Then every definable subset of B^{n} is a disjoint union of GL_{n}(ℤ)-images of products Y × ϑ^{−1}(Z), with |ϑY | = 1. In particular the Grothendieck semiring K_{+}(B) (with respect to the category of all definable sets and functions of B) is generated by the classes of elements Y ⊂ B^{n} with |ϑY | = 1, and pullbacks ϑ^{−1}(Z), Z ⊂ C^{m}.

Proof. By Lemma 3.21, the Grothendieck ring is generated by classes of sets X of the form X = {b ∈ B^{n} : ϑ(b) ∈ W, Nb ∈ Y }. After performing row and column operations on the matrix N, we may assume it is the composition of a projection p : R^{n} → R^{k} with a diagonal k × k integer matrix with nonzero determinant. The composition ϑp(X) is finite; since C is ordered, each element of ϑp(X) is definable, and so we may assume ϑp(X) has one element e. Thus W = {(e) × W′} for some W′, and X = pX × ϑ^{−1}(W′). □

Lemma 3.26. Let T be a theory, and let 0 → A → B →_{ϑ} C → 0 be an exact sequence o f of definable Abelian groups and homomorphisms. If E ≤ M ⊧ T, we will write E_{A} = A(E), etc. Assume:

(1) A, C are orthogonal.

(2) Any parametrically definable subset of B is a Boolean combination of sets Y with ϑ(Y) finite, and of full pullbacks ϑ^{−1}(Z).

(3) C a uniquely divisible Abelian group, and for any E ≤ M ⊧ T, every divisible subgroup containing E_{C} is algebraically closed in C over E.

(4) For any prime p > 0, T ⊧ (∃x ∈ A)(px = 0, x ≠ 0) Let Z ⊂ C^{n} and f : Z → C be definable, and suppose there exists E and E-definable X ⊂ B^{n} and F : X → B lifting f: ϑX = Z, ϑF (x) = f(ϑx). Then there exists a partition of Z to finitely many definable sets Z_{ν}, such that for each ν, for some m ∈ ℤ^{n}, f(x) − ∑_{i=1}^{n} m_{i}x_{i} is constant on Z_{ν}.

The main point is the integrality of the coefficients m_{i}.

Proof. It suffices to show that for any M ⊧ T and any c = (c_{1}, …, c_{n}) ∈ Z(M), there exists m = (m_{1}, …, m_{n}) ∈ ℤ such that f(c)−mc ∈ E^{0}, where E^{0} = dcl(∅) is the smallest substructure of M. For if so, there exists a formula of one variable of sort C, such that T ⊧ (∃^{≤1}z)ψ(z), M ⊧ ψ(f(c) − mc). By compactness there exists a finite set F of such pairs ν = (m, ψ), such that for any M ⊧ T and c ∈ Z(M), for some (m, ψ) ∈ F, M ⊧ ψ(f(c) − mc); the required partition is given by X_{m,ψ} = {z ∈ Z : ψ(f(z) − mz)}.

Fix M and c ∈ Z(M). Let < c > be the smallest divisible subgroup of C(M) containing E_{C}^{0} and c_{1}, …, c_{n}. By (3), < c > is closed under f, so f(c) ∈< c >, i.e. f(c) = ∑ α_{i}c_{i} + d for some α_{i} ∈ ℚ and some d ∈ E_{C}^{0}. The only problem is to show that we can take α_{i} ∈ ℤ.

We will use induction on n. Let K = {β ∈ ℚ^{n} : β · c ∈ E_{C}^{0}}. K is a ℚ-subspace of ℚ^{n}. If K ≠ (0), there exists a primitive integral vector β_{1} ∈ K. β_{1} may be completed to a basis for a ℤ-lattice in ℚ^{n}. Applying a GL_{n}(ℤ) change of variables to B^{n}, we may assume β_{1} = (1, 0, …, 0), i.e. c_{1} ∈ E_{C}^{0}. But then let f′(z_{2}, …, z_{n}) = f(c_{1}, z_{2}, …, z_{n}). Then f′ lifts to a definable function on B^{n} (with parameters, of the form F (b_{1}, y_{2}, …, y_{n})) so by induction, f(c_{1}, …, c_{n}) = f′(c_{2}, …, c_{n}) = ∑_{i≥2} m_{i}z_{i} + d′ for some m_{2}, …, m_{n} ∈ ℤ and d′ ∈ E_{C}^{0}, as required.

Thus we can assume K = (0).

We can find m, m_{i} ∈ ℤ, e ∈ dcl(∅) with

mf(c) = ∑ m_{i}c_{i} + e

If m|m_{i} we are done. We will now derive a contradiction from the contrary assumption that m does not divide each m_{i} in such an equation, with f a liftable function. We may assume that the greatest common divisor of m, m_{1}, …, m_{n}; so there exists a prime dividing m but not (say) m_{1}.

Let g(x) = f(x, c_{2}, …, c_{n})−e/m−∑_{i=2}^{n} m_{i}c_{i}/m; then mg(c_{1}) = m_{1}c_{1}, m does not divide m_{1}, g is E = acl(c_{2}, …, c_{n})-definable and liftable. Since K = (0), by assumption (3), c_{1} ∉ acl(E). Let E′ ⊃ E be such that g lifts to an E′-definable function G′. Enlarging the model if necessary, let c′_{1} realize tp(c_{1}/E), with c′_{1} ∉ E′ (cf. Example 2.4). Therefore there exists E′′ such that E′′, c_{1} and E′, c′_{1} have the same type. In particular g lifts to an E′′-definable function G.

Consider any b_{1} such that ϑ(b_{1}) = c_{1}. Then mϑG(b_{1})−m_{1}ϑ(b_{1}) = 0. So mG(b_{1})−m_{1}b_{1} ∈ A. Let p be prime, p|m but p |̸m_{1}. Let s, r ∈ ℤ be such that sp − rm_{1} = 1, and let h(x) =

sx − \frac{rm}{p}g(x). Then ph(c_{1}) = psc_{1} − rmg(c_{1}) = psc_{1} − rm_{1}c_{1} = c_{1}. Also h is liftable over E′′: indeed if G is E′′-definable and lifts g, then H(x) = sx − \frac{rm}{p}G(x) lifts h.

So pH(b_{1}) = b_{1} + d, some d ∈ A. Let b_{2} = H(b_{1}); then b_{1} = pb_{2} − d, or

b_{2} = H(pb_{2} − d)

Now let c_{2} = h(c_{1}) = ϑ(b_{2}). Then pc_{2} = c_{1}, and so c_{2} ∉ acl(E′′), since by unique divisibility c_{1} ∈ acl(h(c_{1})). By (1), c_{2} ∉ acl(E′′(d)). Let C_{2} = ϑ^{−1}c_{2}. By (2), any E′′(d)-definable set either contains C_{2} or is disjoint from C_{2}. Hence for any y ∈ C_{2}, y = H(py − d).

By (4) there exists 0 ≠ ω_{p} ∈ A with pω_{p} = 0. Let b′_{2} = b_{2} + ω_{p}. Then b_{2} ∈ C_{2}, so b′_{2} = H(pb′_{2} − d). But pb′_{2} = pb_{2}, so b_{2} = b′_{2} and ω_{p} = 0, a contradiction.

□

Remark 3.27. (1) It follows from Lemma 3.26 that a definable bijection between subsets of C^{n} that lifts to subsets of B^{n} is piecewise given by an element of GL_{n}(ℤ) ⋉ C^{n}. (cf. Lemma 3.28.)

(2) The assumption (4) on torsion does not hold in characteristic p > 0 for the sequence k^{∗} → RV → Γ. In this case there is l-torsion for l ≠ p, but no p-torsion, and the corresponding group is GL_{n}(ℤ[1/p]) ⋉ C^{n}.

Note as a corollary that there can be no definable sections of B → C over an infinite definable subset of C.

Lemma 3.28. Let 0 → A → B → C → 0 be as in Lemma 3.26. Let X ⊂ B^{n} be definable, and let f : X → B^{l} be a definable function. X may be partitioned into finitely many pieces X′, such that on each X′

(1) f(x) = Mx + b(x), where M is a l × n-integer matrix and ϑb(x) is constant.

(2) There exists g ∈ GL_{n}(ℤ) such that b ∘ g factors through a projection B^{n} →_{π} B^{k}, where ϑπ(X′) is one point of C^{k}.

Proof. We first prove (1,2) for complete types.

(1) This reduces to l = 1. Let P be a complete type of elements of X. Then on P we have ϑ ∘ f(x) = ∑ m_{i}ϑ(x_{i}) + d for some constant d. ( Lemma 3.26).

Thus f(x) = ∑ m_{i}x_{i} + b(x), where b(x) = f(x) − ∑ m_{i}x_{i}, and ϑb(x) = d is constant.

(2). Let π : B^{n} → B^{k} be a projection such that ϑπ(X) is one point of C^{k}, and with k maximal. So P ⊂ P′ × P′′, P′ ⊂ B^{n−k}, P′′ ⊂ B^{k}, and ϑ(P′′) is a single point of C^{k}, while ϑ(P′) is not contained in any proper hypersurface ∑ n_{i}x_{i} = constant with n_{i} ∈ ℤ. Pick b′′ ∈ P′′. Let γ = (γ_{1}, …, γ_{k}) ∈ ϑ(P′), γ not in any such hypersurface. Let a = (a_{1}, …, a_{k}), ϑ(a_{i}) = γ_{i}, and let a′ be another point with ϑ(a′) = γ. Let e = f(a, b). Then tp(a/b, e) = tp(a′/b, e), so f(a′, b) = e. Thus f(a, b) depends only on b ∈ P′′ and not on a (with (a, b) ∈ P.)

Since (1),(2) hold on each complete type, there exists a definable partition such that they hold on each piece. □

3.4. V-minimality. We assume from now on that T is a theory of C-minimal valued fields, of residue characteristic 0. When using the many-sorted language, we will still say that T is a theory of valued fields when T = T h(F, RV(F)) for some valued field F, possibly with additional structure. A C-minimal T satisfying assumption (3) below will be said to have centered closed balls. If in addition (1), (2) hold, we will say T is V-minimal. Expansions by definition of the language, i.e. the addition of a relation symbol R(x) to the language along with a definition (∀x)(R(x) ⟺ ϕ(x)) to the theory, do not change any of our assumptions. Thus we can assume that T eliminates quantifiers.

(1) Induced structure on RV T contains ACVF(0, 0), and every parametically Tdefinable relation on RV^{∗} is parametrically definable in ACVF(0, 0).

(2) Definable completeness Let A ≤ M ⊧ T, and let W ⊂ 𝔅 be a T_{A}- definable family of closed balls linearly ordered by inclusion. Then ∩W ≠ ∅.

(3) Choosing points in closed balls Let M ⊧ T, A ⊆ VF(M), and let b be an almost A-definable closed ball. Then b contains an almost A-definable point.

T will be called effective if every definable finite disjoint union of balls contains a definable set, with exactly one point in each. A substructure A of a model of T will be called effective if T_{A} is effective.

If every definable finite disjoint union of rv-balls contains a definable set, with exactly one point in each, we can call T rv-effective. However:

Lemma 3.29. Let T be V-minimal. Then T is effective iff it is rv-effective.

Proof. Assume T is rv-effective. Let b be an algebraic ball. If b is closed, it has an algebraic point by Assumption 3.4 3. If b is open, let b̄ be the closed ball surrounding it. Then b̄ has an algebraic point a. Let f(x) = x − a. Then f(b) is an rv-ball, so by rv-effectivity it has an algebraic point a′. Hence a′ + a is an algebraic point of b. □

In general, effectivity is needed for lifting morphisms from RV to VF, not for the “integration” direction.

If T is V-minimal and A is a VF ∪ RV ∪ Γ - generated structure, we will see that T_{A} is V-minimal too. The analog for points in open balls is true but only for VF ∪ Γ-generated substructures; for thin annuli it is true only for VF-generated structures. For this reason the condition on closed balls is more flexible; luckily we will be able to avoid the others.

Lemma 3.30. Let T be a C-minimal theory of valued fields.

Then (1) ⟹ (2) ⟹ (3) ⟹ (4)

(1) T admits quantifier-elimination in a three-sorted language (VF, k, Γ), such that for any basic function symbol F with range VF, the domain is a power of VF; and no relations on k, Γ beyond the field structure on k and the ordered Abelian group structure on Γ.

(2) Every parametrically definable relation on k is parameterically definable in ACF(0); and every parametrically definable relation on Γ is parameterically definable in DOAG.

(3) Assumption 3.4 (1).

(4) k, Γ and RV are stably embedded.

Proof. (1) ⟹ (2) : Let ϕ(a, x) be an atomic formula with paramaters a = (a_{1}, …, a_{n}) from VF and x = (x_{1}, …, x_{m}) variables for the k, Γ sorts. Then ϕ must have the form ψ(t(a), x), where t is a term (composition of function symbols) VF^{∗} → (k∪Γ). So ϕ(a, x) defines the same set as ψ(b, x) where b = t(a). Since every formula is a Boolean combination of atomic ones, (2) follows.

(2) ⟹ (3) : This follows from Corollary 3.22. The assumptions of Lemma 3.26 are satisfied: (1) is automatic since by C-minimality k is strongly minimal and Γ is O-minimal; (2) follows from C-minimality; (3),(4) follow from the assumptions on k, Γ.

(3) immediately implies (4).

□

Lemma 3.31. Let T be a theory of valued fields satisfying Assumption 3.4 (1), such that res induces a surjective map on algebraic points. Then (1) ⟹ (2) ⟹ (3) ⟹ (4):

(1) for any VF-generated substructure A of a model M of T, if Γ(A) ≠ (0), then acl(A) ⊧ T.

(2) for any VF-generated substructure A of a model of T, any T_{A}-definable nonempty finite union of balls contains a nonempty T_{A}-definable finite set.

(3) Assumption 3.4 (3) holds.

(4) Let A be VF-generated, and Y a finite A-definable set of disjoint closed balls. Then there exists an A-definable finite set Z such that |b ∩ Z| = 1 for each b ∈ Y.

Proof. We first show: Claim For any VF-generated A with Γ(A) = (0), res : VF(acl(A)) → k(acl(A)) is surjective.

Proof. It suffices to prove the claim for finitely generated A. For A = ∅ this is true by assumption. Using induction on the number of generators, it suffices to show that if the Claim holds for A_{0} and c ∈ VF then it holds for A = A_{0}(c)).

Since Γ(A) = (0), res is defined and injective on VF(A). If c ∈ acl(A_{0}) there is nothing to prove. Otherwise, by injectivity, res(c) ∉ acl(A_{0}). As a consequence of Assumption 3.4 (1), both dcl and acl agree with the corresponding field-theoretic notions on RV, and in particular on the residue field.

By Lemma 3.20,

k(A_{0}(c)) ⊆ dcl(RV(A_{0}), rv(c)) = dcl(k(A_{0}), res(c)) = k(A_{0})(res(c))

Now if d ∈ k(acl(A)) then d ∈ k and d ∈ acl(A), so by stable embeddedness of k we have d ∈ acl(k(A)); but acl(k(A)) = k(A)^{alg} by Assumptions 3.4 and 1; so d ∈ k(A_{0})(res(c))^{alg} ⊆ res(A_{0}(c)^{alg}). □

Assume (1). If Γ(acl(A)) ≠ (0) then by (1) acl(A) ⊧ T, and in particular every acl(A)- definable ball has a point in acl(A), so (2) holds. Assume therefore that Γ(acl(A)) = 0. Let b be an acl(A)-definable ball. Then b must have valuative radius 0. If some element of b has valuation γ < 0 then all do, and γ ∈ A, a contradiction. So b is the (open or closed) ball of radius 0 around some c ∈ 𝒪. If b is closed, then b = 𝒪 and 0 ∈ b. If b is open, then b = res^{−1}(b′) for some element b′ of the residue field k; in this case b has an acl(A)-definable point by the Claim.

(3) is included in (2), being the case of closed balls.

Assume (3). In expansions of ACVF(0,0), the average of a finite subset of a ball remains within the ball. Thus if Y is a finite A-definable set of disjoint balls, by (3), there exists a finite A-definable set Z_{0} including a representative of each ball in Y. Let Z = {av(b ∩ Z_{0}) : b ∈ Y }, where av(u) denotes the average of a finite set u.

□

Lemma 3.32. When T is a complete theory, definable completeness is true as soon as T has a single spherically complete model M in the sense of Ribenboim and Kaplansky: every intersection of nested closed balls is nonempty.

Proof. Clear. □

Let ACV F^{an} denote any of the rigid analytic theories of [23]. For definiteness, let us assume the power series have coefficients in ℂ((X)). See [14] for variants living over ℤ_{p}.

Lemma 3.33. ACV F (0, 0) is V-minimal and effective. So is ACV F^{an}.

Proof. C-minimality is proved in [24]. Lemma 3.30(1) for ACVF is a version of Robinson’s quantifier elimination; cf. [16].

ACV F^{an} admits quantifier elimination in the sorts (VF, Γ) by [23] Theorem 3.8.2. The residue field sort is not explicit in this language, but one can argue as follows. Let k_{1} be a large algebraically closed field containing ℂ, and let K = ∪_{n≥1}k_{1}((X^{1/n})) be the Puiseux series ring. Then K admits a natural expansion to a model of the theory. K is not saturated, but by C-minimality the induced structure on the residue field is strongly minimal, so k_{1} is saturated. Now any automorphism of k_{1} as a field extends to an automorphism of K as a rigid analytic structure. Thus every K - definable relation on k_{1} is algebraic. (This could be repeated over a larger value group if necessary.) Lemma 3.30 (2) thus holds in both cases, hence Assumption 3.4 (1).

Condition Lemma 3.31 (1) is obviously true for ACVF. For ACV F^{an} it is proved in [24].) It is also evident that these theories have a spherically complete model. Thus by Lemma 3.31 and Lemma 3.32, Assumption 3.4 (3) and (2) hold too.

□

Remarks

(1) Lemma 3.31 (1,2,3) remain true for ACVF in positive residue characteristic, but (4) fails.

(2) ACVF(0,0) also admits quantifier-elimination in the two sorted language with sorts VF, RV; so Assumption 3.4 (1) can also be proved directly, without going through k, Γ as in Lemma 3.30.

(3) (1) is needed for lifting definable bijections of RV to VF, Proposition 6.1, Lemma 6.3. Specifically it implies the truth of assumptions (2) of Lemma 3.21 and (4) of Lemma 3.26. These lemmas are only needed for the injectiveness of the Euler charactersitic and integration maps, not for their construction and main properties. It is also needed for the theory of differentiation and for comparing derivations in VF and RV; indeed even for posing the question, since in general there is no notion of differentiation on RV. The theory of differentiation itself is needed neither for the Euler characteristic nor for integration of definable sets with a Γ-volume form. They are required only for the finer theory introduced here of integration of RV-volume forms.

(4) We know no examples of C-minimal fields where (2) fails.

(5) Beyond effectivity of dcl(∅), (3), imposes a condition on liftability of definable functions from VF to 𝔅^{cl}. Let T_{1} be the theory, intermediate between ACVF(0, 0) and a Lipshitz rigid analytic expansion, generated over ACVF(0, 0) by the relation

val(f(t_{0}x) − y) ≥ val(t_{1})

on 𝒪^{2} where t_{0}, t_{1} are constants with val(t_{1}) >> val(t_{0}) > 0 and f is an analytic function. It appears that balls do not necessarily remain pointed upon adding VFpoints to T_{1}; so (3) is not redundant.

3.5. Definable completeness and functions on the value group. We assume T is Cminimal and definably complete. We show that the property of having centered closed balls is preserved under passage to T_{A} if A is RV, Γ, VF-generated; similarly for open balls if A is Γ, VF-generated. Also included is a lemma stating that every image of an RV-set in VF must be finite; from the point of view of content this belongs to the description of the ”basic geography”, but we need the lemmas on functions from Γ first.

Proposition 3.34. Let M ⊧ T, γ = (γ_{1}, …, γ_{m}) a tuple of elements of Γ(M). Any almost A(γ)-definable ball b contains an almost A-definable ball b′.

Proof. See [16] Prop. 2.4.4. While the Proposition is stated for ACVF there, the proof uses only C-minimality and definable completeness. We review the proof in the case that b ∈ A(γ), i.e. b = f(γ) for some definable function f with domain D ⊆ Γ^{M}.

Let P = tp(γ/A). Let r(γ) be the valuative radius of f(γ). By O-minimality, r is piecewise monotone; since P is a complete type, r is monotone, say decreasing. For a ∈ P let P_{a} = {b ∈ P : b < a}, and for b ∈ P_{a} let f_{a}(b) be the open ball of size r(a) containing f(b). By Lemma 3.15, the valuative radius map rad is finite-to-one on f_{a}(P_{a}); but by definition it is constant, so f_{a}(P_{a}) is finite. Using the linear ordering, f_{a}(P_{a}) is constant on each complete type over a. Pick b_{1} ∈ P, ϵ ∈ Γ with ϵ > 0 but very small (over A(b_{1})), and ϵ′ ∈ Γ with ϵ′ > 0 but ϵ′ very small (over A(b_{1}, ϵ)). Let b_{2} = b_{1} + ϵ, a = b_{2} + ϵ′. Then tp(b_{1}, a/A) = tp(b_{2}, a/A), so f_{a}(b_{1}) = f_{a}(b_{2}). Now if f(b_{1}), f(b_{2}) are disjoint, let δ = val(x_{1} − x_{2}) for (some or any) x_{i} ∈ f(b_{i}). Then r(b_{2}) > δ. Since ϵ′ is very small, r(a) > δ also. So f_{a}(b_{1}), f_{a}(b_{2}) are distinct, a contradiction. Thus f(b_{1}) ⊂ f(b_{2}). Since tp(a/A) = tp(b_{2}/A), we have f(y) ⊂ f(a) for some y ∈ P_{a}. If f(y) ⊂ f(a) for all y ∈ P_{a}, we are done; otherwise let c(a) be the unique smallest element such that f is monotone on (c(a), a). We saw however that f is monotone on (d, c(a)) for some d < c(a), hence also on (d, a), a contradiction. So f is monotone with respect to inclusion. By compactness, this is true on some A-definable interval hence on some interval I containing P.

Let U = ∩_{a∈I}f(a). By definable completeness (2) U ≠ ∅. Clearly U is a ball, and U ⊆ b. □

Lemma 3.35. Let M ⊧ T, γ = (γ_{1}, …, γ_{m}) a tuple of elements of Γ(M). Then any A(γ)- definable ball contains an A-definable ball. If Y is a finite A(γ)-definable set of disjoint balls, then there exists a finite A-definable set Y′ of balls, such that each ball of Y contains a unique ball of Y′.

Proof. This reduces immediately to m = 1. For m = 1, by Proposition 3.34, any almost A(γ)-definable ball b contains an almost A-definable ball b′. Thus given a finite A(γ)-definable set Y of disjoint balls, there exists a finite A-definable set Z of balls, such that any ball of Y contains a ball of Z. Given b ∈ Y, let b′ be the smallest ball containing every sub-ball c of b with c ∈ Z. Then Y′ = {b′ : b ∈ Y } is A(γ)-definable, finite, almost A-definable, and (since b′_{1} is disjoint from b_{2} if b_{1} ≠ b_{2} ∈ Y) each ball of Y contains a unique ball of Y′. Using elimination of imaginaries in Γ, by Example 2.2, being A(γ)-definable and almost A-definable, Y′ is A-definable. □

The following corollary of Lemma 3.35 concerning definable functions from Γ will be important for the theory of integration with an additive character in §11.

Corollary 3.36. Let Y be a definable set admitting a finite-to-one map into Γ^{n}, and let into h be a definable map on Y into VF or VF/𝒪 or VF/ℳ. Then h has finite image.

Proof. One can view h as a function from a subset of Γ^{n} into finite sets of balls. Since a ball whose radius is definable containing a definable ball is itself definable, Lemma 3.35 implies that h(γ) ∈ acl(∅) for any γ ∈ Γ^{n}. By Lemma 2.6, the corollary follows. □

Corollary 3.37. Let Y ⊆ (RV ∪ Γ)^{n} and Z ⊆ VF × Y be definable sets, with Z invariant for the action of ℳ on VF. Then for all but finitely many 𝒪-cosets C, Z ∩ (C × Y) is a rectangle C × Y′.

Proof. Let p : (RV ∪ Γ)^{n} → Γ^{n} be the natural projection, and for γ ∈ Γ^{n} let Z_{γ} be the fiber. For each γ, by Lemma 3.16, there exists a finite F (γ) ⊆ VF/𝒪 such that for any 𝒪-coset C ∉ F (γ), Z_{γ} ∩ (C × Y) is 𝒪-invariant. Now {(u, γ) : u ∈ F (γ)} projects finite-to-one to Γ^{n}, so by Lemma 3.36, this set projects to a finite subset of VF/𝒪. Thus there exists a finite E ⊂ VF/𝒪 such that for any γ, and any 𝒪-coset C ∉ E, Z_{γ} ∩ (C × Y) is 𝒪-invariant. In other words, for any C ∉ E, Z ∩ (C × Y) is 𝒪-invariant. □

Lemma 3.38. Let M ⊧ T, A a substructure of M (all imaginary elements allowed), and let r = (r_{1}, …, r_{m}) be a tuple of elements of RV(M) ∪ Γ(M). Then any closed ball almost defined over A(r) contains a ball almost defined over A.

Proof. This reduces to m = 1, r = r_{1}; moreover using Lemma 3.35, to the case r ∈ RV(M), val_{rv}(r) = γ ∈ A. Let E = {y ∈ RV : val_{rv}(y) = γ}. Then E is a k^{∗}-torsor, and so is strongly minimal within M. If c is almost defined over A(r), there exists an A-definable set W ⊂ E × 𝔅_{cl}, with W (e) = {y : (e, y) ∈ W } finite, and c ∈ W (r). But then W is a finite union of strongly minimals, hence so is the projection P of W to 𝔅_{cl}. But any strongly minimal subset of 𝔅_{cl} is finite (otherwise it admits a definable map onto a segment in Γ; but Γ is linearly ordered and cannot have a strongly minimal segment.) So c ∈ P is almost defined over A. □

Lemma 3.39. Let M ⊧ T, T C-minimal with centered closed balls. Let B be substructure of VF(M) ∪ RV(M) ∪ Γ(M). Then every B-definable closed ball has a B-definable point. If Y is a finite B-definable set of disjoint closed balls, there exists a finite B-definable set Z ⊂ M, meeting each ball of Y in a unique point.

Proof. We may take B to contain a subfield K and be generated over K by finitely many points r_{1}, …, r_{k} ∈ RV. Let Y be a a finite B-definable set of disjoint closed balls, and let b ∈ Y. We may assume all elements of Y have the same type over B. By Lemma 3.38, there exists a closed ball b′ defined almost over K and contained in b. By Assumption 3.4 3 there exists a finite K-definable set Z′ meeting b′ in a unique point. Let Y′ = {b′′ ∈ Y : b′′ ∩ Z′ ≠ ∅}, and Z = {av(Z′ ∩ b′′) : b′′ ∈ Y′}. Then Z meets each ball of Y′ in a unique point, and Z, Y′ are B-definable. As for Y \ Y′, it may be treated inductively. □

Corollary 3.40. Let M ⊧ T, T C-minimal with centered closed balls, and effective. Let B be an almost Γ-generated substructure. Then T is effective.

Proof. Same as the proof of Lemma 3.39, using Lemma 3.34 in place of Lemma 3.38. □

Lemma 3.41. Let Y be a T - definable set admitting a finite-to-one map into RV^{n}. Let g : Y → VF^{m} be another definable map. Then g(Y) is finite.

Proof. It suffices to prove this for T_{A}, where A ⊧ T. We may also assume m = 1. We will use the equivalence (3) ⟺ (4) of Lemma 2.6. If g(Y) is infinite, then by compactness there exists a ∈ g(Y) a ∉ acl(A). But for some b we have a = g(b), so if c = f(b), we have c ∈ RV^{n}, a ∈ acl(c). Thus it suffices to show:

(*) If a ∈ VF, c ∈ RV^{n} and a ∈ acl(A(c))), then a ∈ acl(A).

This clearly reduces to the case n = 1, c ∈ RV. Let d = val_{rv}(c), A′ = acl(A(d)). Then c lies in an A′-definable strongly minimal set S (namely S = val_{rv}^{−1}(d)). Using Lemma 2.6 in the opposite direction, since a ∈ acl(A′(c))) there exists a finite-to-one map f : S′ → S and a definable map g′ : S′ → VF with a ∈ g′(f^{−1}(S′)). By Corollary 3.13, g′(f^{−1}(S′)) is finite. Hence a ∈ acl(A(d)). But then by Lemma 3.36, a ∈ acl(A). □

In particular, there can be no definable isomorphism between an infinite subset of RV^{n} and one of VF^{m}.

Lemma 3.42. Let M ⊧ T, T C-minimal with centered closed balls, and let A be a substructure of M. Write A_{VF} for the field elements of A, A_{RV} for the RV-elements of A.

Let c ∈ RV(M), and let A(c) = dcl(A ∪ {c}). Then A(c)_{VF} ⊂ (A_{VF})^{alg}, and rv(A(c)_{VF}) ∩ A_{RV} = rv(A_{VF}).

Proof. Let e ∈ A(c)_{VF}. Then e = f(c) for some A-definable function f : W → VF, W ⊆ RV. By Lemma 3.41, the image of f is finite, e ∈ acl(A). This proves the first point. Now if d ∈ RV_{A} and rv^{−1}(d) has a point in A(c), then it has a point in (A_{VF})^{alg}, by Assumption 3.4

(3)  □

3.6. Transitive sets in dimension one. Let b be a closed ball in a valued field. Then the set Aff(b) of maximal open sub-balls of b has the structure of an affine space over the residue field. We will now begin using this structure. Without it, more general transitive annuli (missing more than one ball) could exist.

Lemma 3.43. Let X ⊆ VF be a transitive T_{B}-definable set, where B is some set of imaginaries. Then X is a finite union of open balls of equal size, or a finite union of closed balls of equal size, or a finite union of thin annuli.

Proof. By C-minimality, X is a finite Boolean combination of balls. There are finitely many distinct balls b_{1}, …, b_{n} that are almost contained in X (i.e. b_{i} \ X is contained in a finite union of proper sub-balls of b_{i}) but such that no ball larger than b_{i} is almost contained in X. These b_{i} must be disjoint. If some of the b_{i} have different type than the others, their union (intersected with X) will be a proper B-definable subset of X. Thus they all have the same type over B; in particular they have the same radius β.

Consider first the case where the balls b_{i} are open, then b_{i} ⊆ X: otherwise b_{i} \X is contained in a unique smallest ball c_{i}; say c_{i} has radius α; then α > β. Let b′_{i} be the open ball of radius (1/2)(α + β) around c_{i}; then ∪_{i}b′_{i} is a B-definable proper subset of X, a contradiction. Thus in the case of open balls, X ⊇ ∪_{i}b_{i} and therefore X = ∪_{i}b_{i}.

If the balls b_{i} are closed, let c_{ij} be a minimal finite set of sub-balls of b_{i} needed to cover b_{i} \ X. The same argument shows that no c_{ij} has radius < β. Thus all c_{ij} are elements of the set V_{i} of open sub-balls of b_{i} of radius β. Now V_{i} is a k-affine space, and if there is more than one c_{ij} then over acl(B), V_{i} admits a bijection with k; so there is a finite B- definable set of bijections V_{i} → k; since any finite definable subset of k is contained in a strictly bigger one, the union of the pullbacks gives a B-definable subset of V_{i} properly containing the c_{ij}, leading to a proper B-definable subset of X. Thus either b_{i} ⊆ X (and then X = ∪_{i}b_{i}), or else b_{i} \ c_{i} ⊆ X for a unique maximal open sub-ball c_{i}. Now ∪c_{i} intersects X in a proper subset, which must be empty. So in this case, X = ∪_{i}(b_{i} \ c_{i}). □

Let X be a transitive B-definable set. Call Y ⊆ X potentially transitive if there exists B′ ⊃ B such that Y is B′-definable and B′-transitive. Let ℱ(X) be the collection of all proper potentially transitive subsets Y of X. Let ℱ_{max}(X) be the set of maximal elements of ℱ(X).

Lemma 3.44. (1) If X is an open ball, ℱ_{max}(X) = ∅ (2) If X is a closed ball, ℱ_{max}(X) = {X \ Y : Y ∈ Aff(X)}

(3) If X is a thin annulus X′ \ Y with X′ closed, then ℱ_{max}(X) = Aff(X) \ {Y }

Proof. Any element of ℱ(X) must be a ball or a thin annulus, so the lemma follows by inspection □

Lemma 3.45. Let b be a transitive closed ball (respectively thin annulus). Let Y = Aff(b) be the set of maximal open sub-balls of b. Then the group of automorphisms of Y over k is definable, acts transitively on Y, and in fact contains G_{a}(k) (respectively G_{m}(k)).

If b, b′ are transitive definable closed balls, and F : b → b′ a definable bijection, let F_{∗} : Y (b) → Y (b′) be the induced map. Then F_{∗} is a homomorphism of affine spaces, i.e. there exists a vector space isomorphism F_{∗∗} : V (b) → V (b′) between the corresponding vector spaces, and F_{∗}(a + v) = F_{∗}(a) + F_{∗∗}(v). If b = b′ then F_{∗∗} = Id.

Proof. Y = Aff(b) is transitive, and there is a k- affine space structure on Y (respectively a k-vector space structure on V = Y′ ∪^{.} {0}.) Let G = Aut(Y/k) be the subgroup of the group Aff = (G_{m} ⋉ G_{a})(k) of affine transformations of Y that preserve all definable relations. By definition, this is an intersection of definable subgroups of Aff. However there is no infinite descending chain of definable subgroups of Aff, so G is definable.

If G is finite, then Y ⊆ acl(k), and it follows (cf. §2.1.5 that there are infinitely many algebraic points of Y, contradicting transitivity. Thus G is an infinite subgroup of (G_{m} ⋉ G_{a})(k) such that the set of fixed points Y^{G} is empty. So G must contain a translation, and by strong minimality it must contain G_{a}(k). Similarly in the case of the annulus G is an infinite definable subgroup of G_{m}(k), so it must equal G_{m}(k).

As for the second statement, F induces a group isomorphism Aut(Y (b)/k) → Aut(Y (b′)/k), and hence an isomorphism G_{a}(k) → G_{a}(k), which must be multiplication by some γ ∈ k^{∗}. Since G_{a}(k) acts by automorphisms on (Y (b), Y (b′)), any definable function Y (b) → Y (b′) commutes with this action and hence has the specified form. If b = b′ then Y (b) = Y (b′); now if F_{∗∗} ≠ Id then F_{∗} would have a fixed point, contradicting transitivity. □

Lemma 3.46. Let b be a transitive T_{B}-definable closed (open) ball. Let F be a B-definable function, injective on b. Then F (b) is a closed (open) ball.

Proof. By Lemma 3.43, since F (b) is also transitive, it is either a closed ball, or an open ball, or a thin annulus. We must rule out the possibility of a bijection between such sets of distinct types.

Consider the collection ℱ_{max}(b) defined above. Any definable bijection between b and b′ clearly induces a bijection ℱ_{max}(b) → ℱ_{max}(b′). By Lemma 3.44, the bijective image of an open ball is an open ball.

Let b be a closed ball, b′ = b′′ \ b′′′ a closed ball minus an open ball, A = ℱ_{max}(b) ≃ Aff(b) A′ = ℱ(b′) ≃ Aff(b′′) \ {b′′′}, G = Aut(A/k), G′ = Aut(A′/k). Then a definable bijection A → A′ would give a definable group isomorphism G → G′. But by Lemma 3.45, G′ = G_{m}(k) while G contains G_{a}(k), so no such isomorphism is possible (say because G_{m}(k) has torsion points.)

Thus the three types are distinct. □

We will see later that there can be no definable bijection between an open and a closed ball, whether transitive or not.

Lemma 3.47. Let b be a transitive ball. Then every definable function on b into RV ∪ Γ is constant. If b is a transitive thin annulus, every definable function on b into k ∪ Γ is constant. More generally, this is true for definable functions into definable cosets C of k^{∗} in RV that contain algebraic points.

Proof. When a ball b is transitive, it is actually finitely primitive. For if E is a B-definable equivalence relation with finitely many classes, then exactly one of these classes is generic (i.e. is not contained in a finite union of proper sub-balls of b.) This class is B-definable, hence must equal b.

Thus a definable function on b with finite image is constant.

Let F be a definable function on b into Γ. If F is not constant, then for some γ ∈ Γ, F^{−1}(γ) is a proper subset of b; it follows that some finite union of proper sub-balls of b is γ-definable. By Lemma 3.35, it follows that some such finite union is already definable, a contradiction.

Thus it suffices to show that functions into a single coset C = val_{rv}^{−1}(γ) of k^{∗} are constant on b.

Assume first that b is open, or a properly infinite intersection of balls. By Lemma 3.19 definable functions on b into C are generically constant; but then by transitivit they are constant.

Now suppose b is closed, or a thin annulus. Let Y be the set of maximal open sub-balls b′ of b. Each b′ ∈ Y is transitive over T_{b′}, so F |b′ is constant. Thus F factors through Y.

In the case of the annulus, by Lemma 3.45, G_{m}(k) acts transitively on Y by automorphisms over k. This suffices to rule out nonconstant functions into k. More generally, if a coset C of k^{∗} has algebraic points, then Aut(C/k) is finite. Since Aut(Y/k) is transitive, it follows that if f : Y → C is definable then f(Y) is finite. But Y is finitely primitive, so f(Y) is a point.

Assume finally that b is a closed ball. Using Lemma 3.45, we can view G_{a}(k) as a subgroup of Aut(Y/k). Aut(C/k) is contained in G_{m}(k). Let S = Aut(Y × C/k) ∩ (G_{a}(k) × G_{m}(k)). Then S projects onto G_{a}(k). By strong minimality, S ∩ (G_{a}(k) × (0)) is either G_{a}(k) or a finite group. In the first case, S = G_{a} × T for some T ≤ G_{m}. In the latter, S is the graph of an finite-to-one homomorphism G_{a} → T; but this is impossible. Thus G_{a} × (0) ≤ S and G_{a} acts transitively on Y by automorphisms fixing C; it follows that F is constant. □

3.7. Resolution and finite generation.

Lemma 3.48. Let A ≤ B be substructures of a model of T. Assume B is finitely generated over A. Then RV(B) is finitely generated over RV(A). Also, if RV(A) ≤ C ≤ RV(B) then C is finitely generated over RV(A).

Proof. Suppose Γ(B) has infinitely many elements, ℚ-linearly independent modulo Γ(A). By Lemma 3.1, they are algebraically independent. By Lemma 3.20, they lift to algebraically independent elements of B over A, contradicting the assumption of finite generation. Thus rk_{Γ}Γ(B)/Γ(A) < ∞. It is thus clear that any substructure of Γ(B) containing Γ(A) is finitely generated over Γ(A). So it suffices to show that RV(B) is finitely generated over A ∪ Γ(B); replacing A by A ∪ Γ(B), we may assume Γ(B) = Γ(A). In this case RV(B) ⊂ RES. See [17] Proposition 7.3 for a proof stated for ACVF_{A}, but valid in the present generality. Here is a sketch. One looks at B = A(c) with c ∈ VF. If c ∈ acl(A) then the Galois group Aut(acl(A)/A(c)) has finite index in Aut(acl(A)/A). Hence the same is true of their images in Aut(acl(A) ∩ RV), and since RV is stably embedded (by clause (1) of the definition of V -minimality) it follows that there exists a finite subset C′ of A(c) ∩ RV such that any automorphism of acl(A) fixing A(C′) fixes A(c) ∩ RV. By Galois theory for saturated structures (§2.1.3) C′ generates A(c) ∩ RV over A.

On the other hand if c ∉ acl(A), then tp(c/ acl(A)) agrees with the generic type over A of either a closed ball, an open ball, or an infinite intersection of balls. In the latter two cases, RES(A) = RES(B) using Lemma 3.19. In the case of a closed ball b, let b′ be the unique maximal open sub-ball of b containing c. Then b′ ∈ A(c), and tp(c/A(b′)) is generic in the open ball b′. Thus by Lemma 3.17, RES(B) = RES(A(b′)) so it is 1-generated. □

Recall 𝔅 = 𝔅^{o} ∪ 𝔅^{cl} is the sort of closed and open balls.

We require a variant of a result from [17] on canonical resolutions. We state it for 𝔅-generated structures, but it can be generalized to arbitrary ACVF-imaginaries ([16]).

The proposition and corollaries will have the effect of allowing free use of the technology constructed in this paper over arbitrary base. (cf. Proposition 8.3.)

For this proposition, we allow 𝔅 (and Γ) as sorts, in addition to VF and RV; so that a structure is a subset of 𝔅, Γ of a model of T, closed under definable functions.

Assume for simplicity that T has quantifier elimination. (cf. 3.4)

Let us call a structure A resolved if any ball and any thin annulus defined over acl(A) has a point over acl(A).

Lemma 3.49. Let T be V-minimal. Let M ⊧ T, and A be a substructure of M. Then (1),(2) are equivalent; if Γ(A) ≠ (0) then (3) is equivalent to both.

(1) A is effective and VF(acl(A)) → Γ(A) is surjective.

(2) A is resolved.

(3) acl(A) is an elementary submodel of M.

Proof. Clearly (3) implies (1) and (2) implies (1). To prove that (1) implies (3) it suffices to show that every definable ϕ(x) of T_{A} in one variable, with a solution in M, has a solution in A. If x is an RV-variable it suffices to show that ϕ(rv(y)) has a solution; so we may assume x is a VF- variable, so ϕ defines D ⊆ VF. By C-minimality D is a finite Boolean combination of balls. D can be written as a finite union of definable sets of the form ∪_{j=1}^{m}D_{j} \ E_{j}, where for each j, D_{j} is a closed ball, and E_{j} a finite union of maximal open sub-balls of D_{j}, or D_{j} is an open ball and E_{j} is a proper sub-ball of D_{j}, or E_{j} = ∅, or D_{j} = K. In the third case, by effectivity there exists a finite set meeting each D_{j} in a point; since A = acl(A), this finite set is contained in A; so D(A) ≠ ∅, as required. In the first and second cases, there exists similarly a finite set Y meeting each E_{j}. Since A = acl(A), Y ⊆ A. By picking a point and translating by it, we may assume 0 ∈ E_{j} for some j. Say E_{j} has valuative radius α; picking a point d ∈ A with val(d) = α and dividing, we may assume α = 0. Now in the open case any element of valuation 0 will be in D_{j}. In the closed case, the image of E_{j} under res is a finite subset of the residue field; pick some element ā of k(A) outside this finite set; by effectivity, pick a ∈ A with res(a) = ā; then a ∈ D. In the fourth case, we use the assumption that Γ(A) ≠ (0). This proves (3).

It remains to show that (1) implies (2). Let b be a thin annulus defined over acl(∅); so b = b′ \ b′′ for a unique closed ball b′ and maximal open sub-ball b′′. By effectivity, b′′ has an algebraic point, so translation we may assume 0 ∈ b′′. In this case the assumption that VF(acl(A)) → Γ(A) is surjective gives a point of b′ \ b′′. □

If T_{0} is V-minimal, A is a finitely generated structure (allowing 𝔅, or even ACV F - imaginareis), and T = (T_{0})_{A}, we will call T a finitely generated extension of a V-minimal theory.

Remark 3.50. If A is effective, then A is VF ∪ Γ-generated. If A is resolved, then A is VF-generated.

Proposition 3.51. Let T be V-minimal.

(1) There exists an effective structure E_{eff} admitting an embedding into any effective structure E. We have RV(E_{eff}), Γ(E_{eff}) ⊆ dcl(∅).

(2) There exists a resolved E_{rslv} embedding into any resolved structure E. We have k(E_{rslv}), Γ(E_{rslv}) ⊆ dcl(∅). In fact C(E_{rslv}) ⊆ dcl(∅) for any cosets C of k^{∗} in RV that contain algebraic points.

(3) Let A be a finitely generated substructure of a model of T, in the sorts VF ∪ 𝔅. Then (1),(2) hold for T_{A}.

Proof. (1) Let (b_{i})_{i<λ} enumerate the definable balls. Define a tower of VF-generated structures A_{i}, and a sequence of balls b_{i}, as follows. Let A_{0} = dcl(∅); if κ is a limit ordinal, let A_{κ} = ∪_{i<κ}A_{i}. Assume A_{i} has been defined. If possible, let b_{i} be an A_{i}-definable, A_{i}-transitive ball, not a point; and let c_{i} be any point of b_{i}. If no such ball b_{i} exists, the construction ends, and we let E_{eff} = A_{i} for this i.

Suppose E is any effective substructure of a model of T. We can inductivelly define a tower of embeddings f_{i} : A_{i} → E. At limit stages κ let f_{κ} = ∪_{i<κ}f_{i}. Given f_{i} with A_{i} ≠ E, let b′_{i} be the image under f of b_{i}. By effectivity, b′_{i} has a point c′_{i} ∈ E. Since b_{i} is transitive over A_{i}, the formula x ∈ b_{i} generates a complete type; so tp(c_{i}/A_{i}) is carried by f to tp(c′_{i}/A′_{i}). Thus there exists an embedding f_{i+1} : A_{i+1} → E extending f_{i}, and with c_{i} ↦ c′_{i}.

Each A_{i} is VF-generated; by Lemma 3.31 (3) ⟹ (4), the process can only stop when A_{i} = E_{eff}. This shows that E_{eff} embeds into E, and at the same time that the construction of E_{eff} itself must halt at some stage (of cardinality ≤ |T|.)

By construction, E_{eff} is VF-generated; and hence T_{E_{eff}} is V-minimal. Moreover there are no E_{eff}-definable E_{eff}-transitive balls (except points). In other words all E_{eff}-definable balls are centered. By V-minimality (3) every closed ball has a definable point, so every centered ball has one. So E_{eff} is effective.

It remains only to show that RV(E_{eff}), Γ(E_{eff}) ⊆ dcl(∅). We show inductively that RV(A_{i}), Γ(A_{i}) ⊆ dcl(∅). At limit stages this is trivial, and at successor stages it follows from Lemma 3.47.

(2) Identical to (1), but using thin annuli as well as balls. If a thin annulus is not transitive, it contains a proper nonempty finite union of balls, so by V-minimality it contains a proper nonempty finite set. Hence the construction of the A_{i} stops only when A_{i} is resolved.

(3) Let A_{0} = (A ∩ (VF ∪ Γ)). A is generated over A_{0} by some b_{1}, …, b_{n} ∈ 𝔅 with b_{i} of valuative radius γ_{i} ∈ A_{0}. Since T_{A_{0}} is V-minimal, we may assume T = T_{A_{0}} and A is generated by b_{1}, …, b_{n}, with γ_{i} definable.

Let J be a subset of {1, …, n} of smallest size such that acl({b_{j} : j ∈ J}) = acl({b_{1}, …, b_{n}}). By minimality, no b_{j} is algebraic over {b_{j′} : j′ ∈ J, j′ ≠ j}. Let j ∈ J, and let Y_{j} be the set of balls of radius γ_{j}; then Y_{j} is a definable family of disjoint balls. By Lemma 3.8 for T′ = T_{<{b_{j′}:j′∈J,j′≠j}>}, b_{j} is transitive in T′_{b_{j}}, i.e. in T_{<b_{j′}:j′∈J>}; hence b_{j} is transitive over acl(b_{1}, …, b_{n}) = acl(A). Let us now show, using induction on |J|, that Π_{j∈J}b_{j} is transitive over A. Let c_{j} ∈ b_{j}. By Lemma 2.10 the ({b_{j′} : j′ ∈ J, j′ ≠ j}) remain algebraically independent over < c_{j} >. Thus by induction, Π_{j≠j′}b_{j′} is transitive over A(c_{j}); since b_{j} is transitive over A, Π_{j∈J}b_{j} is too. Let A′ = A(c_{j} : j ∈ J). Claim If B is a VF ∪ Γ-generated structure containing A, then A′ embeds into B over A.

Proof. Since B is VF ∪ Γ-generated, every ball of T_{B} is centered, in particular b_{j} has a point c′_{j} defined over T_{B}. Let c′ = (c′_{j} : j ∈ J). By transitivity of Π_{j∈J}b_{j} we have tp(c/A) = tp(c′/A). So A′ embeds into B.

Note that A′ is almost VF ∪ Γ-generated; indeed since γ_{i} is definable, b_{i} ∈ dcl(c_{i}) so A′ ⊆ acl((c_{j})_{j∈J}). Thus T_{A′} is V-minimal. So (1,2) applies and prove (3).

□

See Lemma 3.60 for a uniqueness statement.

Corollary 3.52. Let f : VF → (RV ∪ Γ)^{∗} be a definable map.

(1) There exists a definable f̃: RV → (RV∪Γ)^{∗} such that for any x ∈ RV, for some x ∈ VF with rv(x) = x, f̃(x) = f(x).

(2) Let Ω = VF/ℳ. There exists a definable map f̃ : Ω → (RV ∪ Γ)^{∗} such that for any x ∈ Ω, for some x ∈ VF with x + ℳ = x, f̃(x) = f(x).

Proof. (1) In view of Lemma 2.3, it suffices to show that for a given complete type P ⊆ RV, there exists such a function f̃ on P. We fix a ∈ P, and show the existence of c ∈ dcl(a) such that for some a with rv(a) = a, f(a) = c.

By Proposition 3.51, there exists an effective substructure A with a ∈ A and (RV ∪ Γ)(A) = (RV ∪ Γ)(< a >). Thus the open ball rv^{−1}(a) has an A-definable point a. Set c = f(a); since f(a) ∈ RV(A) = RV(< a >) we have c = f̃_{P} (a) for some definable function f̃_{P}. Clearly f̃_{P} satisfies the lemma for the input a, hence for any input from P.

The proof of (2) is identical, using Lemma 3.51 (3). □

Corollary 3.53. Let T be V-minimal. Assume every definable point of Γ lifts to an algebraic point of RV. Then there exists a resolved structure E_{rslv} such that E_{rslv} can be embedded into any resolved structure E, and RV(E_{rslv}), Γ(E_{rslv}) ⊆ dcl(∅). If A is a finitely generated substructure of a model of T, in the sorts VF ∪ 𝔅, the same is true for T_{A}.

Proof. Under the assumption of the corollary, the conclusion of Proposition 3.51 implies RV(E_{rslv}) ⊆ dcl(∅). □

Remark 3.54. It is easy to see using the description of imaginaries in [16] that in a resolved structure, any definable ACVF− imaginary is resolved. In other words, if A is a resolved, and ∼ is a definable equivalence relation on a definable set D, then D(A) → (D/ ∼)(A) is surjective.

If A is only effective, then there exists γ ∈ Γ(A)^{n} such that for any t with val(t) = γ, (D/ ∼)(A) ⊆ dcl(D(A)/ ∼, t); this can be seen by embedding D/ ∼ into B_{n}(K)/H for an appropriate H ≤ B_{n}(𝒪), and splitting B_{n} = T_{n}U_{n}.

3.8. Dimensions. We define the VF-dimension of a T_{M}-definable set X to be the smallest n such that for some n, X admits a T_{M}-definable map with finite fibers into VF^{n} × (RV ∪ Γ)^{∗}.

By essential bijection Y → Z we mean a bijection Y_{0} → Z_{0}, where dim_{VF}(Y \Y_{0}), dim_{VF}(Z \ Z_{0}) < dim_{VF}(Y) = dim_{VF}(Z); and where two such maps are identified if they agree away from a set of dimension < dim_{VF}(Y).

We say that a map f : X → VF^{n} has RV-fibers if there exists g : X → (RV ∪ Γ)^{∗} with (f, g) injective.

Lemma 3.55. Let X ⊆ VF^{n} × (RV ∪ Γ)^{∗} be a definable set. Then (1) X has VF-dimension ≤ n iff there exists a definable map f : X → VF^{n} with RV-fibers.

(2) If it exists, the map f is “unique up to isogeny”: if f_{1}, f_{2} : X → VF^{n} have RV-fibers, then there exists a definable h : X → Z ⊆ VF^{n} × (RV ∪ Γ)^{∗} and g_{1}, g_{2} : Z → VF^{n} with finite fibers, such that f_{i} = g_{i}h.

Proof. (1) If f : X → VF^{n} has RV-fibers, let g be as in the definition of RV-fibers; then (f, g) : X → VF^{n} × (RV ∪ Γ)^{∗} is injective, so certainly finite-to-one. If ϕ : X → VF^{n} × RV^{∗} is finite-to-one, by Lemma 3.9, each fiber ϕ^{−1}(c) admits a c-definable injective map ψ_{c} : ^{−1}(c) → RV^{∗}. By Lemma 2.3 we can find θ : X → VF^{n} → RV^{∗} that is injective on each ϕ-fiber. Let f(x) = (ϕ, θ). This proves the equivalence.

(2) Now suppose f_{1}, f_{2} : X → VF^{n} both have RV- fibers.

Let h(x) = (f_{1}(x), f_{2}(x)), Z′ = h(X), and define g_{i} : Z′ → VF^{n} by g_{1}(x, y) = x, g_{2}(x, y) = y. Then g_{i} has finite fibers. Otherwise we can find a ∈ X such that f_{1}(a) ∉ acl(f_{2}(a)) (or vice versa.) But for any a ∈ X, we have f_{1}(a) ∈ acl(f_{2}(a), c) for some c ∈ (RV∪Γ)^{∗}. By Lemma 3.41, f_{1}(a) ∈ acl(f_{2}(a)), a contradiction. By Lemma 3.9 (cf. Lemma 2.3), there exists a definable bijection between Z′ and a subset Z of VF^{n} × RV^{∗}. Replacing Z by Z′ finishes the proof of the lemma. □

Corollary 3.56. Let f : X → RV ∪ Γ, X_{a} = f^{−1}(a). Then dim(X) = max_{a} dim X_{a}.

Proof. Let n = max_{a} dim X_{a}. For each a there exist definable functions g_{a} : X_{a} → VF^{n} and h_{a} : X_{a} → (RV ∪ Γ)^{∗} with (g_{a}, f_{a}) injective on X_{a}. Thus by the compactness argument of Lemma 2.3, there exists definable functions g : X → VF^{n} and h : X_{a} → (RV ∪ Γ)^{∗} such that (g, h) is injective when restricted to each X_{a}. But then clearly (g, h, f) is injective, so dim(X) ≤ n. The other inequality is obvious. □

We continue to assume T is V-minimal.

Lemma 3.57. Let a, b ∈ VF. If a ∈ acl(b) \ acl(∅) then b ∈ acl(a).

Proof. Suppose b ∉ acl(a). Let A_{0} = Γ(acl(a, b)). Then by Lemma 3.36, b ∉ acl(A_{0}(a)).

Let C be the intersection of all acl(A_{0})-definable balls such that b ∈ C, and let C′ be the union of all acl(A_{0})-definable proper sub-balls of C. Let B = ∩_{i}{B_{i}} be the set of all balls defined over acl(A_{0}(a)) with b ∈ B_{i}, and let B′ = ∪_{j}{B′_{j}} be the union of all acl(A_{0}(a))-definable proper sub-balls of B.

Since a ∈ acl(b), we have a ∈ acl(b′) for all b′ ∈ B \ B′, outside some proper sub-ball. It follows by compactness that for some i, j, a ∈ acl(b′) for all b′ ∈ B_{i} \ B′_{j}. Say i = j = 1, B′_{1} ⊂ B_{1}. By Example 3.57, a ∈ acl(A_{0}(f_{1})) where f_{1} ∈ 𝔅 codes the ball B_{1}.

If B_{1} is a point, we are done. Othewise B_{1} has valuative radius α_{1} < ∞ defined over A_{0}. It follows that if B_{1} ⊇ C then B_{1} is acl(A_{0})-definable; but then a ∈ acl(A_{0}), contradicting the assumption. Since B_{1} meets P nontrivially, we have therefore B_{1} ⊂ C. Similarly B_{1} cannot contain any ball in C′ since it is not acl(A_{0})-definable, but it cannot be contained in C′ since B_{1} ∩ P ≠ ∅. so B_{1} ∩ C′ = ∅. Thus B_{1} ⊂ P.

Let B̄_{1} be the closed ball of radius α_{1} containing B_{1}, and let e_{1} be the corresponding element of 𝔅_{cl}. Since B̄_{1} is almost definable over A_{0}(a), it follows from V-minimality

that there exists an almost A_{0}(a)-definable point c(a) in B̄_{1}. Now if a ∈ acl(A_{0}(e_{1})), then B̄_{1} contains an A_{0}(e_{1})-definable finite set F_{1} = F_{1}(e_{1}). But since B_{1} is a proper subset of P, e_{1} ∉ acl(A_{0}). This contradicts Lemma 3.8. So a ∉ acl(A_{0}(e_{1})).

Nevertheless we have seen that a ∈ acl(A_{0}(f_{1})). Thus B_{1} ≠ B̄_{1}, so B_{1} is a maximal open sub-ball of B̄_{1}. Let b_{1} be the point of Aff(B̄_{1}) representing B_{1}. Then a ∈ acl(b_{1}). It follows that tp(a/acl(A_{0}(e_{1}))) is strongly minimal, contradicting Lemma 3.13. We have obtained a contradction in all cases; so b ∈ acl(a). □

Since the lemma continues to apply over any VF-generated structure, algebraic closure is a dependence relation in the sense of Steinitz (also called a pre-matroid or combinatorial geometry, cf. [33].) Define the VF-transcendence degree of a finitely generated structure B to be the maximal number of elements of VF(B) that are algebraically independent over VF(A). This is the size of any maximal independent set, and also the minimal size of a subset whose algebraic closure includes all VF-points. Hence:

Corollary 3.58. The VF dimension of a definable set D is the maximal transcendence degree of < b >. □

We can now obtain a strengthening of Lemma 3.41, and a uniqueness statement in Proposition 3.51:

Corollary 3.59. Let Y be a T - definable set admitting a finite-to-one map f into 𝔅^{n}. Let g : Y → VF^{m} be a definable map. Then g(Y) is finite.

Proof. We may assume m = 1. We will use the equivalence (3) ⟺ (4) of Lemma 2.6. If g(Y) is infinite, then by compactness there exists a ∈ g(Y) a ∉ acl(A). But for some b we have a = g(b), so if c = f(b), we have c ∈ 𝔅^{n}, a ∈ acl(c). Thus it suffices to show:

(*) If a ∈ VF, c ∈ 𝔅^{n} and a ∈ acl(A(c))), then a ∈ acl(A).

This clearly reduces to the case n = 1, c ∈ 𝔅. Let γ be the valuative radius of c. As follows from Corollary 3.36, it suffices to show that a ∈ acl(A(γ)). So in (*) we may assume γ ∈ A.

Finally, to prove (*) (using again the equivalence of Lemma 2.6), we may enlarge A, so we may assume A ⊧ T.

Since γ ∈ A, c ∈ dcl(A(e)) for any element e of the ball c. Thus a ∈ acl(A(e)). Suppose a ∉ acl(A); then by exchange for algebraic closure in VF, e ∈ acl(A(a)). Thus any two elements of the ball c are algebraic over each other. By Example 2.4, c has finitely many points; which is absurd. This contradiction shows that a ∈ acl(A). □

Lemma 3.60. (cf. Proposition 3.51.) Let T be a finitely generated extension of an effective V-minimal theory. Then if E_{1}, E_{2} are effective and both embed into any effective E, then they are finitely generated, and E_{1} ≃ E_{2}.

Proof. The finite generation is clear. Since E_{1}, E_{2} embed into each other, they have the same VF-transcendence degree We may assume E_{1} ≤ E_{2}. But then by Lemma 3.58, E_{2} ⊆ acl(E_{1}). By Lemma 3.9, E_{2} ⊆ dcl(E_{1}, F) for some finite F ⊆ RV^{∗} ∩ dcl(E_{2}). But RV(E_{1}) = RV(E_{2}), so F ⊆ dcl(E_{1}), and thus E_{2} = E_{1}.

Remark 3.61. The analogous statement is true for resolved structures. Note that if F is a finite definable subset of RV^{n} then automatically the coordinates of the points of F lie in cosets of k^{∗} that have algebraic points.

□

Remark The hypothesis of Lemma 3.60 can be slightly weakened to: T is finitely generated over a V-minimal theory, and there exists a finitely generated effective E.

Example 3.62. In ACVF, when X ⊆ VF^{n}, the VF-dimension equals the dimension of the Zariski closure of X. This is proved in [35]. The idea of the proof: the VF-dimension is clearly bounded by the Zariski dimension. For the opposite inequality, in the case of dimension 0, if X is a finite A-definable subset of VF, then using quantifier elimination there exists a nonzero polynomial f with coefficients in A, such that f vanishes on X. In general, if a definable X ⊆ VF^{n} has VF dimension < n, one can reduce to the case where all fibers of the projection pr : X → prX ⊂ VF^{n−1} are finite, then X is not Zariski dense in VF^{n}, using the 0-dimensional case.

The RV-dimension of a definable set X ⊆ RV^{∗} is the smallest integer n (if any) such that X admits a parametrically definable finite-to-one map into RV^{n}. More generally for X ⊆ (RV∪Γ)^{∗}, dim_{RV}(X) is the smallest integer n (if any) such that X admits a parametrically definable finiteto-one map into (RV ∪ Γ)^{n}

Note that RV is one-dimensional, but Γ and every fiber of val_{rv} are also one-dimensional. In this sense RV ∪ Γ-dimension is not additive; model-theoretically it is closer to weight than to rank. We do have dim(X × Y) = dim(X) + dim(Y).

Dually, if a structure B is RV-generated over a substructure A, we can define the weight of B/A to be the least n such that B ⊆ acl(A, a_{1}, …, a_{n}), with a_{i} ∈ RV.

For subsets of RV, RV-dimension can be viewed as the size of a Steinitz basis with respect to algebraic closure. One needs to note that the exchange principle holds:

Lemma 3.63 (exchange). Let a, b_{1}, …, b_{n} ∈ RV; assume a ∈ acl(A, b_{1}, …, b_{n}) \ acl(A, b_{1}, …, b_{n−1}). Then b_{n} ∈ acl(A, b_{1}, …, b_{n−1}, a).

Proof. We may take n = 1, b_{n} = b, and A = acl(A). Let α = val_{rv}(a) ∈ Γ, β = val_{rv}(b). If β ∈ A then Γ(A(a, b)) = Γ(A(b)) = Γ(A). The first equality is true since a ∈ acl(A(b)) so A(a, b) ⊂ acl(A(b)), and using the stable embeddedness of Γ (§2.1.5) and the linear ordering on Γ. The second equality follows from Lemma 3.10. So if β ∈ A then a, b lie in A-definable strongly minimal sets, cosets of k^{∗}, and the lemma is clear.

Assume β ∉ A. If α ∈ A, then tp(a/A) is strongly minimal, and tp(a/A) implies tp(a/A(b)) by Lemma 3.10; ; but then a ∈ acl(A), contradicting the assumption. So α, β ∉ A; from the exchange principle in Γ it follows that A′ := acl(A, α) = acl(A, β). Moreover a ∉ acl(α) by Lemma 3.11 and Lemma 2.6. By the previous case, b ∈ acl(A′, a), so b ∈ acl(A, a). □

Lemma 3.64. A definable X ⊆ RV^{n} has RV-dimension n iff it contains an n-dimensional definable subset of some coset of k^{∗}^{n}.

Proof. Assume X has RV-dimension n. Then there exists (a_{1}, …, a_{n}) ∈ X with a_{1}, …, a_{n} algebraically independent. Let c ∈ Γ; then since a_{n} ∉ acl(a_{1}, …, a_{n−1}), it follows as in the proof of Lemma 3.63 that a_{n} ∉ acl(a_{1}, …, a_{n−1}, c). This applies to any index, so a_{1}, …, a_{n} remain algebraically independent over c; and inductively we may add to the base any finite number of elements of Γ. Let c_{i} = val_{rv}(a_{i}), and let A′ = A(c_{1}, …, c_{n}). Then a_{1}, …, a_{n} are algebraically independent over A′, and the lie in X′ = X ∩ Π_{i=1}^{n}rv^{−1}(c_{i}); thus X′ is an n-dimensional definable subset of a coset of k^{∗}^{n}. □

Definition 3.65. VF[n, ·] be the category of definable subsets of VF^{∗} × RV^{∗} of dimension ≤ n. Morphisms are definable maps.

Let X ∈ Ob VF[n, ·]. By Lemma 3.55, there exists a definable f : X → VF^{n} with RV-fibers; and the maximal RV-dimension of a fiber is a well-defined quantity, depending only on the isomorphism type of X (but not on the choice of f). In particular, the subcategory of definable sets of maximal fiber dimension 0 will be denoted VF[n].

Definition 3.66. We define RV[n, ·] to be the category of definable pairs (U, f), with U ⊆ RV^{∗}, f : U → RV^{n}. If U, U′ ∈ Ob RV[n, ·], a morphism h : U → U′ is a definable map, such that U′′ = {(f(u), f′(h(u))) : u ∈ U} has finite-to-one first projection to RV^{n}. RV[n] is the full subcategory of pairs (U, f) with f : U → RV^{n} finite-to-one.

RES[n] is the full subcategory of RV [n] whose objects are pairs (U, f) ∈ Ob RV[n] such that val_{rv}(U) is finite, i.e. U ⊆ RES^{∗}.

Remark 3.67. (1) For X, Y Ob RV[n], any definable bijection X → Y is in

Mor_{RV[n]}(X, Y).

(2) The forgetful map (X, f) ↦ X is an equivalence of categories between RV[n] and the category of all definable subsets of RV^{∗} of RV-dimension ≤ m, with all maps between them. The presentation with f is nonetheless useful for defining 𝕃.

By Remark 3.67 K_{+}(RV[m]) is isomorphic to the Grothendieck semigroup of definable subsets of RV^{∗} of RV-dimension ≤ m. If dim(X) ≤ m, let [X]_{m} denote the class [X]_{m} = [(X, f)]_{m} ∈ RV[m], where f : X → RV^{∗} is any finite-to-one definable map.

Unlike the case of VF[n, ·] or RV[n], for (U, f) ∈ Ob RV[n, ·] the map f cannot be reconstructed from U alone, even up to isogeny, so it must be given as part of the data. We view (U, f) as a cover of f(U) with “discrete” fibers.

We denote

RV[≤ N, ·] := ⊕_{0≤n≤N}RV[n, ·], RV[≤ N] = ⊕_{0≤n≤N}RV[n]
RV[∗, ·] := ⊕_{0≤n}RV[n, ·], RV[∗] := ⊕_{0≤n}RV[n] RES[∗] := ⊕_{0≤n}RES[n]

We have natural multiplication maps K_{+}RV[k, ·]×K_{+}RV[l, ·] → K_{+}[k+l, ·], ([(X, f)], [(Y, g)]) ↦ [(X×Y, f×g)]. This gives a semiring structure to K_{+}(RV[∗]). This differs from the Grothendieck ring K_{+}(RV).

3.8.1. Alternative description of RV[≤ N, ·]. An object of RV[≤ N, ·] thus consists of a formal sum ∑_{n=0}^{N} X_{n} of objects X_{n} = (X_{n}, f_{n}) of RV[n, ·]. This can be explained from another angle if one adds a formal element ∞ to RV, and extends rv to VF by rv(0) = ∞. Define a function f[k] by f[k](x) = (f_{n}(x), ∞, …, ∞) (N − k times). If X = (X, f), let X[k] = (X, f[k]). Then ∑_{n=0}^{N} X_{n} can be viewed as the disjoint union ∪_{i=0}^{N}X_{i} × {∞}[N − i]. The rv pullback is then a set of VF-dimension N, invariant under multiplication by 1 + ℳ; the sum over dimensions ≤ N is necessary to ensure that any such invariant set is obtained. (Cf. Lemma 4.9.) From this point of view, an isomorphism is a definable bijection preserving the function “number of finite coordinates”. We will use RV[≤ N, ·] or RV_{∞}[N, ·] interchangeably.

Lemma 3.68. Let X, X′ ∈ Ob RV[n, ·], and assume a bijection g : X′ → X lifts to G : 𝕃X′ → 𝕃X. Then g ∈ Mor_{RV[n,·]}(X′, X).

Proof. We only have to check the isogeny condition, i.e. that f(g(a)) ∈ acl(f′(a)) for a ∈ X′ (and dually). By Lemma 3.42, for x ∈ ρ_{X′}^{−1}(a), G(x)_{VF} ∈ acl(x_{VF}), i.e. the VF-coordinates of G(x) are algebraic over those of x. Thus f(g(a)) ∈ acl(x_{VF}). This is true for any x ∈ ρ_{X′}^{−1}(a), so f(g(a)) ∈ acl(a). □

4. Descent to RV: objects

We assume T is C-minimal with centered closed balls. We will find a very restricted set of maps that transform any definable set to a pullback from RV. This is related to Denef’s cell decomposition theorem; since we work in C-minimal theories it takes a simpler form. Recall that this assumption is preserved under passage to T_{A}, when A is a (VF, RV, Γ)-generated substructure of a model of T ( Lemma 3.39).

Recall that RV = VF^{×}/(1 + ℳ), rv : VF^{×} → RV the quotient map. Let RV_{∞} = RV ∪ {∞}, and define rv(0) = ∞. We will also write rv for the induced map rv^{n} : (VF^{×})^{n} → (RV)^{n}.

Definition 4.1. Fix n. Let 𝒞^{0} be the category whose objects are the definable subsets of VF^{n} × RV_{∞}^{∗}, and whose morphisms are generated by the inclusion maps together with functions of one of the following types:

(1) Maps

(x_{1}, …, x_{n}, y_{1}, …, y_{l}) ↦ (x_{1}, …, x_{i−1}, x_{i} + a, x_{i+1}, …, x_{n}, y_{1}, …, y_{l})

with a = a(x_{1}, …, x_{i−1}, y_{1}, …, y_{l}) : VF^{i−1} × RV_{∞}^{l} → VF an A- definable function of the coordinates y, x_{1}, …, x_{i−1}.

(2) Maps (x_{1}, …, x_{n}, y_{1}, …, y_{l}) ↦ (x_{1}, …, x_{n}, y_{1}, …, y_{l}, rv(x_{i}))

The above functions are called elementary admissible transformations over A; a morphism in 𝒞_{A}^{0} generated by elementary admissible transformations over A will be called an admissible transformation over A. Taking l = 0, we see that all A-definable additive translations of VF^{n} are admissible.

Analogously, if Y is a given definable set, one defines the notion of a Y -family of admissible transformations.

If e ∈ RV and T_{e} is an A(e)-admissible transformation, then there exists an A-admissible T such that ι_{e}T_{e} = T ι_{e}, where ι_{e}(x_{1}, …, x_{n}, y_{1}, …, y_{l}) = (x_{1}, …, x_{n}, e, y_{1}, …, y_{l}). This is easy to see for each generator and follows inductively.

Informally, note that admissible maps preserve volume for any product satisfying Fubini’s theorem of translation invariant measures on VF and counting measures on RV.

We will now see that any X ⊂ VF^{n} is a finite disjoint union of admissible transforms of pullbacks from RV. We begin with n = 1.

Lemma 4.2. Let T be C-minimal with centered closed balls. Let X be a definable subset of VF. Then X is the disjoint union of finitely many definable sets Z_{i}, such that for some admissible transformations T_{i}, and definable subsets H_{i} of RV_{∞}^{l_{i}}, T_{i}Z_{i} = {(x, y) : y ∈ H_{i}, rv(x) = y_{l_{i}}}

If X is bounded, H_{i} is bounded below; in fact for any h ∈ H_{i}, val_{rv}(h) ≥ val(x) for some x ∈ X.

Here VF will be considered a ball of valuative radius −∞, points as balls of valuative radius ∞.

Proof. We may assume X is a finite union of disjoint balls of the same valuative radius α ∈ Γ∪{±∞}, each minus a finite union of proper sub-balls, since any definable set is a finite union of definable sets of that form.

Case 1 X is a closed ball.

In this case, by the assumption of centered closed balls, X has a definable point a. Let T (x) = x − a. Then T X \ {0} is the pullback of a subset of Γ, the semi-infinite interval [α, ∞) (where α is the valuative radius of X.) Thus T X = rv^{−1}(H) where H = val_{rv}^{−1}([α, ∞))∪{∞}.

Case 2 X is an open ball.

Let X be the surrounding closed ball of the same radius α, and as in Case 1 let a ∈ X be an definable point, T (x) = x−a. If 0 ∈ T X then T X = rv^{−1}(H) where H = val_{rv}^{−1}((α, ∞))∪{∞}. If 0 ∉ T X, then T X = rv^{−1}(H) where H = rv(T X) is a singleton of RV.

Case 3 X = C \ F is a ball with a single hole, the closed ball F.

Let β be the valuative radius of F. Let a ∈ F be a definable point, T (x) = x − a. Then T X = rv^{−1}(H), H = val_{rv}^{−1}(I), where I is the open interval (α, β)) of Γ in case C is closed, the half-open interval [α, β) when C is open.

Case 4 X = C \ ∪_{j∈J}F_{j} is a closed ball, minus a finite union of maximal open sub-balls.

As in Case 1, find T_{1} such that 0 ∈ T_{1}X. Then T_{1}X is the union of the maximal open sub-ball S of radius α, with rv^{−1}(H), where H = rv(X \ S). S can be treated as in Case 2. Here H is a subset of val_{rv}^{−1}(α), consisting of val_{rv}^{−1}(α) minus finitely many points.

Cases 3a,4a: X is a union of m balls (perhaps with holes) of types 1-4 above.

Here we use induction on m; we have m balls C_{j} covering X. Let E be the smallest ball containing all C_{j}. As we may assume m > 1, E must be a closed ball; and each C_{j} is contained in some maximal open sub-ball M_{j} of E. By the choice of E, not all C_{j} can be contained in the same maximal open ball of E. Let a ∈ E be a definable point, T_{1}(x) = x − a. If 0 ∈ T_{1}C_{j} for some j, the lemma is true by induction for this C_{j} and for the union of the others, hence also for X. Otherwise, F = rv(T_{1}(X)) is a finite set, with more than one element. For b ∈ F, let Y_{b} = T_{1}X ∩ rv^{−1}(b). By Lemma 2.3, we can in fact find a definable Y whose fiber at b is Y_{b}. By induction again, there exists an admissible transformation T_{b} such that T_{b}(Y) is a pullback of the required form. Let T_{2}(x) = (x, rv(x)), T_{3}((x, b)) = ((T_{b}(x), b)). Then T_{3}T_{2}T_{1} solves the problem.

General subsets of VF

Let β ≥ α be the least size (i..e greatest element of Γ) such that some ball of radius β contains more than one hole of X. Let {C_{j} : j ∈ J} be the balls of radius β around the holes W of X, and let C = ∪_{j∈J}C_{j}. Then X = (X \C) ∪^{.} (C \W). Now X \C has fewer holes than X, so it can be dealt with inductively. Thus we may assume X = C \ W; and any proper sub-ball of C of less than maximal size contains at most one hole of X. We may assume the {C_{j}} form a single Galois orbit; so they each contain two or more holes of X. Since these holes are not contained in a proper sub-ball of C_{j}, each C_{j} must be closed, and the maximal open sub-balls of C_{j} separate holes. Let D_{j,k} be the maximal open sub-balls of C_{j} containing a hole F_{j,k}. Let F̄_{j,k} be the

smallest closed ball containing F_{j,k}. Then X = (C\∪_{j,k}D_{j,k}) ∪^{.} ∪_{j,k}(D_{j,k}\F̄_{j,k}) ∪^{.} ∪_{j,k} F̄_{j,k}\F_{j,k}).

The second summand in this union falls into Case 3a, the first and third (when non-empty) into case 4a.

□

Remark If we allow arbitrary Boolean combinations (rather than disjoint unions only), we can demand in Lemma 4.2 that the sets H_{i} be finite. More precisely, let X be a definable subset of VF. Then there exist definable sets Z_{i}, admissible transformations T_{i}, and finite definable subsets H_{i} of RV_{∞}^{l_{i}} such that:

X is a Boolean combination of the sets Z_{i}, and T_{i}Z_{i} is one of the following:

(1) VF

(2) (0) × H_{i}

(3) b_{i} × H_{i}, with b_{i} a definable ball containing 0

(4) {(x, y) : y ∈ H_{i}, rv(x) = f_{i}(y)}, for some definable function f_{i} : H_{i} → RV_{∞} Corollary 4.3. Let X ⊆ VF × RV^{∗} be definable. Then there exists a definable ρ : X → RV^{∗} and c : RV^{∗} → VF, c′ : RV^{∗} → RV_{∞}, c′′ : RV^{∗} → RV^{∗} such that every fiber ρ^{−1}(α) has the form (c(α) + rv^{−1}(c′(α))) × {c′′(a)}. Moreover c has finite image.

Proof. The finiteness of the image of c is automatic, by Lemma 3.41. The corollary is obviously true for sets of the form 𝕃(H, h) = {(x, u) ∈ VF × H : rv(x) = h(u)}; take ρ(x, u) = (rv(x), u). If the statement holds for T X where T is an admissible transformation, then it holds for X. If true for two disjoint sets, it is also true for their union (add to ρ a map to {1, −1} ⊆ k^{∗} whose fibers are the two sets.) Hence by Lemma 4.2 is is true for all definable sets. □

Corollary 4.4. Let T be V-minimal, X ⊆ VF and let f : X → RV ∪ Γ be a definable function. Then there exists a definable finite partition of X = ∪_{i=1}^{m}X_{i} such that either f is constant on X_{i}, or else X_{i} is a finite union of balls of equal radius (possibly missing some sub-balls), there is a definable set F_{i} meeting each of the balls b in a single point, and for x ∈ X_{i}, letting n(x) be the point of F_{i} nearest x, for some function H, f(x) = H(rv(x − n(x))).

Proof. The conclusion is so stated that it suffices to prove it over acl(∅), i.e. we may assume every almost definable set is definable; cf. §2.1.4. By compactness it suffices to show that for each complete type p, f|p has the stated form. Let b be the intersection of all balls containing p. If b is transitive then by Lemma 3.47 f|p is constant. Otherwise by V-minimality b contains a definable point, and so we may assume 0 ∈ b. It follows that rv(p) is infinite. Thus by Lemma 3.20, f factors through rv. □

Proposition 4.5. Let T be C-minimal with centered closed balls, and let X be a definable subset of VF^{n} × RV^{l}. Then X can be expressed as a finite disjoint union of A-definable sets Z, with each Z of the following form. For some A- admissible transformation T, A-definable subset H of RV_{∞}^{l^{∗}}, and map of indices ν ∈ {1, …, n} ↦ ν′ ∈ {1, …, l^{∗}},

T Z = {(a, b) : b ∈ H, rv(a_{ν}) = b_{ν′} (ν = 1, …, n)}

If X projects finite-to-one to VF^{n}, then the projection of H to the primed coordinates 1′, …, n′ is finite to one.

If X is bounded, then H is bounded below in RV_{∞}.

Proof. By induction on n; the case n = 0 is trivial. Let pr : X → prX be the projection of X to VF^{n−1} × RV^{l}, so that X ⊂ VF × prX.

Let pr^{∗}(Y) = {v : (∃y ∈ Y)(x, y) ∈ Y }. For any c ∈ prX, according to Lemma 4.2, we can

write pr^{∗}(c) = ⋃^{•}_{i=1}^{k}Z_{i}(c), where

T_{i}(c)Z_{i}(c) = {(a, b) : b ∈ H_{i}(c), rv(a) = b_{1′}}

for some A(c)-admissible T_{i}(c), A(c)-definable Z_{i}(c), and H_{i}(c) ⊆ RV = RV^{1′}. We can write Z_{i}(c) = {x : (x, c) ∈ Z_{i}}, H_{i}(c) = {x : (x, c) ∈ H_{i}} for some definable Z_{i} and H_{i} ⊂ VF^{n−1} × RV^{1′}. By compactness, as in Lemma 2.3, one can asssume that the Z_{i}(c), H_{i}(c), T_{i}(c) are uniformly definable: there exists a partition of prX into finitely many definable sets Y, and for each Y families Z_{i}, H_{i}, T_{i} over Y of definable sets and admissible transformations over Y, such that the integer k is the same for all c ∈ Y, and the Z_{i}(c), H_{i}(c), T_{i}(c) are fibers over c of

Z_{i}, H_{i}, T_{i}. In this case, pr^{∗}(Y) = ⋃^{•}_{i=1}^{k}Z_{i}. We can express X as a disjoint union of the various pr^{∗}(Y); so we may as well assume prX = Y and X = Z_{1}. Let T_{1} be such that ι_{c}T_{1}(c) = T_{1}ι_{c}. Then

T_{1}X = {(a, c, b) : (c, b) ∈ H_{1}, rv(a) = b_{1′}}

Any admissible transformation is injective and so commutes with disjoint unions.

Now by induction, H_{1} itself is a disjoint union H_{1} = ⋃^{•}_{j=1}^{k′}Z_{j}, with

T′_{i}Z′_{i} = {(d, b) : b ∈ H′_{i}, rv(d_{ν}) = d_{ν′} (ν = 2, …, n)}

Notational remarks: Here d = (d_{2}, …, d_{n}) are the VF-coordinates of c above. The ′ depends on i but we will not represent this notationally

Let T_{i}^{∗}(a, d, b) = (a, T′_{i}(d, b)), i.e T_{i}^{∗} does not touch the first coordinate. Note that T_{i}^{∗} also does not move the 1′ coordinate, since in general admissible transformations can only add RV coordinates but not change existing ones. Let

Z_{i} = {x : T_{1}(x) = (a, d, b), (d, b) ∈ Z′_{i}, rv(a) = b_{1′}}

Then (as one sees by applying T_{1}) X = ⋃^{•}_{i=1}^{k}Z_{i}, and if T_{i} = T_{i}^{∗}T_{1}, we have:

T_{i}Z_{i} = {(a, d′, b′) : (d′, b′) ∈ T′_{i}Z′_{i}, rv(a) = b′_{1′}} = {((a, d′, b′) : b ∈ H′_{i}, rv(a) = b_{1′}, rv(d_{ν}) = b_{ν′}}

As for the finiteness of the projection, if X admits a finite-to-one projection to VF^{n}, so does each Z in the statement of the Proposition, and hence the isomorphic set T Z. We have H ⊂ RV^{n+l}, π : RV^{n+l} → RV^{n}, so T Z = {(a, b, b′) : (b, b′) ∈ H, rv(a) = b′}. For fixed a, this yields an a-definable finite-to-one map T Z′(a) = {b′ : (a, b, b′) ∈ T Z} → VF^{n}. By Lemma 3.41), T Z′(a) is finite. Now fix b and suppose (b, b′) ∈ H with b′ not algebraic over b. Then for generic a ∈ rv^{−1}(b), b′ is not algebraic over b, a. Yet (a, b, b′) ∈ T Z and so b′ ∈ T Z′(a), a contradiction.

The statement on boundedness is obvious from the proof; if X ⊆ {x : val(x) ≥ −γ}^{n} × RV^{m}, then H is bounded below by −γ in each coordinate. □

4.0.2. A remark on more general base structures.

Lemma 4.6. Let T be V-minimal, A an 𝔅-generated substructure of a model of T. Let X be a T_{A}- definable subset of VF^{n} × RV^{l}. Then there exist T_{A}-definable subsets Y_{i} ⊂ RV^{m_{i}} and (projection) maps f_{i} : Y_{i} → RV^{n}, a disjoint union Z of

Z_{i} = Y_{i} ×_{f_{i},rv} VF^{n}

and a nonempty A-definable family ℱ of admissible transformations X → Z. ℱ will have an A′-point for any VF ∪ RV ∪ Γ-generated structure containing A.

.

Proof. We may assume A is finitely generated. By Proposition 3.51 there exists an almost VF ∪ Γ-generated A′ ⊃ A embeddable over A into any VF ∪ Γ-generated structure containing A, and with RV(A′) = RV(A). By Proposition 4.5, the required objects Y_{i}, f_{i} exist over A′. But since RV is stably embedded, this data is defined over RV(A′) ⊆ A. The admissible transformations X → Z = ∪^{.} (Y_{i}×_{f_{i},rv} VF^{n}) exist over A′; so one can find a definable set D with an A′-point, and such that any element of D codes an admissible transformation X → Z. □

Remark In fact, arbitrary ACVF-imaginaries may be allowed here.

Example 4.7. ℱ need not have an A-rational point. For instance if A consists of an element of VF/ℳ, i.e. an open ball c, then we can take Y = Y_{1} to be the point 0 ∈ RV (since c can be transformed to ℳ); but there is no A-definable bijection of c with ℳ.

4.0.3. A statement in terms of Grothendieck groups. Recall definitions 3.65, 3.66.

Definition 4.8. Define 𝕃 : Ob RV[n, ·] → Ob VF[n, ·] by:

𝕃(X, f) = (VF^{×})^{n} ×_{rv^{n},f} X ⊂ VF^{n} × RV^{m}

where VF^{×} = VF \ {0}.

For X = ∑_{i} X_{i} ∈ RV[∗], we let 𝕃(X) be the disjoint sum ∑_{i} 𝕃(X_{i}) over the various components in RV[i].

Let ρ denote the natural map 𝕃(X, f) → X.

Lemma 4.9. The image of 𝕃 : ObRV[≤ n, ·] → Ob VF[n, ·] meets every isomorphism class of VF[n, ·].

Proof. For X ⊆ RV^{∗} and f : X → RV_{∞}, define rv(0) = ∞ and

𝕃(X, f) = VF^{n} ×_{rv^{n},f} X ⊂ VF^{n} × RV^{m}

Then in the statement of Proposition 4.5, we have T Z = 𝕃(H, h) where h is the projection to the primed coordinates. For x ∈ H, let s(x) = {i : h_{i}(x) = ∞}. For w ⊆ {1, …, n}, let H_{w} = {x ∈ H : s(x) = w}. Let H̄_{w} = (H_{w}, h′_{w}) where h′_{w} = (h_{i})_{i∉w}. Then H̄_{w} ∈ RV[|w|, ·], and 𝕃(H_{w}, h|H_{w}) ≃ 𝕃(H̄_{w}). So 𝕃(H, h) ≃ 𝕃(∑_{w} H̄_{w}). □

4.0.4. A restatement in terms of VF alone. This restatement will not be used later in the paper.

Definition 4.10. Let A be a subfield of VF. Let 𝒞_{A}^{1}(n, l) be the category of definable subsets of VF^{n} × (VF^{×})^{l}, generated by composition and restriction to subsets by maps of one of the following types:

(1) Maps

(x_{1}, …, x_{n}, y_{1}, …, y_{l}) ↦ (x_{1}, …, x_{i−1}, x_{i} + a, x_{i+1}, …, x_{n}, y_{1}, …, y_{l})

with a = a(x_{1}, …, x_{i−1}, y_{1}, …, y_{l}) : VF^{i+l−1} → VF an A- definable function of the coordinates y, x_{1}, …, x_{i−1}.

(2) Maps (x_{1}, …, x_{n}, y_{1}, …, y_{l}) ↦ (x_{1}, …, x_{n}, y_{1}, …, y_{i−1}, x_{i}y_{i}, y_{i+1}, …, y_{l}) : X → Y assuming x_{i} ≠ 0 on X, and that this function takes X into Y.

Remark 4.11. The morphisms in this category are measure preserving with respect to Fubini products of invariant measures (additively for VF, multiplicatively for VF^{×}), viz. dx_{1} ∧ ⋯ ∧ dx_{n} ∧ \frac{dy_{1}}{y_{1}} ⋯ ∧ \frac{dy_{l}}{y_{l}}

Lemma 4.12. Let T be C-minimal with centered closed balls, X a definable subset of VF^{n}. Then X can be expressed as a disjoint union of A-definable sets Z with the following property. For some l ∈ ℕ, there exists an 𝒞_{A}^{1}(n, l)-transformation T and a definable subset H of RV_{∞}^{n} × RV^{l}, such that

T (Z × ((1 + ℳ))^{l}) = rv^{−1}(H)

Moreover, the projection of H to RV_{∞}^{n} is finite-to-one.

If val(x) is bounded below, then val(H) may be taken to be bounded below in the RVcoordinates, and bounded in the RV_{∞}-coordinates.

Proof. Proposition 4.5 □

5. V-minimal geometry: Continuity and differentiation

We work with a V-minimal theory.

5.1. Images of balls under definable functions. Proposition 5.1. Let X, Y be definable subsets of VF, and let F : X → Y be a definable bijection. Then there exists a partition of X to finitely many definable equivalence classes, such that for any open ball b contained in one of the classes, F (b) is an open ball; and dually, if F (b) is an open ball, so is b.

Proof. It suffices to show that such a partition exists over acl(∅); for any finite almost definable partition has a finite definable refinement (cf. discussion of Galois theory in §2.1.3). Thus as in §2.1.4 we may assume every almost definable set is named.

We will show that if p is a complete type, and b is an open sub-ball of p, then F (b) is an open ball; and that if b′ is an open sub-ball of F (p), then b is an open ball. From this it follows by compactness that there exists a definable D_{p} containing p with the same property; by another use of compactness, finitely many D_{p} cover X; it then suffices to choose any partition, such that any class is contained in some D_{p}.

When p has a unique solution, the assertion is trivial. When p is the generic type of a closed ball, or of VF, or of a transitive open or ∞-definable ball, for any α ∈ Γ, p remains complete over < α >. In the transitive cases, this follows from Lemma 3.47, while in the centered closed case it follows from Lemma 3.18.

Thus all open sub-balls b_{t} of p of any radius α have the same type over < α >; hence they are all transitive over < t >, where t ∈ K/ℳ_{α}, where ℳ_{α} = {x : val(x) > α} ( Lemma 3.8, with Q = p.) Thus by Lemma 3.46, F (b_{t}) is an open ball.

The remaining case is that p is the generic type of a centered open or ∞-definable ball b_{1}. So b_{1} contains a definable proper sub-ball b_{0}. If b is an open sub-ball of p, of radius α, then b ∩ b_{0} = ∅; let b̄ be the smallest closed ball of containing b and b_{0}. Then b is contained in the generic type of b̄, and so by the case of closed balls, F (b) is an open ball. □

Remark 5.2. When X ⊆ VF × RV^{n}, by a ball contained in X we will mean a subset of X of the form b × {e}, where b ∈ 𝔅 and e ∈ RV^{n}. With this understanding, the proposition extends immediately to such sets X. Indeed for each e ∈ RV^{n}, according to the proposition there is a finite partition of X(e) with the required property; as in Lemma 2.3 these can be patched to form a single partition of X.

Remark 5.3. When X ⊆ VF there exists a finite set of points F (not necessarily A-definable) such that F (b) is an open ball whenever b is an open ball disjoint from F. (This does not extend to X ⊆ VF × RV^{∗}.)

Indeed by Proposition 5.1 there is a finite number of closed and open balls b_{i} and points, such that F (b) is an open ball for any open ball b that is either contained in or is disjoint from each b_{i}. Now let c_{i} be a point of b_{i}. If b is an open ball and no c_{i} ∈ b, then b must be disjoint from, or contained in, each b_{i}; otherwise b contains b_{i}, hence c_{i}.

5.2. Images of balls, II.

Lemma 5.4. Let X, Y be balls, and F : X → Y a definable bijection taking open balls to open balls. Then for all x, x′ ∈ X

val(F (x) − F (x′)) = val(x − x′) + v_{0}

where v_{0} is the difference of the valuative radii of X, Y.

Proof. Translating by some a ∈ X and by F (a) ∈ Y, we may assume 0 ∈ X, 0 ∈ Y, F (0) = 0; and by multiplying we may assume and both X, Y have valuative radius 0, i.e. X = Y = 𝒪. Let M(α) = {x : val(x) < α}. Then F (M(α)) = M(β) for some β = β(α). β is an increasing definable surjection from {α ∈ Γ : α > 0} to itself; it must have the form β(α) = mα for some rational m > 0. By Lemma 3.26, we have m ∈ ℤ. Now reversing the roles of X, Y and using F^{−1} will transform m to m^{−1}, so m^{−1} ∈ ℤ also, i.e. m = ±1. Since m > 0, we have m = 1. □

Lemma 5.5. Let X be a transitive open or closed ball (or infinite intersection of balls), and F : X → Y a definable bijection. Then there exists definable e_{0} ∈ RV such that for x ≠ x′ ∈ X, rv(F (x) − F (x′)) = e_{0}rv(x − x′).

Proof. We first show a weaker statement: Claim For some definable e_{0} : Γ → RV rv(F (x) − F (x′)) = e_{0}(val(x − x′))rv(x − x′) for all x ≠ x′ ∈ X.

Proof. Fix a ∈ X. For δ ∈ Γ, let b_{δ} = b_{δ}(a), the closed ball around a of valuative radius δ. Consider those b_{δ} with b_{δ} ⊆ X. As we saw in the proof of Lemma 5.1, as any a ∈ X is generic, b_{δ} is transitive in T_{b_{δ}}. By Lemma 3.45,rv(F (x)−F (a)) = f_{a}(δ)rv(x−a), where val(x−a) = δ, and f_{a}(δ) is a function of a and δ. But then f_{a} is a function Γ → RV, so by Lemma 3.11 it takes finitely many values v_{1}, …, v_{n}. Let Y_{i} = f_{a}^{−1}(v_{i}). Y_{i} has a canonical code e_{i} ∈ Γ^{∗}, consisting of the endpoints of the intervals making up Y_{i}. Using the linear ordering on Γ, each individual e_{i} is definable from the set {e_{i}}_{i}, and hence from a; thus v_{i} = f_{a}(Y_{i}) is also definable from a. So f_{a} f_{a} is definable from (e_{i}, v_{i})_{i}. (This last argument could have been avoided by quoting elimination of imaginaries in RV ∪ Γ.) However as X is transitive, every definable function X → (RV ∪ Γ) is constant, and so f_{a} = f_{b} for any a, b ∈ X.) Let e_{0}(δ) = f_{a}(δ). □

We now have to show that the function e_{0} of the Claim is constant. Using the O-minimality of Γ, it suffices to show for any definable δ ∈ dom(e_{0})

1) If e_{0}(δ) = e then e_{0}(γ) = e for sufficiently small γ > δ.

and if δ is not a minimal element of dom(e_{0})), then also:

2) if e_{0}(γ) = e for sufficiently large γ < δ, then e_{0}(δ) = e.

To determine e_{0}(δ) it suffices to know rv(F (x) − F (x′)) and rv(x − x′) for one pair x, x′ with val(x − x′) = δ. Thus in 1) we may replace X by a closed sub-ball Y of valuative radius δ, and in 2) by any closed sub-ball Y of X of valuative radius < δ. Since such closed balls Y are transitive (over their code), we may assume X is a closed ball.

Fix a ∈ X. Pick a generic c (over a) with rv(c) = e.

To prove 1), note that type of such c is generic in an open ball, whereas the elements of X are generic in a closed ball; these generic types are orthogonal by Lemma 3.19; so X remains transitive in T_{c}. Thus we may assume (by passing to T_{c}) that c is definable.

Let q_{a} be the generic type of the closed ball {x : val(a − x) ≥ δ}. For x ⊧ q_{a}, let v_{0} = val(F (a) − F (x) − c(a − x)) − val(c).

By the definition of e, val(F (a) − F (x) − c(a − x)) > val(F (a) − F (x)), so we have

(1) v_{0} + val(c) = val(F (a) − F (x) − c(a − x)) > val(F (a) − F (x)) = val(c(a − x)) = δ + val(c)

If δ < γ < v_{0}, find x, x′ ⊧ q_{a} with val(x − x′) = γ. Then val(F (x) − F (x′) − c(x − x′)) ≥ v_{0} + val_{rv}(e) > γ + val_{rv}(e) = val(c(x − x′)), so rv(F (x) − F (x′)) = rv(c(x − x′)) showing that e_{0}(γ) = rv(c) = e. This proves 1).

For 2), let Q_{0} = {γ : γ < δ}, Q_{0}^{def} the set of definable elements of Q_{0}, and Q = {γ ∈ Q_{0} : (∀y ∈ Q_{0}^{def})(γ > y)}. So Q is a complete type of elements of Γ. For γ ∈ Q, according to Lemma 3.17, the formula val(x−a) = γ generates a complete type q_{γ;a}(x). By Lemma 3.47) X is transitive over γ, so the formula x′ ∈ X generates a complete T_{γ}-type. Thus by transitivity a complete T_{γ}-type q_{γ}(x, x′) is generated by: x, x′ ∈ X, val(x − x′) = γ; namely (a, b) ⊧ q_{γ} iff b ⊧ q_{γ;a}.

For some definable v_{0}, for (a, x) ⊧ q_{γ} we have, as in 1):

(2)  val(F (a) − F (x) − c(a − x)) = v_{0}(γ) + val(c) > γ + val(c)

If we show that v_{0}(γ) > δ we can finish as in 1).

Now v_{0}(γ) = mγ + γ_{0} for some definable γ_{0} ∈ Γ, and some rational m. Letting γ → δ in (2) gives mδ + γ_{0} ≥ δ. If m < 0 then v_{0}(γ) = mγ + γ_{0} > mδ + γ_{0} ≥ δ so we are done; hence we may take m ≥ 0.

By Lemma 3.47 RV(< ∅ >) = RV(< a >); by Lemma 3.20, when x ⊧ q_{γ;a}, RV(< a, x >) is generated over RV(< a >) by rv(a − x).

In particular on q_{γ,a}, x ↦ rv(F (a) − F (x) − c(a − x)) is a function of rv(a − x). This function lifts v_{0} to a function on RV; hence by Lemma 3.26, m ∈ ℤ. (This, and m ≥ 0, are simplifications rather than essential points.) We have:

val((F (a) − F (x) − c(a − x))(a − x)^{−m}) = γ_{0}

By Lemma 3.47, (RV ∪ Γ)(< a >) = (RV ∪ Γ)(< ∅ >). By Lemma 3.17, then val_{rv}^{−1}(γ_{0}) ∩ dcl(a, x) = val_{rv}^{−1}(γ_{0})∩dcl(a). Thus val_{rv}^{−1}(γ_{0})∩dcl(a, x) = val_{rv}^{−1}(γ_{0})∩dcl(∅). So rv((F (a)− F (x) − c(a − x))(a − x)^{−m}) ∈ dcl(∅); i.e.

rv((F (a) − F (x) − c(a − x))(a − x)^{−m}) = e_{1}

for some definable e_{1}. As in 1) we may assume there exists a definable c_{1} with rv(c_{1}) = e_{1}. Thus for (a, x) ⊧ q_{γ},

(3) val((F (a) − F (x) − c(a − x) − c_{1}(a − x)^{m})) > val(F (a) − F (x) − c(a − x)) = v_{0}(γ) + val(c)

Let x′ ⊧ q_{γ,a} be generic over {γ, a, x}, so in particular val(x−x′) = val(x−a) = val(a−x′) = γ. We have

val((F (a) − F (x′) − c(a − x′) − c_{1}(a − x′)^{m})) > val(F (a) − F (x) − c(a − x)) = v_{0}(γ) + val(c)

Subtracting from (3) we obtain:

(4) val((F (x′)−F (x)−c(x′ −x)−c_{1}[(a−x)^{m})−(a−x′)^{m}]) > v_{0}(γ)+val(c) = val(c_{1}(a−x′)^{m})

But since (x, x′) ⊧ q_{γ}, by (3) we have

(5) val((F (x) − F (x′) − c(x − x′) − c_{1}(x − x′)^{m})) > v_{0}(γ) + val(c) = val(c_{1}(x − x′)^{m})

Comparing (4), (5) (and subtracting val(c_{1})) we see that

val((a − x)^{m} − (a − x′)^{m} − (x′ − x)^{m}) > val((x − x′)^{m}) = val((a − x′)^{m}) = val((a − x)^{m})

Let u = (a − x′)/(x′ − x); then (a − x)/(x′ − x) = u + 1, val(u) = 0 = val(u + 1), and val((u + 1)^{m} − u^{m} − 1) > 0. If U = res(u) we get (U + 1)^{m} = U^{m} + 1. Since the residue characteristic is 0 this forces m = 1. (Note that U is generic.) So v_{0}(γ) = γ + γ_{0}.

From (2), γ + γ_{0} + val(c) > γ + val(c), or γ_{0} > 0. But δ − γ_{0} ∈ Q_{0}^{def}, so since γ ∈ Q we have γ > δ − γ_{0}, or v_{0}(γ) = γ + γ_{0} > δ. As noted below (2) this proves the lemma. □

Remark 5.6. In ACVF(p,p), the Claim of Lemma 5.5 remains true, but it is possible for e_{0} to take more than one value; consider x − cx^{p} on a closed ball of valuative radius 0, where val(c) < 0.

Lemma 5.7. Let X be a transitive open ball, and let F : X → X be a definable bijection. Then rv(F (x) − F (y)) = rv(x − y) for all x ≠ y ∈ X.

Proof. This follows from the second assertion in Lemma 3.45, and from Lemma 5.5. □

At this point, Lemma 5.1 may be improved.

Definition 5.8. Call a function G on an open ball nice if for some e_{0}, for all x ≠ x′ ∈ prX, rv(G(x) − G(x′)) = e_{0}rv(x − x′).

Proposition 5.9. Let X, Y be definable subsets of VF, and let F : X → Y be a definable bijection. Then there exists a partition of X to finitely many definable classes, such that on any open ball b contained in one of the classes, F (b) is an open ball, and F |b is nice.

Proof. The proof of Proposition 5.1 goes through verbatim, only quoting Lemma 5.5 along with Lemma 3.46. □

A definable translate of a ball rv^{−1}(α) will be called a basic 1-cell. Thus Corollary 4.3 states that every fiber of ρ is a basic 1-cell. By a basic 2-cell we mean a set of the form:

X = {(x, y) : x ∈ prX, rv(y − G(x)) = α

where prX is a basic 1-cell, and G is nice.

Corollary 5.10. Let X ⊆ VF^{2} be definable. Then there exists a definable ρ : X → RV^{∗} such that every fiber is a basic 2-cell.

Proof. Let X(a) = {y : (a, y) ∈ X}. By Corollary 4.3 there exist an a-definable ρ_{a} : X(a) → RV^{∗} and functions c, c′ such that every fiber ρ_{a}^{−1}(α) is a basic 1-cell rv^{−1}(c′(a, α)) + c(a, α). By Lemma 2.3 we can glue these together to a function ρ_{1} : X → RV^{∗} with ρ_{a}(y) = ρ_{1}(a, y). Let ρ_{2}(x, y) = (ρ_{1}(x, y), c′(x, ρ(x, y))). Then any fiber D of ρ_{2} has the form {(x, y) : x ∈ pr_{1}D, rv(y − G_{D}(x)) = α where G_{D}(x) = c(x, α), α depending on the fiber D. Combining ρ_{2} with a function whose fibers yield a partition as in Proposition 5.9, we may assume G takes open balls to open balls (cf. Remark 5.2). Now apply Corollary 4.3 to prX to obtain a map ρ′ : prX → RV^{∗} with nice fibers. □

5.3. Limits and continuity. We now assume T is a C-minimal theory of valued fields, satisfying Assumption 3.4 (1).

Let V be a VF-variety. By “almost all a” we will mean: all a away from a set of smaller VF-dimension.

Lemma 5.11. Let g be a definable function on a ball around 0. Then either valg(x) → −∞ as val(x) → ∞, or there exists a unique b ∈ VF such that b = lim_{x→0,x≠0} g(x), i.e.

(∀ϵ ∈ Γ)(∃δ ∈ Γ)(0 ≠ x & val(x) > δ ⟹ val(g(x) − b) > ϵ

Proof. Let p be the generic type of an element of large valuation; so c ⊧ p|A iff val(x) > Γ(A). and let q = tp(g(c)/A) where c ⊧ p|A. By Remark 3.5, q coincides with the generic type of P over A where P is a closed ball, an open ball, or an infinite intersection of balls, or P = VF. The last case means that valg(x) → −∞. The existence of g shows that p, q are non-orthogonal, so it follows from Lemma 3.19 that the first case is impossible.

We begin by reducing to the case P is centered. Assume therefore that P is transitive. For b ∈ P, let q_{b} = tp(g(c′)/A(b)) where c′ ⊧ p|A(b). If q_{b} includes a proper b-definable sub-ball P_{b} of P, or a finite union of such balls, we may take them all to have the same radius α(b); so α(b) is b-definable. By Lemma 3.47, α is constant. If as b varies there are only finitely many ball P_{b}, then P is after all centered. If not, then there are two disjoint P_{b}, P_{b′}; but this is absurd since if c′′ ⊧ p|A(b, b′) then g(c′′) ∈ P_{b} ∩ P_{b′}. Thus q_{b} cannot include a proper sub-ball P_{b} of P; so q_{b} is just the generic type of P, over A(b). Moving from A to A(b) we may thus assume that P is centered.

So P is a centered open or infinitely-definable ball; therefore it has a proper definable subball b. If y ∉ b, write val(b − y) for the constant value of val(c − y), c ∈ b. By the definition of a generic type of P, val(b − g(c)) ∉ Γ(A). Now val(b − g(c)) ∈ Γ(A(c)) = Γ(A) ⊕ ℚval(c) (by (2) of the definition of V-minimality), and val(c) > Γ(A); it follows that val(b − g(c)) < Γ(A) or val(b − g(c)) > Γ(A). The first case is again the case of P = VF, while the second implies that P is an infinite intersection of balls P_{i}, whose radius is not bounded by any element of Γ(A). In other words, P = {b}. Unwinding the definitions shows that b = lim_{x→0,x≠0} g(x). □

Remark In reality, the transitive case considered in the proof above cannot occur.

By an (open, closed) polydisc, we mean a product of (open, closed) balls. Let B be a closed polydisc. Let M ⊧ T. Let b ∈ B(M), a ∈ B(acl(∅)). Write b → a if for any definable γ ∈ Γ, and each coordinate i, val(b_{i} − a_{i}) > γ. Let p_{0} be the type of elements of Γ greater than any given definable element. Then Lemma 5.11 can also be stated thus: given a definable g on a ball B_{0} around 0 into B, there exists b ∈ dcl(∅) such that if val(t) ⊧ p, then (t, g(t)) → (0, b).

Stated this way, the lemma generalizes to functions defined on a finite cover of B_{0}:

Lemma 5.12. Let B_{0} be a ball around 0, and B a closed polydisc, both 0-definable. Let t ∈ B_{0} have val(t) ⊧ p_{0}, and let a ∈ acl(t), a ∈ B. Then there exists b ∈ B, b ∈ acl(∅) with (t, a) ↦ (0, b).

Proof. The proof of Lemma 5.11 goes through. □

The following is an analogue of a result of Macintyre’s for the p-adics. By the boundary of a set X, we mean the closure minus the interior of X.

Lemma 5.13. (1) Any definable X ⊆ VF^{n} of dimension n contains an open polydisc. (2) Any definable function VF^{n} → RV ∪ Γ is constant on some open polydisc.

(3) The boundary of any definable X ⊆ VF^{n} has dimension < n.

Proof. Given (1), (3) follows since the boundary is definable; so it suffices to prove (1,2). For a given n, (2) follows from (1): by Lemma 3.56, the fibers of the function cannot all have dimension < n.

For n = 1, (1) is immediate from C-minimality. Assume (1-2) true for n and let X ⊆ VF×VF^{n} be a definable set of dimension n+1. For any a ∈ VF^{n} such that X_{a} = {b : (a, b) ∈ X} contains an open ball, let γ(a) be the infimum of all γ such that X_{a} contains an open γball. By (2) for n, γ takes a constant value γ_{0} on some polydisc U; pick γ_{1} > γ_{0}. Let X′ = {(u, z) ∈ X : u ∈ U, &(∀z′)(val(z − z′) > γ_{0} ⟹ (u, z′) ∈ X}. Then dim(X′) = n + 1. Now consider the projection (u, z) ↦ z. For some c ∈ VF, the fiber X′_{c} = {u : (u, c) ∈ X′} must have dimension n. By induction, X′_{c} contains a polydisc V. Now clearly V × B_{γ_{1}}^{o}(c) ⊆ X. □

If x = (x_{1}, …, x_{n}), x′ = (x′_{1}, …, x′_{n}), write val(x − x′) for min val(x_{i} − x′_{i}). Say a function F is δ-Lipschitz at x if whenever val(x−x′) is sufficiently large, val(F (x)−F (x′)) > δ+val(x−x′). Say F is locally Lipschitz on X if for any x ∈ X, for some δ ∈ Γ, F is δ-Lipschitz at x.

Lemma 5.14. Let F : X ⊆ VF^{n} → VF be a definable function. Then F is continuous away from a subset X′ of dimension < n. Moreover F is locally Lipschitz on X \ X′.

Proof. Let X′ be the (definable) set of points x where F is not Lipschitz. We must show that X′ has dimension < n (in this case, by Lemma 5.13, the closure of X′ has dimension < n too.) Suppose otherwise. For n = 1 the lemma follows from Lemmas 5.1 and 5.4. Let π_{i} : X′ ⊆ VF^{n} → VF^{n−1} be the projection along the i’th coordinate axis. Let Y be the set of b ∈ VF^{n−1} such that π_{i}^{−1}(b) is infinite, or equivalently contains a ball; it is a definable set. For b ∈ Y, let

D_{i}(b) = {x ∈ π_{i}^{−1}(b) : (∃δ ∈ Γ)(F |π_{i}^{−1}(b) is δ-Lipschitz near x}

By the case n = 1, π_{i}^{−1}(b) \ D_{i}(b) is finite. Thus if D_{i} = ∪_{b∈Y} D_{i}(b), then π_{i} has finite fibers on X \ D_{i}, so dim(X \ D_{i}) < n. Let X^{∗} = ∩_{i}D_{i}, and for x ∈ X^{∗} let δ(x) be the infinimum of all such Lipschitz constants δ (for all n projections.) By Lemma 5.13, δ is constant on some open polydisc U ⊆ X^{∗}. Let δ′ be greater than this constant value. Then at any x ∈ U, the restriction of F to a line parallel to an axis is δ′-Lipschitz. It follows immediately (using the ultrametric inequality) that F is δ′-Lipschitz on U; but this contradicts the definition of X′. □

Remark 5.15. Via Assumption 3.4 (1), we used the existence of p-torsion points in the kernel of RV → Γ for each p. In ACV F (p, p) this fails; one can still show that F is locally logarithmically Lipschitz, i.e. for some rational α > 0, for any x ∈ X \ X′, for sufficiently close x′, val(F (x) − F (x′)) > δval(x − x′).

5.4. Differentiation in VF. Let F : VF^{n} → VF be a definable function, defined on a neighborhood of a ∈ VF^{n}. We say that F is differentiable at a if there exists a linear map L : VF^{n} → VF such that for any γ ∈ Γ, for large enough δ ∈ Γ, if val(x_{i}) > δ for each i, x = (x_{1}, …, x_{n}), then val(F (a + x) − F (a) − Lx) > δ + γ. If such an L exists it is unique, and we denote it dF_{a}.

Lemma 5.16. Let F : X ⊆ VF^{n} → VF^{m} be a definable function. Then each partial derivative is defined at almost every a ∈ X.

Proof. We may assume n = m = 1. Let g(x) = (F (a+x)−F (a))/x. By Lemma 5.4, for almost every a, for some δ ∈ Γ, for all x with val(x) sufficiently large, val(F (a+x)−F (a)) = δ +val(x); so valg(x) is bounded. By Lemma 5.11, and Proposition 5.1, g(x) approaches a limit b ∈ VF as x → 0 (with x ≠ 0); the lemma follows. □

Corollary 5.17. Let F : VF^{n} → VF be a definable function. Then F is continuously differentiable away from a subset of dimension < n

Proof. F has partial derivatives almost everywhere, and these are continuous almost everywhere, so the usual proof works. □

Lemma 5.18. Let X ⊆ VF^{n} × RV^{m} be definable, pr : X → VF^{n} the projection. Then for almost every p ∈ VF^{n}, there exists an open neighborhood U of p and H ⊆ RV^{m} such that pr^{−1}(U) = U × H. If h : X → VF, then for almost all x ∈ X, h is differentiable with respect to each VF-coordinate.

Proof. For x ∈ VF^{n}, let H(x) = {h ∈ RV^{m} : (x, h) ∈ X}. By Corollary 3.24, Lemma 2.8, there exists H′ ⊆ RV^{m}×RV^{l}×Γ^{k} such that for any x ∈ VF^{n}, there exists a unique y = f(x) ∈ RV^{l}×Γ^{k} with H(x) = H′(y). By Lemma 5.13, f is locally constant almost everywhere. Thus for almost all x, for some neighborhood U of x, for all x′ ∈ U, H(x) = H(x′); so pr^{−1}U = U × H(x). The last assertion is immediate. □

This allows the definition of partial derivatives of a definable map F : X → VF (almost everywhere); we just take them with respect to the VF-coordinates, ignoring the RV-coordinates.

Given h : X → VF^{n}, h′ : X′ → VF^{n} with RV-fibers, and a definable map F : X → X′, we define the partials of F to be those of h′ ∘ F. Then the differential dF_{x} exists at almost every point x ∈ X by Corollary 5.17, and we denote the determinant by Jcb, and refer to it as usual as the Jacobian.

Definition 5.19. Let X, X′ ∈ VF[n, ·] and let F : X → X′ be a definable bijection. F is measure preserving if rvJcb(x) = 1 for almost all x ∈ X. VF_{vol}[n, ·] is the subcategory of VF[n, ·] with the same objects, and whose morphisms are the measure preserving morphisms of VF[n, ·].

Let VF_{vol} be the category whose objects are those of VF[n, ·], and whose morphisms X → Y are the essential bijections f : X → Y that are measure preserving.

5.5. Differentiation and Jacobians in RV. Let X, Y be definable sets, together with finiteto-one definable maps f_{X} : X → RV^{n}, f_{Y} : Y → RV^{n}. Here X, Y can be subsets of RV^{∗} or of RV^{∗} × VF^{∗}, etc.; the notion of Jacobian will not depend on the particular realization of X, Y.

Let h : X → Y be a definable map.

The notion of Jacobian will depend not only on h, X, Y but also on f_{X}, f_{Y}; to emphasize this we will write h : (X, f_{X}) → (Y, f_{Y}).

We first define smoothness. When A = f_{X}(X), B = f_{Y} (Y) are definable subsets of k^{n}, we say h, X, Y are smooth if A, B are Zariski open, {(f_{X}(x), f_{Y} (h(x))) : x ∈ X} ∩ (A × B) = Z for some nonsingular Zariski closed set Z ⊂ A × B, and the differentials of the projections to A and to B are isomorphisms at any point z ∈ Z. In this case, composing the inverse of one of these differentials with the other, we obtain a linear isomorphism T_{a}(A) → T_{b}(B) for any a = f_{X}(x), b = f_{Y} (h(x)); since T_{a}(A) = k^{n} = T_{b}(B), this linear isomorphism is given by an invertible matrix, whose determinant is the Jacobian J.

In general, to define smoothness of X, Y at (x, y = h(x)), we restrict to the cosets of (k^{∗})^{n} containing x and y, translate multiplicatively by x and y respectively, and pose the same condition.

Any X, Y, h are smooth outside of a set E where E ∩ C has dimension < n for each coset C of (k^{∗})^{n}. Equivalently (by Lemma 3.64) E has RV-dimension < n.

Assume now that X, Y, h are smooth. Define

Jcb_{RV}(h)(q) = Π(f_{X}(q))^{−1}Π(f_{Y} (q′))J(1, 1) ∈ RV

where Π(c_{1}, …, c_{n}) = c_{1} · · · · · c_{n}.

At times it is preferable not to use a different translation at each point of a coset of (k^{∗})^{n}. The Jacobian Jcb_{RV}(h) of h at q ∈ X can also be defined as follows. Let q′ = h(q), γ = val_{rv}(q), γ′ = val_{rv}(q′) ∈ Γ^{n}. Pick any c, d ∈ RV^{n} with val_{rv}(c) = γ, val_{rv}(d) = γ′ (one can take c = f_{X}(q), d = f_{Y} (q′). ) Let

W (γ, γ′) = {a : f_{X}(a) ∈ val_{rv}^{−1}(γ), f_{Y} (h(a)) ∈ val_{rv}^{−1}(γ′))}
H′ = {(c^{−1}f_{X}(a), d^{−1}f_{Y} (h(a))) : a ∈ W }

Since f_{X}, f_{Y} are finite-to-one, H′ ⊂ (k^{∗}^{n})^{2} both projections of H′ to k^{∗}^{n} are finite-to-one, and H′ is nonsingluar by the smoothness of (X, Y, h). We can thus define the Jacobian J′ of H′ at any point. We have:

Jcb_{RV}(h)(q) = Π(c)^{−1}Π(d)J′(qc^{−1}, q′d^{−1}) ∈ RV

We also define Jcb_{Γ}(h)(q) = ∑ γ′ − ∑ γ ∈ Γ (writing Γ additively). Note that this depends only on the value of h at q. We have:

val_{rv}Jcb_{RV}(h)(q) = Jcb_{Γ}(h)(q)

Example 5.20. Jacobian of maps on Γ. If X̄, Ȳ ⊂ Γ^{n}, we saw that a definable map f̄: X̄ → Ȳ lifts to RV iff it is piecewise given by an element of GL_{n}(ℤ) composed with a translation. Assume f̄ is given by a matrix M ∈ GL_{n}(ℤ), let X = val_{rv}^{−1}(X̄), Y = val_{rv}^{−1}(Ȳ), and let f : X → Y be given by the same matrix, but multiplicatively. Then X, Y, f are smooth, and

J(f)(x) = Π(y)Π(x)^{−1} det M

where y = f(x), and det(M) = ±1.

5.5.1. Alternative: Γ-weighted polynomials. We have seen that the geometry on val_{rv}^{−1}(γ) (γ ∈ Γ^{n}) translates to the geometry on (k^{∗})^{n}, but this is true for the general notions and not for specific varieties; a definable subset of C(γ) = val^{−1}(γ) does not correspond canonically to any definable subset of val^{−1}(0). An invariant approach is therefore useful. Let Γ_{0} = Γ(< ∅ >). X = (X_{1}, …, X_{n}) be variables, γ = (γ_{1}, …, γ_{n}) ∈ Γ_{0}^{n}, and let ν = (ν(1), …, ν(n)) ∈ ℕ^{n} denote a multi-index. By a γ-weighted monomial we mean an expression a_{ν}X^{ν} with a_{ν} a definable element of RV, such that val_{rv}(a_{ν}) + ∑ ν(i)γ_{i} = 0 ∈ Γ. Let Mon(γ, ν) be the set of γ−-weighted monomials of exponent ν, together with 0. Then Mon(γ, ν) \ {0} is a copy of val_{rv}^{−1}(−(a_{ν}) + ∑ ν(i)γ_{i}); so Mon(γ, ν) is a one-dimensional k-space. In particular addition is defined in Mon(γ, ν). We also have a natural multiplication Mon(γ, ν) × Mon(γ, ν′) → Mon(γ, ν + ν′). Let R[X; γ] = ⊕_{ν∈ℕ^{n}}Mon(γ, ν). This is a finitely generates graded k-algebra. It may be viewed as an affine coordinate ring of C[γ]; but the ring of the product C[γ, γ′] is R[X, X′; (γ, γ′)], in general a bigger ring than R[X, γ]⊗_{k}R[X′, γ′]. Nevertheless, a Zariski closed subset of C(γ) corresponds to a radical ideal of R[X′; γ]. In this way notions such as smoothness may be attributed to closed or constructible subsets of any C(γ) in an invariant way.

Definition 5.21. Let X, Y ∈ Ob RV[n, ·] and let h : X → Y be a definable bijection. h is measure preserving if Jcb_{Γ}h(x) = 0 for all x ∈ X, and Jcb_{RV}h(x) = 1 for all x ∈ X away from a set of RV-dimension < n. If only the first condition holds, we say h is Γ-measure preserving.

For X, Y ∈ RV[≤ n, ·], we say h : X → Y is measure preserving if this is true of the RV[n]-component of h.

RV_{vol}[n, ·] (respectively RV_{Γ−vol}[n, ·]) is the subcategory of RV[n, ·] with the same objects, and whose morphisms are the measure preserving (respectively Γ-measure preserving) definable bijections.

RV_{vol}[≤ n, ·] = ⊕_{k<n}RV_{Γ−vol}[k, ·] ⊕ RV_{vol}[n, ·].

Note that when X, Y ∈ Ob RV[n, ·], a bijection h : X → Y is Γ-measure preserving iff it leaves invariant the sets S_{γ} = {(a_{1}, …, a_{n}) : ∑_{i=1}^{n} val_{rv}(a_{i}) = γ}.

5.6. Comparing the derivatives. Consider a definable function F : VF → VF lying above f : RV → RV, i.e. rvF = frv. The fibers of the map rv : VF → RV above k, for instance, are open balls of valuative radius 0, whereas the derivative is defined on the scale of balls of radius r for r → +∞. Thus the comparison between the derivatives of F and f is not tautological. Nevertheless one obtains the expected relation almost everywhere.

While this case of the affine line would suffice (using the usual technique of partial derivatives), it is easier to place oneself in the more general context of curves. More precisely we consider definable sets C together with finite-to-one maps f : C → RV. Let 𝕃C and ρ : 𝕃C → C be as above.

In the following lemma, H′, h′ denote respectively the VF, RV-derivatives of functions H, h defined on objects of VF[1], RV[1] respectively.

Proposition 5.22. Let C_{i} ⊆ RV^{∗} be definable sets, f_{i} : C_{i} → RV finite-to-one definable maps (i = 1, 2). Let h : C_{1} → C_{2} be a definable bijection, and let H : 𝕃C_{1} → 𝕃C_{2} be a lifting of h, i.e. ρH = hρ. Then

(1) For all but finitely many c ∈ C_{1}, h is differentiable at c, H is differentiable at any x ∈ 𝕃c, and rvH′(x) = h′(rv(x)).

(2) For all c ∈ C_{1}, H is differentiable at a generic x ∈ 𝕃c, and valH′(x) = (val_{rv}h′)(x) = val(f_{2}(h(x))) − val(f_{1}(x)).

Proof. (1) Let Z′ be the set of x ∈ 𝕃C_{1} such that H is not differentiable at x (a finite set) or that rv(H′(x)) ≠ h′(rv(x)). We have to show that ρ(Z′) ⊆ C is finite, or equivalently that f_{1} ∘ ρ(Z′) is finite. Otherwise there exists c ∈ ρ(Z′) with c ∉ acl(A). By Lemma 3.20, the formula rv(x) = f_{1}(c) generates a complete type q over A(c); it defines a transitive open ball b_{c} over A(c). Since ρ ∘ H = ρ ∘ h, we have H(c, y) = (c, H_{c}(y)) for some A(c)-definable bijection H_{c} of b_{c}. By Lemma 5.5, for some e_{0} ∈ RV, rv(H(u) − H(v)) = e_{0}rv(u − v) for all u, v ∈ b_{c}; so rv((H(u) − H(v))/(u − v)) = e_{0}. Since H is differentiable almost everywhere on b_{c} ( Lemma 5.17) and b_{c} is transitive, it is differentiable at every point. Clearly rvH′(u) = e_{0}, contradicting the definition of Z′.

(2) follows from Lemma 5.4 □

Corollary 5.23. Let X ∈ Ob RV[n], F : 𝕃X → VF^{n} a definable function, f : 𝕃X → RV^{n} a definable function. Assume rvF (x) = f(rv(x)). Then Proposition 5.22 applies for each partial derivative of F. In particular,

• For all c ∈ X away from a set of smaller dimension, for all x ∈ 𝕃c, F is differentiable at x, f is differentiable at c, and rvJcb(F)(x) = Jcb_{RV}(f)(x).

• For all c ∈ X, for generic x ∈ 𝕃c, F is differentiable at x, and valJcb(F)(x) =

(Jcb_{Γ}f)(x)

□

Corollary 5.24. Let X, Y ∈ Ob RV[≤ n], f ∈ Mor_{RV[≤n]}(X, Y), F ∈ Mor_{VF_{vol}[n]}(𝕃X, 𝕃Y). Assume rvF (x) = f(rv(x)). Then f ∈ Mor_{RV_{vol}[n]}(X, Y). □

Proof. Follows from Corollary 5.23.

6. Lifting functions from RV to VF

Proposition 6.1. Let T be an effective V-minimal theory. Let X ⊂ RV^{k} be definable and let ϕ_{1}, ϕ_{2} : X → RV^{n} be two definable maps with finite fibers. Then there exists a definable bijection F : X ×_{ϕ_{1},rv} (VF^{×})^{n} → X ×_{ϕ_{2},rv} (VF^{×})^{n}, commuting with the natural projections to X.

Proof. Let A = dcl(∅) ∩ (VF ∪ Γ). If b ∈ dcl(∅) ∩ RV, then viewed as a ball b has a point a ∈ A; since the valuative radius of b is also in A, we have b ∈ dcl(A). Thus ϕ_{1}, ϕ_{2}, X are ACVF_{A}- definable. Any ACV F_{A}-definable bijection F is a fortiori T-definable; so the Proposition for ACVF_{A} implies the Proposition for T. Moreover ACVF_{A} is V-minimal and effective, since any algebaic ball of ACVF_{A} is T_{A}-algebraic and hence has a point in VF(A)^{alg}. Thus we may assume T = ACV F_{A}.

The proof will be asymmetric, concentrating on ϕ_{1}X.

We may definably partition X, and prove the proposition on each piece.

Consider first the case where ϕ_{1} : X → U and ϕ_{2} : X → V are bijections to definable subsets U, V ⊆ (k^{∗})^{k}. Our task is to lift the bijection f = ϕ_{2}ϕ_{1}^{−1} to VF^{n}. A definable subset of k^{n} (such as ϕ_{i}(X)) is a disjoint union of smooth varieties. We thus consider a definable bijection f : U → V between k varieties U ⊂ k^{n} and V ⊂ k^{n}. Induction on dim(U) will allow us to remove a subset of U of smaller dimension. Hence we may assume U is smooth, cut out by h = (h_{1}, …, h_{l}), T U = Ker(dh), f = (f_{1}, …, f_{n}) where f_{i} are regular on U (defined on an open subset of k^{n}), and df is injective on T U at each point of U. Thus the common kernel of dh_{1}, …, dh_{l}, df_{1}, …, df_{n} equals 0.

It follows that at a generic point of U (i.e. every point outside a proper subvariety), if Q is a sufficiently generic n × l matrix of elements of A (or integers), and we let f′_{i} = f_{i} + Qh, then the common kernel of df′_{1}, …, df′_{n} vanishes. Note that f_{i}|U = f′_{i}|U. Let W be a smooth variety contained in f(U) and whose complement in f(U) is a constructible set of dimension smaller than dim(U). Replacing U by f^{−1}(W), we may assume f(U) is also a smooth variety.

Let Ũ = res^{−1}(U). Lift each f′_{i} to a polynomial F_{i} over 𝒪, with definable coefficients. This is possible by effectiveness of ACVF_{A}. Obtain a regular map F, whose Jacobian is invertible at points of Ũ. We have res ∘ F = f ∘ res. Since f is 1-1 on U, the invertibility of dF implies that F is 1-1 on Ũ. Moreover, by Hensel’s lemma, F : rv^{−1}(U) → rv^{−1}(W) is bijective.

Next consider the case where in place of a bijection f : U → V we have a finite-to-finite correspondence f̃ ⊂ U × V (where U = ϕ_{1}(X), V = ϕ_{2}(X)), f̃ = {(ϕ_{1}(x), ϕ_{2}(x)) : x ∈ X}. We may take f̃ ⊂ U × V to be a subvariety, unramified and quasi-finite over U and over V; and we can take U, V to be smooth varieties. As before we can lift f̃ to a correspondence F̃ ⊂ Ũ × Ṽ, such that F̃ ∩rv^{−1}(u)×rv^{−1}(v) is a bijection rv^{−1}(u) → rv^{−1}(v) whenever (u, v) ∈ f̃. It follows that a bijection X ×_{ϕ_{1},rv} (VF^{×})^{n} → X ×_{ϕ_{2},rv} (VF^{×})^{n} is given by: (x, y) ↦ (x, y′) iff (y, y′) ∈ F̃.

If ϕ_{1} : X → U and ϕ_{2} : X → V are bijections to definable subsets U, V, each contained in a single coset of (k^{∗})^{k}, say U ⊆ C(γ), V ⊆ C(γ′) for some γ, γ′ ∈ Γ^{k} (cf. §5.5.1.) Let Z = (Z_{1}, …, Z_{k}) be variables, R[Z; γ] be the subring of VF[Z] consisting of polynomials ∑ a_{ν}Z^{ν}, with val(a_{ν}) + ∑_{i=1}^{k} ν(i)γ_{i} = 0, and a_{ν} a definable element of VF. There is a natural homomorphism R′[Z; γ] → R[Z; γ], where R[Z; γ] is the coordinate ring of C(γ). By effectivity, this homomorphism is surjective. The proof now proceeds in exactly the same way as above.

This proves the proposition in case val_{rv}ϕ_{i}(X) consists of one point.

Next assume val_{rv}ϕ_{2} consists of one point, and val_{rv}ϕ_{1}(X) is finite. So ϕ_{1}(X) lies in the union of finitely many cosets (C(a) : a ∈ E), with E finite.

For a ∈ E, A(a) remains almost VF, Γ-generated; since the proposition is true for ϕ_{1}^{−1}C(a) (definable in T_{A(a)}), then so by the one-coset case an appropriate isomorphism F exists; and the finitely many F obtained this way can then be glued together, to yield a map defined over A.

The case of val_{rv}ϕ_{1}, val_{rv}ϕ_{2} both finite is treated similarly.

This proves the existence of a lifting in case val_{rv}ϕ_{i}(X) is finite. Now for the general case. Claim Let P ⊂ X be a complete type. Then there exists a definable D with P ⊂ D ⊂ X , and definable functions θ on val_{rv}(ϕ_{1}(D)) and θ′ on val_{rv}(ϕ_{2}(D)) such that for x ∈ D, θ(val_{rv}(ϕ_{1}(x))) = val_{rv}ϕ_{2}(x), θ′(val_{rv}(ϕ_{2}(x))) = val_{rv}ϕ_{1}(x).

Proof. Let a ∈ P, γ_{i} = val_{rv}(ϕ_{i}(a)). Then γ_{2} is definable over some points of ϕ_{1}^{−1}val_{rv}^{−1}(γ_{1}). But val_{rv}^{−1}(γ_{1}) is a coset of k^{∗}, and ϕ_{1} is finite-to-one, so ϕ_{1}^{−1}val_{rv}^{−1}(γ_{1}) is orthogonal to Γ. Thus γ_{2} is algebraic over γ_{1}. Since Γ is linearly ordered, γ_{2} is definable over γ_{1}; so γ_{2} = θ(γ_{1}) for some definable θ. Similarly γ_{1} = θ′(γ_{2}). Clearly θ restricts to a bijection val_{rv}ϕ_{1}P → val_{rv}ϕ_{2}P, with inverse θ′. By Lemma 2.7 there exists a definable D with θϕ_{1} = ϕ_{2}, ϕ_{1} = θ′ϕ_{2} on D.

Now by compactness, there exist finitely many (D_{i}, θ_{i}, θ′_{i}) as in the claim with ∪_{i}D_{i} = X. We may cut down the D_{i} successively, so we may assume the union is disjoint. But in this case the proposition reduces to the case of each individual D_{i}, so we may assume X = D. Let B_{i} = val_{rv}ϕ_{i}(X). Given b ∈ B_{1}, let X_{b} = (val_{rv}ϕ_{1})^{−1}(b). Then by the case already considered there exists an A(b)-definable F_{b} : X_{b} ×_{ϕ_{1},rv} (VF^{×})^{n} → X_{b} ×_{ϕ_{2},rv} (VF^{×})^{n}. Let F = ∪_{b∈B_{1}}F_{b}. By Lemma 2.3, F : X ×_{ϕ_{1},rv} (VF^{×})^{n} → X ×_{ϕ_{2},rv} (VF^{×})^{n} is bijective. (see discussion in §2.1.1.)

□

We note a corollary:

Lemma 6.2. Let T be V-minimal and effective, and let A be an almost (VF, Γ)-generated structure. Then A is effective.

Proof. By Lemma 3.29 it suffices to show A is rv-effective. Note that if A ⊆ acl(∅), then T is rv-effective iff T_{A} is rv-effective (see proof of Lemma 3.31 (2 → 3).) Thus it suffices to show that if A_{0} = acl(A_{0}), a ∈ VF ∪ Γ, and T′ = T_{A_{0}} is effective then so is T′(a). The case a ∈ Γ is included in Corollary 3.40, so assume a ∈ VF. Let P be the intersection of all A_{0}-definable balls containing a. If P is transitive over A_{0}, then by Lemma 3.47 we have RV(A_{0}(a)) = RV(A_{0}), so rv-effectivity remains true trivially. Otherwise P is centered over A_{0}, hence has an A_{0}-definable point, and by translation we may assume 0 ∈ P. a is then a generic point of P over A_{0}. Let c ∈ RV(A_{0}(a)); we must show that rv^{−1}c is centered over A_{0}(a). By Lemma 3.20, if c ∈ RV(A_{0}(a)) then c = f(d) for some A_{0}-definable function f : RV → RV, where d = rv(a). By Lemma 6.1 there exists an A_{0}-definable function F : VF → VF lifting f. Then F (d) ∈ rv^{−1}(c). □

6.0.1. Base change: summary. Base change from T to T_{A} preserves V-minimality, effectiveness and being resolved, if A is VF-generated; V-minimality and effectiveness, if A is RV-generated; V-minimality, if A is Γ-generated. (Lemmas 6.2, 3.39, 3.40. The resolved case follows using 3.49.)

Though the notion of a morphism g : (X_{1}, ϕ_{1}) → (X_{2}, ϕ_{2}) does not depend on ϕ_{1}, ϕ_{2}, recall that the RV-Jacobian of g is defined with reference to these finite-to-one maps.

Lemma 6.3. Let T be V-minimal and effective. Let X_{i} ⊂ RV^{k_{i}} be definable and let ϕ_{i} : X → RV^{n} be definable maps with finite fibers; let g : X_{1} → X_{2} be a definable bijection. Assume given in addition a definable function δ : X_{1} → RV, such that:

(1) val_{rv}δ(x) = Jcb_{Γ}g(x) for all x ∈ X_{1}.

(2) δ(x) = Jcb_{RV}g(x) for almost all x ∈ X_{1} (i.e. all x outside a set of dimension < n.)

Then there exists a definable bijection G : X_{1} ×_{ϕ_{1},rv} (VF^{×})^{n} → X_{2} ×_{ϕ_{2},rv} (VF^{×})^{n} such that ρ_{2} ∘ G = g ∘ ρ_{1}, where ρ_{i} is the natural projections to the X_{i}, and such that for any x ∈ X_{1} ×_{ϕ_{1},rv} (V F^{∗})^{n}, G is differentiable at x, and rv(Jcb(G)(x)) = δ(x)

Proof. We follow closely the proof of Proposition 6.1. As there, we may assume T = ACV F_{A}, with A be an almost (VF, Γ)-generated substructure.

We first assume that val_{rv}ϕ_{1}(X_{1}) is a single point of Γ^{n}

Then X_{1} can be definably embedded into k^{N} for some N, and it follows from the orthogonality of k and Γ that the image of X_{1} in Γ under any definable map is finite. Thus ϕ_{2}X_{2} is contained in finitely many cosets (C(a) : a ∈ S) of (k^{∗})^{n}; by partitioning X_{1} working in T_{A(a)}, we may assume ϕ_{2}X_{2} is contained in a single coset. (cf. Lemma 2.3).

As in Proposition 6.1, we may assume ϕ_{i}X ⊆ k^{n}, and indeed that ϕ_{1}X = U, ϕ_{2}X = V are smooth varieties. If dim(U) = n, then the lift constructed in Proposition 6.1 satisfies rv(Jcb(G))(x) = Jcb_{RV}g(x) for x ∈ X ×_{ϕ_{1},rv} VF^{n}; thus by assumption (2), we have rv(Jcb(G)))(x) = δ(x) for almost all x. The exceptional points have dimension < n, and may be partitioned into smooth varieties of dimension < n. Thus we are reduced to the case dim(U) < n. We prove it by induction on dim(U). In this case choose any lifting G_{0}. We have an error term e(x) = rv(Jcb(G_{0}))(x)^{−1}δ(x). Now A(x) is almost VF, Γ-generated, and so balls rv^{−1}(y) contain definable points; thus e(x) = rvE(x) for some definable E : (X ×_{ϕ_{1},rv} VF^{n}) → VF. Since U is a smooth subvariety of k^{n} of positive codimension, some regular h on k^{n} vanishes on V, while some partial derivative (say h_{1}) vanishes only on a smaller-dimensional subvariety. By induction may assume h_{1} vanishes nowhere. Lift h to H; so H_{1} lifts h_{1}. Compose G_{0} with a map fixing all coordinates but the first, and multiplying the first coordinate by E(x)H(y)/H_{1}(y). (Here x = g^{−1}(y).) Where h vanishes, this has Jacobian E(x); so the composition has RV- Jacobian δ(x) as required.

Now in general. For any γ ∈ Γ^{n} let X_{1}(γ) = {x ∈ X_{1} : val_{rv}ϕ_{1}(x) = γ}, X_{2}(γ) = g(X_{1}(γ)). By definition of Jcb_{RV} and Jcb_{Γ}, Jcb_{RV}(g|X_{2}(γ)) = Jcb_{RV}(g)|X_{2}(γ) and likewise Jcb_{Γ}. By the case already analyzed (for the sets X_{1}(γ), X_{2}(γ) defined in ACV F_{A(γ)}) there exists an A(γ)- definable bijection G_{γ} : X_{1}(γ) ×_{ϕ_{1},rv} (V F^{×})^{n} → X_{2}(γ) ×_{ϕ_{2},rv} (V F^{×})^{n} with rv(Jcb(G_{γ})(x)) = δ(x). As in Lemma 2.3 one can extend the G_{γ} by compactness to definable sets containing γ, cover X_{1} by finitely many such definable sets, and glue together to obtain a single function G with the same property.

□

Remark Assume Id_{X} : (X, ϕ_{1}) → (X, ϕ_{2}) has Jacobian 1 everywhere. Then it is possible to find F that is everywhere differentiable, of Jacobian precisely equal to 1. At the before the point where Hensel’s lemma is quoted, it is possible to multiply the function by J(F)^{−1} (not effecting the reduction, since J(F) ∈ 1 + ℳ.) Then one obtains on each such coset a function of Jacobian 1, and therefore globally.

Example Let ϕ_{2}(x) = ϕ_{1}(x)^{m}. A definable bijection

X ×_{ϕ_{1},rv} (VF^{×})^{n} → X ×_{ϕ_{2},rv} (VF^{×})^{n}

is given by (x, y) ↦ (x, y^{m}). (If rv(u) = ϕ(x)^{m}, there exists a unique y with rv(y) = ϕ(x) and y^{m} = u.) Example 6.4. Proposition 6.1 need not remain valid over an RV-generated base set. Let A = dcl(c)}, c a transcendental point of k. Let f_{1}(y) = y, f_{2}(y) = 1, 𝕃(Y, f_{i}) := VF ×_{rv,f_{i}} Y = {(x, y) ∈ RV × Y : rv(x) = y}. Then 𝕃(Y, f), 𝕃(Y, f′) are both open balls; over any field A′ containing A, they are definably isomorphic, using a translation. But these balls are not definably isomorphic over A.

7. Special bijections and RV-blowups

We work with a V-minimal theory T. Recall the lift 𝕃 : RV[≤ n, ·] → VF[n, ·], with ρ_{X} : 𝕃X → X. Our present goal is an intrinsic description in terms of RV of the congruence relation: 𝕃X ≃ 𝕃Y.

A will denote a (VF, Γ, RV)-generated substructure of a model of T. Note that T_{A} is also V-minimal (Corollary 3.39) so any lemma proved for T under our assumptions can be used for any T_{A}.

The word “definable” below refers to T. The categories VF, RV[∗] defined below thus depend on T; when necessary, we will denote them VF_{T}, etc. When T has the form T = T_{A}^{0} for fixed T^{0} but varying A, we write VF_{A}, etc.

7.1. Special bijections. Let X ⊆ VF^{n+1} × RV^{m} be ∼_{rv}-invariant. Say

X = {(x, y, u) ∈ VF × VF^{n} × RV^{m} : (rv(x), rv(y), u) ∈ X̄}

(We allow x to be any of the n + 1 coordinates, y the others.)

Let s(y, u) be a definable function into VF with ∼_{rv}-invariant domain of definition

dom(s) = {(y, u) : (rv(y), u) ∈ S̄}

and θ(u) a definable function on pr_{u}(dom(s)) into RV, such that (s(y, u), y, u) ∈ X and rv(s(y, u)) = θ(u) for (y, u) ∈ dom(s). Note that θ is uniquely defined (given s) if it exists. Let

X_{1} = {(x, y, u) ∈ X : (rv(y), u) ∈ S̄, rv(x) = θ(u)}, X_{2} = X \ X_{1}
X′_{1} = {(x, y, u) ∈ VF × dom(s) : val(x) > val_{rv}θ(u)}

and let X′ = X′_{1} ∪^{.} X_{2}. Also define e_{s} : X′ → X to be the identity on X_{2}, and

e_{s}(x, y, u) = (x + s(y, u), y, u)

on X′_{1}. Definition 7.1. e_{s} : X′ → X is a definable bijection, called an elementary bijection. □ Lemma 7.2. (1) If X is ∼_{rv}-invariant, so is X′. If X → VF^{n+1} is finite-to-one, the same

is true of X′.

(2) If X_{i} = 𝕃X̄_{i}, X′_{1} = 𝕃X̄′_{1}, then X̄′_{1} is isomorphic to (RV^{>0} ∪^{.} {1}) × S̄, while X̄_{1} is isomorphic to S̄.

(3) If the projection X → VF^{n+1} has finite fibers, then so do the projection dom(s) → VF^{n}, and also the projection S̄ → RV^{n}, (y′, u) ↦ y′.

(4) e_{s} has partial derivative matrix I everywhere, hence has Jacobian 1. Thus if F : X → Y is such that rvJcbF factors through ρ_{X}, then rvJcb(F ∘ e_{s}) factors through ρ_{X′}.

Proof. (1,4) are clear. The first isomorphism of (2) is obtained by dividing x by θ(u), the second is evident. For (3), note that if (y, u) ∈ dom(s) then (s(y, u), y, u) ∈ X so by the assumption u ∈ acl(y, s(y, u)). But for fixed y, {s(y, u) : u ∈ dom(s)} is finite, by Lemma 3.41. So in fact (y, u) ∈ dom(s) implies u ∈ acl(y). Hence (y′, u) ∈ S̄ implies u ∈ acl(y) for any y with rv(y) = y′, so (fixing such a y) {u : (y′, u) ∈ S̄} is finite for any given y′. □

A special bijection is a composition of elementary bijections and auxiliary bijections (x_{1}, …, x_{n}, u) ↦ (x_{1}, …, x_{n}, u, rv(x_{1}), …, rv(x_{n})).

An elementary bijection depends on the data s of a partial section of X → VF^{n} × RV^{m}. Conversely, given s, if rv(s(y, u)) depends only on u we can define θ(u) = rv(s(y, u)) and obtain a special bijection. If not, we can apply an auxiliary bijection to X ⊆ VF^{n} × RV^{m}, and obtain a set X′ ⊆ VF × RV^{m+n}, such that rv(x) = pr_{m+1}(u) for (x, u) ∈ X′. For such a set X′, the condition for existence of θ is automatic and we can define an elementary bijection X′′ → X′ based on s, and obtain a special bijection X′′ → X as the composition.

The classes of auxiliary morphisms and elementary morphisms are all closed under disjoint union with any identity morphism, and it follows that the class special morphisms under disjoint unions.

7.2. Special bijections in one variable and families of RV-valued functions. We consider here special bijections in dimension 1. An elementary bijection X′ → X in dimension one involves a finite set B of rv-balls, and a set of “centers” of these balls (i.e. a set T containing a unique point t(b) of each b ∈ B), and translates each ball so as to be centered at 0 (while fixing the RV coordinates.) We say that X′ → X blows up the balls in B, with centers T.

Given a special bijection h′ : X′ → X, let F n^{RV}(X; h′) be the set of definable functions X → RV of the form H(ρ_{X′}((h′)^{−1}(x))), where H is a definable function. This is a finitely generated set of definable functions X → RV. There will usually be no ambiguity in writing F n^{RV}(X, X′ → X) instead.

Note that while a special bijection is an isomorphism in VF, an asymmetry exists If e : X′ → X is a special bijection, then F n^{RV}(X, X) ⊆ F n^{RV}(X, X′ → X), usually properly.

What is the effect on F n^{RV} of passing from X′ to X′′, where X′′ → X′ is a special bijection? The auxiliary bijections have no effect. Assume rv is already a coordinate function of X′. Consider an elementary bijection e_{s} : X′′ → X′. Let B = {(x, u) ∈ X′ : u ∈ dom(s)}. Then the characteristic function 1_{B} lies in F n^{RV}(X′, Id_{X′}); so 1_{B} ∘ (h′)^{−1} lies in F n^{RV}(X, h′). Using this, we see that F n^{RV}(X′, e_{s}) is generated over F n^{RV}(X′, Id_{X′}) by the function B → RV, (x, u) ↦ rv(x−s(u)). (Extended by 0 outside B.) Thus if h′′ = h′ ∘e_{s} : X′′ → X, F n^{RV}(X, h′′) is generated over F n^{RV}(X, h′) by the composition of the function (x, u) ↦ rv(x − s(u)) with (h′)^{−1}.

Conversely, if B is a finite union of open balls whose characteristic function lies in F n^{RV}(X, h′), and if there exists a definable set T of representatives (one point t(b) in each ball b of B), and a function ϕ = (ϕ_{1}, …, ϕ_{n}), ϕ_{i} ∈ F n^{RV}(X, h′), with ϕ injective on T, then one can find a special bijection X′′ → X′ with composition h′′ : X′′ → X, such that F n^{RV}(X, h′′) is generated over F n^{RV}(X, h′) by y ↦ rv(y − t(y)), y ∈ b ∈ B. Namely let dom(s) = ϕ(T), and for u ∈ dom(s) set s(u) = h′^{−1}(t) if t ∈ T and ϕ(t) = u. In this situation, we will say that the balls in B are blown up by X′′ → X′, with centers T. Let θ(u) = rv(s(u)). Because X′ → X may already have blown up some of the balls in B, F n^{RV}(X, h′′) is generated over F n^{RV}(X, h′) by the restriction of y ↦ rv(y − t(y)) to some sub-ball of b, possibly proper. Nevertheless, we have: Lemma 7.3. The function y ↦ rv(y − t(y)) on B lies in F n^{RV}(X, h′′).

Proof. This follows from the following more general Claim Let c ∈ VF, b ∈ 𝔅 be definable, with c ∈ b. Let b′ be an rv-ball with c ∈ b′. Then the function rv(x − c) on b is generated by its restriction to b′, rv, and the characteristic function of b.

Proof. Let x ∈ b \ b′. From rv(x) compute val(x). If val(x) < val(c), rv(x − c) = rv(c). If val(x) > val(c), rv(x − c) = rv(x). When val(x) = val(c), but x ∉ b′, rv(x − c) = rv(x) − rv(c). Recall here that val_{rv}^{−1}(γ) is the nonzero part of a k-vector space; subtraction, for distinct elements u, v, can therefore be defined by u − v = u(u^{−1}v − 1). □

Thus any special bijection can be understood as blowing up a certain finite number of balls (in a certain sequence and with certain centers.) We will say that a special bijection X′′ → X′ is subordinate to a given partition of X if if each ball blown up by X′′ → X′ is contained in some class of the partition.

It will sometimes be more convenient to work with the sets of functions F n^{RV}(X, h) than with the special bijections h themselves.

We observe that any finite set of definable functions X → RV is contained in F n^{RV}(X; h) for some X′, h:

Lemma 7.4. Let X ⊆ VF × RV^{∗} be ∼_{rv}-invariant , and let f : X → (RV ∪ Γ) be a definable

map. Then there exists a ∼_{rv}-invariant X′ ⊆ VF × RV^{∗} a special bijection h : X′ → X, and a definable function t such that t ∘ ρ_{X′} = f ∘ h. Moreover, if X = ∪_{i=1}^{m}P_{i} is a finite partition of X into sets whose characteristic functions factor through ρ, we can find X′ → X subordinate to this partition.

Proof. Say X ⊆ VF × RV^{m}; let π : X → VF, π′ : X → RV^{m} be the projections. Applying an auxiliary bijection, we may assume rv(π(x)) = pr_{m}π′(x), i.e. rv(π(x)) agrees with one of the coordinates of π′(x). We now claim that there exists a finite F′ ⊆ RV^{m}, such that away from π^{′−1}(F′), f factors through π′. To prove this, it suffices to show that if p is a complete type of X and π′_{∗}p is non-algebraic (i.e. not contained in a finite definable set), then f|p factors through π′; this follows from Lemma 3.20.

We can thus restrict attention to π′^{−1}(F′); our special bijections will be the identity away from this. So we may assume π′(X) is finite. Recall that (since an auxiliary bijection has been applied) rv is constant on each fiber of π′. In this case there is no problem relativizing to each fiber of π′, and then collecting them together ( Lemma 2.3), we may assume in fact that π′(X) consists of a single point {u}. In this case the partition (since it is defined via ρ) will automatically be respected.

The rest of the proof is similar to Lemma 4.2. We consider first functions f with finite support F (i.e. f(x) = 0 for x ∉ F,) and prove the analog of the statement of the lemma for them. If F = {0} × {u} then F = ρ^{−1}({(0)} × {u})) so the claim is clear. If F = {(x_{0}, u)}, let s : {u} → VF, s(u) = x_{0}. Applying e_{s} returns us to the previous case. If F = F_{0} × {u} has more than one point, we use induction on the number of points. Let s(u) be the average of F_{0}. Apply the special bijection e_{s}. Then the result is a situation where rv is no longer constant on the fiber. Applying a auxiliary bijection to make it constant again, the fibers of F → RV^{m+1} become smaller.

The case of the characteristic function of a finite union of balls is similar (following Lemma 4.2.)

Now consider a general function f. Having disposed of the case of characteristic function, it suffices to treat f on each piece of any given partition. Thus we can assume f has the form of Corollary 4.4, f(x) = H(rv(x − n(x))). Translating by the n(x) as in the previous cases, we may assume n(x) = 0. But t then again f factors through ρ and rv, so one additional auxiliary bijection suffices. □

Corollary 7.5. Let X, Y ⊆ VF^{n} × RV^{∗}, and let f : X → Y be a definable bijection. Then there exists a special bijection h : X′ → X, and t such that ρ_{Y} ∘ (f ∘ h) = t ∘ ρ_{X′}

It can be found subordinate to a given finite partition, factoring through ρ_{X}. □

We wish to obtain a symmetric version of Corollary 7.5. We will say that bijections f, g : X → Y differ by special bijections if there exist special bijections h_{1}, h_{2} with h_{2}g = fh_{1}. We show that every definable bijection between ∼_{rv}-invariant objects differs by special bijections from a ∼_{rv}-invariant bijection.

Lemma 7.6. Let X ⊆ VF × RV^{m}, Y ⊆ VF × RV^{m′} be definable, ∼_{rv}-invariant; let F : X → Y be a definable bijection. Then there exist special bijections h_{X} : X′ → X, h_{Y} : Y′ → Y, and an ∼_{rv}-invariant bijection F′ : X′ → Y′ with F = h_{Y} F′h_{X}^{−1}. I.e. F differs from an ∼_{rv}-invariant morphism by special bijections.

Proof. It suffices to find h_{X}, h_{Y} such that F n^{RV}(X, h_{X}) = F ∘ F n^{RV}(Y, h_{Y}); for then we can let F′ = h_{Y}^{−1}F h_{X}.

Let X = ∪_{i=1}^{m}P_{i} be a partition as in Proposition 5.1. By Lemma 7.4, there exist X_{0}, Y_{1} and special bijections X_{0} → X, Y_{1} → Y, such that the characteristic functions of the sets P_{i} (respectively the sets F (P_{i})) are in F n^{RV}(X, X_{0} → X)(respectivelyF n^{RV}(Y, Y_{1} → Y)).

By Corollary 7.5, one can find a special X_{1} → X_{0} such that F n^{RV}(X, X_{1} → X) contains F ∘ F n^{RV}(Y, Y_{1} → Y). By another application of the same, one can find a special bijection Y_{∗} → Y_{1} subordinate to {F (P_{i})} such that

(6)  F n^{RV}(Y, Y_{∗} → Y) ⊇ F^{−1} ∘ F n^{RV}(X, X_{1} → X)

Now Y_{∗} is obtained by composing a sequence Y_{∗} = Y_{m} → … → Y_{1} of elementary bijections and auxiliary bijections. We define inductively X_{m} → … → X_{2} → X_{1}, such that

(7)  F n^{RV}(Y, Y_{k} → Y) ∘ F ⊆ F n^{RV}(X, X_{k} → X)

Let k ≥ 1. Y_{k+1} is obtained by blowing up a finite union of balls B of Y, with definable set T of representatives such that some ϕ ∈ F n^{RV}(Y, Y_{k} → Y) is injective on T; and then F n^{RV}(Y, Y_{k+1} → Y) is generated over F n^{RV}(Y, Y_{k} → Y) by ψ, where for y ∈ b ∈ B ψ(y) = rv(y − t(b)) ( Lemma 7.3). By the choice of the partition {P_{i}}, F^{−1}(B) is also a finite union of balls.

Now F^{−1}(B), with center set F^{−1}(T), can serve as data for a special bijection: the requirement about the characteristic function of B and the injective function on T being in F n^{RV} are satisfied by virtue of Lemma 7.3. We can thus define X_{k+1} → X_{k} so as to blow up F^{−1}(B) with center set F^{−1}(T). By Lemma 5.4, rv(F (x) − F (x′)) is a function of rv(x − x′) (and conversely) on each of these balls, so F n^{RV}(X, X_{k+1} → X) is generated over F n^{RV}(X, X_{k}) by ψ ∘ F. Hence (7) remains valid for k + 1.

Now by (6), F n^{RV}(X, X_{1} → X) ⊆ F n^{RV}(Y, Y_{∗} → Y) ∘ F; since the generators match at each stage, by induction on k ≤ m

(8)  F n^{RV}(X, X_{k} → X) ⊆ F n^{RV}(Y, Y_{m} → Y) ∘ F

By (7) and (8) for k = m, F n^{RV}(X, X_{m} → X) = F n^{RV}(Y, Y_{∗} → Y) ∘ F □

For the sake of possible future refinements, we note that the proof of Lemma 7.6 shows also:

Lemma 7.7. Let X ⊆ VF × RV^{m}, Y ⊆ VF × RV^{m′} be definable, ∼_{rv}-invariant; let F : X → Y be a definable bijection. If a Proposition 5.1 - partition for F has characteristic functions factoring through ρ_{X}, ρ_{Y}, and if F is ∼_{rv}-invariant, then for any special bijection h′_{X} : X′ → X

there exists a special bijection h′_{Y} : Y′ → Y′ such that (h′_{Y})^{−1}F h′_{X} is ∼_{rv}-invariant. □

7.3. Several variables. We will show now in general that any definable map from an ∼_{rv}- invariant object to RV factors through the inverse of a special bijection, and the standard map ρ.

Lemma 7.8. Let X ⊆ VF^{n} × RV^{m} be ∼_{rv}-invariant, and let ϕ : X → (RV ∪ Γ). Then there exists a special bijection h : X′ → X, and a definable function τ such that τ ∘ ρ_{X′} = ϕ ∘ h.

Proof. By induction on n. For n = 0 we can take X′ = X, since ρ_{X} = Id_{X}.

For n = 1 and X ⊆ VF, by Lemma 7.4, there exists μ = μ(X, ϕ) ∈ ℕ such that the lemma holds for some h that is a composition of μ elementary and auxiliary bijections. It is easy to verify the semi-continuity of μ with respect to the definable topology: if X_{t} is a definable family of definable sets, so that X_{b} is A(b)-definable, and μ(X_{b}, ϕ|X_{b}) = m, then there exists a definable set D with b ∈ D and such that if b′ ∈ D, then μ(X_{b′}, ϕ|X_{b′}) ≤ m.

Assume the lemma known for n and suppose X ⊆ VF × Y, with Y ⊆ VF^{n} × RV^{m}. For any b ∈ Y, let X_{b} = {x : (x, b) ∈ X} ⊆ VF; so X_{b} is A(b)-definable.

Let μ = max_{b} μ(X_{b}, ϕ|X_{b}). Consider first the case μ = 0. Then ϕ|X_{b} = τ_{b} ∘ ρ|X_{b}, for some A(b)-definable function τ_{b} : RV^{m} → (RV ∪ Γ). By stable embededness and elimination of imaginaries in RV ∪ Γ, there exists (§2.1.5) a canonical parameter d ∈ (RV ∪ Γ)^{l}, and an A-definable function τ, such that τ_{b}(t) = τ(d, t); and d itself is definable from τ_{b}, so we can write d = δ(b) for some definable δ : Y → (RV∪Γ)^{l}. Using the induction hypothesis for (Y, δ) in place of (X, ϕ), we find that there exists a ∼_{rv}-invariant Y′ ⊆ VF^{n} ×RV^{∗}, a special h_{Y} : Y′ → Y, and a definable τ_{Y}, such that τ_{Y} ∘ ρ_{Y′} = δ ∘ h_{Y}. Let X′ = X ×_{Y} Y′, h(x, y′) = (x, h_{Y} (y′)). An elementary bijection to Y determines one to X, where the function s does not make use of the first coordinate; so h : X′ → X is special. In this case the lemma is proved: ϕ ∘ h(x, y′) = ϕ(x, h_{Y} (y′)) = τ(δ(h_{Y} (y′)), ρ(x, y)) = τ(τ_{Y} (ρ_{Y′}(y′)), ρ(x, y)).

Next suppose μ > 0. Applying an auxiliary bijection, we may assume that for some definable function (in fact projection) p, rv(x) = p(u) for (x, y, u) ∈ X. For each b ∈ Y (M) (with M any model of T_{A}) there exists an elementary bijection h_{b} : X′_{b} → X_{b}, such that μ(X′_{b}, ϕ|X′_{b}) < μ; h_{b} is determined by s_{b}, θ_{b},with s_{b} ∈ rv(s_{b}) = θ_{b}, and (s_{b}, θ_{b}) ∈ X. (The u-variables have been absorbed into b.) By compactness, one can take s_{b} = s(b) and θ_{b} = θ′(b) for some definable functions s, θ′. By the inductive hypothesis applied to (Y, θ′), as in the previous paragraph, we can assume θ′(y, u) = θ(u) for some definable θ. Applying the special bijection with data (s, θ) now amounts to blowing up (s_{b}, θ_{b}) uniformly over each b, and thus reduces the value of μ. □

Question 7.9. Is Proposition 7.6 true in higher dimensions?

Corollary 7.10. Let X ⊆ VF^{n} × RV^{m} be definable. Then every definable function ϕ : X → Γ factors through a definable function X → RV^{∗}.

Proof. By Lemma 4.5 we may assume X is ∼_{rv}-invariant; now the corollary follows from Lemma 7.8.

(It is convenient to note this here, but it can also be proved with the methods of §3; the main point is that on the generic type of a ball with center c, every function into RV ∪ Γ factors through rv(x − c); while on a transitive ball, every function into RV ∪ Γ is constant.)

Consider pairs (X′, f′) with X′, f′ : X′ → VF^{n} definable, and such that f′ has RV-fibers. A bijection g : X′ → X′′ is said to be relatively unary (with respect to f′, f′′) if it commutes with n − 1 coordinate projections, i.e. pr_{i}f′′g = pr_{i}f′ for all but at most one value of i.

Given X ⊆ VF^{n} × RV^{m}, we view it as a pair (X, f) with f the projection to VF^{n}. Thus for X, Y ⊆ VF^{n} × RV^{∗}, the notion: F : X → Y is relatively unary is defined.

Note that the elementary bijections are relatively unary, as are the auxiliary bijections.

Lemma 7.11. Let X, Y ⊆ VF^{n} × RV^{∗}, and let F : X → Y be a definable bijection. Then F can be written as the composition of relatively unary morphisms of VF[n, ·].

Proof. We have X with two finite-to-one maps f, g : X → VF^{n} (the projection, and the composition of F with the projection Y → VF^{n}.) We must decompose the identity X → X into a composition of relatively unary maps (X, f) → (X, g).

Begin with the case n = 2; we are given (X, f_{1}, f_{2}) and (X, g_{1}, g_{2}). Claim There exists a definable partition of X into sets X_{ij} such that (f_{i}, g_{j}) : X → VF^{2} is finite-to-one.

Proof. Let a ∈ X. We wish to show that for some i, j, a ∈ acl(f_{i}(a), g_{j}(a)). This follows from the exchange principle for algebraic closure in VF: if a ∈ acl(∅), there is nothing to show. Otherwise g_{j}(a) ∉ acl(∅) for some j; in this case either a ∈ acl(f_{1}(a), g_{j}(a)) or f_{1}(a) ∈ acl(g_{j}(a)), and then a ∈ acl(f_{2}(a), g_{j}(a)). The Claim follows by compactness.

Let h : X′ → X be a special bijection such that the characteristic functions of X_{ij} are in F n^{RV}(X, X′ → X). ( Lemma 7.8). Since h is composition of relatively unary bijections, we may replace X by X′ (and f_{i}, g_{i} by f_{i} ∘ h, g_{i} ∘ h respectively.) So we may assume the characteristic function of X_{ij} is in F n^{RV}(X, X), i.e. X_{ij} ∈ V F r[n]. But then it suffices to treat each X_{ij} separately; say X_{11}. In this case the identity map on X takes

(X, f_{1}, f_{2}) ↦ (X, f_{1}, g_{1}) ↦ (X, g_{2}, g_{1}) ↦ (X, g_{2}, g_{1} − g_{2}) ↦ (X, g_{1}, g_{1} − g_{2}) ↦ (X, g_{1}, g_{2})

where each step is relatively unary.

When n > 2, we move between (X, f_{1}, …, f_{n}) and (X, g_{1}, …, g_{n}), by partitioning, and on a given piece replacing each f_{i} by some g_{j}, one at a time. □

7.4. RV-blowups. We now define the RV-counterparts of the special bijections, which will be called RV-blowups. These will not be bijections; the kernel of the homomorphism 𝕃 : K_{+}[RV] → K_{+}[VF] will be seen to be obtained by formally inverting RV-blowups. Let RV_{∞}^{>0} = {x ∈ RV : val(x) > 0 ∪ {∞}} ⊆ RV_{∞}. In the RV[≤ 1]-presentation, RV_{∞}^{>0} = [RV^{>0}]_{1} + [1]_{0}. (cf. §3.8.1).

Definition 7.12. (1) Let Y = (Y, f) ∈ Ob RV_{∞}[n, ·] be such that f_{n}(y) ∈ acl(f_{1}(y), …, f_{n−1}(y)), and f_{n}(y) ≠ ∞. Let Y′ = Y × RV_{∞}^{>0}. For (y, t) ∈ Y′, define f′ = (f′_{1}, …, f′_{n}) by: f′_{i}(y, t) = f_{i}(y) for i < n, f′_{n}(y, t) = tf_{n}(y). Then Ỹ = (Y′, f′) is an elementary blowup of Y. It comes with the projection map Y′ → Y.

(2) Let X = (X, g) ∈ Ob RV_{∞}[n, ·], X = X′ ∪^{.} X′′, g′ = g|X′, g′′ = g|X′′, and let ϕ :

Y → (X′, g′) be an RV_{vol}-isomorphism. Then the RV- blowup X̃_{ϕ} is defined to be Ỹ + (X′′, g′′) = (Y′ ∪^{.} X′′, f′ ∪^{.} g′′). It comes with b : Y′ ∪^{.} X′′ → X, defined to be the identity on X′′, and the projection on Y′. X′ is called the blowup locus of b : X̃_{ϕ} → X.

An iterated RV-blowup is obtained by finitely many iterations of RV- blowups.

Since blowups in the sense of algebraic geometry will not occur in this paper, we will say ‘blowup’ for RV-blowup.

Remark 7.13. In the definition of an elementary blowup, dim_{RV}(Y) < n. For such Y, ϕ : Y → (X′, g′) is an RV_{vol}[≤ n, ·]-isomorphism iff it is an RV_{Γ−vol}-isomorphism (Definition 5.21).

Lemma 7.14. (1) Let Y′ be an elementary blowup of Y. Y′ is RV_{vol}[n, ·]- isomorphic to Y′′ = (Y′′, f′′), with Y′′ = {(y, t) ∈ Y × RV_{∞} : val_{rv}(t) > f_{n}(y)}, f′′(y, t) = (f_{1}(y), …, f_{n−1}(y), t).

(2) An elementary blowup Y′ of Y is RV_{∞}[n, ·]- isomorphic to (Y × RV_{∞}, f′) for any f′ isogenous to (f_{1}, …, f_{n}, t).

(3) Up to isomorphism, the blowup depends only on blowup locus. In other words if X, X′, g, g′ are as in Definition 7.12, and ϕ_{i} : Y_{i} → (X′, g′) (i = 1, 2) are isomorphisms, then X̃_{ϕ_{1}}, X̃_{ϕ_{2}} are X-isomorphic in RV_{vol}[n, ·].

Proof. (1) The isomorphism is given by (y, t) ↦ (y, tf_{n}(y))

(2) The identity map on Y × RV is an RV_{∞}[n, ·] isomorphism.

(3) Let ψ_{0} = ϕ_{2}^{−1}ϕ_{1}, and define ψ_{1} : Y_{1} × RV_{∞}^{>0} → Y_{2} × RV_{∞}^{>0} by ψ(y, t) = (ψ_{0}(y), t). The sum of the values of the n coordinates of Ỹ_{i} is then (∑_{i<n} val_{rv}f_{i})+(val_{rv}(t)+val_{rv}f_{n}) in both cases. Since by assumption ψ_{0} : Y_{1} → Y_{2} is an RV_{vol}-isomorphism, it preserves ∑_{i≤n} val_{rv}f_{i} and so ψ_{1} too is an RV_{Γ−vol}-isomorphism; thus Jcb_{RV}(ψ_{1}) ∈ k^{∗} a.e. Let θY_{1} → k^{∗} be a definable map such that θ = Jcb_{RV}(ψ_{1}) almost everywhere. Define ψ : Y_{1} ×RV_{∞}^{>0} → Y_{2} ×RV_{∞}^{>0} by ψ(y, t) = (ψ_{0}(y), t/θ(y)). Then one computes immediately that Jcb_{RV}(ψ) = 1, so ψ is an RV_{vol}[n, ·]-isomorphism, and hence so is ψ ∪^{.} Id_{X′′} : X̃_{ϕ_{1}} → X̃_{ϕ_{2}}

□

Here is a coordinate-free description of RV-blowups; we will not really use it in the subsequent development.

Lemma 7.15. (1) Let Y = (Y, g) ∈ Ob RV_{∞}[n, ·], with dim(g(Y)) < n; let f : Y → RV^{n−1} be isogenous to g. Let h : Y → RV be definable, with h(y) ∈ acl(g(y)) for y ∈ Y, and with ∑(g) = ∑(f) + val_{rv}(h). Let Y′ = Y × RV_{∞}^{>0}, and f′(y, t) = (f(y), th(y)). Then Y′ = (Y′, f′) with the projection map to Y is a blowup.

(2) Let Y′′ → Y be a blowup with blowup locus Y. Then there exist f, h such that with Y′ as in (3), Y′′, Y′ are isomorphic over Y.

Proof. (1) Since dim_{RV}(g(Y)) < n, Id_{Y} : (Y, (f, h)) → (Y, g) is an RV_{vol}-isomorphism. Use this as ϕ in the definition of blowup.

(2) With notation as in Definition 7.12, let h = g_{n} ∘ ϕ^{−1}, f = (g_{1}, …, g_{n−1}) ∘ ϕ^{−1}.

Definition 7.16. For 𝒞 = RV[≤ n, ·] or 𝒞 = RV_{vol}[≤ n, ·], let I_{sp}[≤ n] be the set of pairs (X_{1}, X_{2}) ∈ Ob 𝒞 such that there exist iterated blowups b_{i} : X̃_{i} → X_{i} and an 𝒞- isomorphism F : X̃_{1} → X̃_{2}.

When n is clear from the context, we will just write I_{sp}.

Definition 7.17. Let 1_{0} denote the one-element object of RV[0]. Given a definable set X ⊆ RV^{n} let X_{n} denote (X, Id_{X}) ∈ RV[n], and [X]_{n} the class in K_{+}(RV[n]). Write [1]_{1} for [{1}]_{1} (where {1} is the singleton set of the identity element of k.)

Lemma 7.18. Let 𝒞 = RV[≤ n, ·] or 𝒞 = RV_{vol}[≤ n, ·].

(1) Let f : X_{1} → X_{2} be an 𝒞- isomorphism, and let b_{1} : X̃_{1} → X_{1} be a blowup. Then there exists a blowup b_{2} : X̃_{2} → X_{2} and a 𝒞- isomorphism F : X̃_{1} → X̃_{2} with b_{2}F = fb_{1}.

(2) If b : X̃ → X is a blowup, then so are b ∪^{.} Id : X̃ ∪^{.} Z → X ∪^{.} Z and (b × Id) : X̃ × Z → X × Z.

(3) Let b_{i} : X̃_{ϕ_{i}} → X be a blowup (i = 1, 2). Then there exist blowups b′_{i} : Z_{i} → X̃_{ϕ_{i}} and an isomorphism F : Z_{1} → Z_{2} such that b_{2}b′_{2}F = b_{1}b′_{1}.

(4) Same as (1,2,3) for iterated blowups.

(5) I_{sp} is an equivalence relation. It induces a semi-ring congruence on K_{+}RV[∗, ·], respectively K_{+}RV_{vol}[∗, ·].

(6) As a semi-ring congruence on K_{+}RV[∗, ·], I_{sp} is generated by ([1]_{1}, [RV^{>0}]_{1} + 1_{0}).

Proof. (1) This reduces to the case of elementary blowups. If 𝒞 = RV_{vol}[n, ·] then the composition f ∘ b_{1} is already a blowup. If 𝒞 = RV[≤ n, ·], it is also clear using Lemma 7.14 (2).

(2) From the definition of blowup.

(3) If b_{1} is the identity, let b′_{1} = b_{2}, b′_{2} = Id, F = Id. Similarly if b_{2} is the identity. If X = X′ ∪^{.} X′′ and the statement is true above X′ and above X′′, then by glueing it is true also above X. We thus reduce to the case that b_{1}, b_{2} both are blowups with blowup locus equal to X. But then by Lemma 7.14 (3), there exists an isomorphism F : X̃_{ϕ_{1}} → X̃_{ϕ_{2}} over X. Let b′_{1} = b′_{2} = Id.

(4) For (1,2) the induction is immediate. For (3), write k-blowup as shorthand for “an iteration of k blowups”. We show by induction on k_{1}, k′ a more precise form: Claim If X_{1} → X is a k_{1}-blowup, and X′ → X is a k′ blowup, then there exists an k′-blowup Z′_{1} → X_{1} a k_{1}-blowup Z′ → X, and an RV_{vol}[n, ·]- isomorphism Z′_{1} → Z′ over X.

If k_{1} = k′ = 1 this is (3). So say k′ > 1. The map X′ → X is a composition X′ → X_{2} → X, where X_{2} → X is a k′ − 1-blowup and X′ → X_{2} is a blowup. By induction there is a k′ − 1 blowup Z_{1} → X_{1} and a k_{1}-blowup Z_{2} → X_{2} and an RV_{vol}[n, ·]- isomorphism Z_{1} → Z_{2} over X.

By induction again there is a blowup and Z′_{2} → Z_{2}, a k_{1}-blowup Z′ → X′ an RV_{vol}[n, ·]- isomorphism Z′ → Z_{2} over X_{2}. By (1) there exists a blowup Z′_{1} → Z_{1} and an RV_{vol}[n, ·]- isomorphism Z′_{1} → Z′_{2}, making the Z_{1}, Z_{2}, Z′_{1}, Z_{2}-square commute. So Z_{1} → X_{1} is a k′- blowup, Z′ → X′ is a k_{1}-blowup, and we have a composed isomorphism Z′_{1} → Z′_{2} → Z′ over X.

(5) If (X_{1}, X_{2}), (X_{2}, X_{3}) ∈ I_{sp}, there are iterated blowups X′_{1} → X_{1}, X′_{2} → X_{2} and an isomorphism X′_{1} → X′_{2}; and also X′′_{2} → X_{2}, X′_{3} → X_{3} and X′′_{2} → X_{3}. Using (3) for iterated blowups, there exist iterated blowups X̂_{2}′ → X′_{2}, X̂_{2}′′ → X′′_{2}, and an isomorphism X̂_{2}′ → X̂_{2}′. By (1) for iterated blowups there are iterated blowup X̂_{1} → X′_{1}, X̂_{3} → X_{3} and isomorphisms X̂_{1} → X̂_{2}′, X̂_{2}′′ → X̂_{3}, with the natural diagrams commuting. Composing we obtain X̂_{1} → X̂_{3}, showing that (X_{1}, X_{3}) ∈ I_{sp}. Hence I_{sp} is an equivalence relation.

Isomorphic objects are I_{sp}-equivalent, so an equivalence relation on the semiring K_{+}𝒞 is induced. If (X_{1}, X_{2}) ∈ I_{sp} then by (2), (X_{1} ∪^{.} Z, X_{2} ∪^{.} Z) ∈ I_{sp}, and (X_{1} × Z, X_{2} × Z) ∈ I_{sp}. It follows that I_{sp} induces a congruence on the semiring K_{+}𝒞.

(6) We can blow up 1_{1} to RV^{>0}_{1} + 1_{0}, so ([1]_{1}, [RV^{>0}]_{1} + 1_{0}) ∈ I_{sp}. Conversely, under the conditions of Definition 7.12, let Y^{−} = [(Y, f_{1}, …, f_{n−1})]; then [Y] = [(Y, f_{1}, …, f_{n−1}, 0)] = [Y^{−}] × [1]_{1} by Lemma 7.14, and we have:

[X̃_{Y}] = [Y]_{n−1} + [Y]_{n−1} × [RV^{>0}]_{1} + [X′′] ≅_{I_{sp}} [Y] × [1]_{1} + [X′′] = [X]

modulo the congruence generated by ([1]_{1}, [RV^{>0}]_{1} + 1_{0}).

□

We now relate special bijections to blowing ups. Given X = (X, f), X′ = (X′, f′) ∈ RV[n, ·], say X, X′ are strongly isomorphic if there exists a bijection ϕ : X → X′ with f′ = ϕf. Strong isomorphisms are always in RV_{vol}[n, ·].

Up to strong isomorphism, an elementary blowup of (Y, f) can be put in a different form: ˜(Y) ≃ (Y′′, f′′), Y′′ = {(z, y) : y ∈ Y, val_{rv}(z) > val_{rv}f_{n}(y)}, f_{i}(z, y) = f_{i}(y) for i < n, f_{n}(z, y) = z. The strong isomorphism Y′′ → Y′ is given by (z, y) ↦ (y, z/f_{n}(y)). This matches precisely the definition of special bijection, and makes evident the following lemma.

Lemma 7.19. Let 𝒞 = RV_{∞}[n, ·] or RV_{vol}[≤ n, ·].

X, Y are strongly isomorphic over RV^{n} iff 𝕃X, 𝕃Y are isomorphic over the projection to VF^{n}.

{(1) \atop (2)} Let X, X′ ∈ RV[≤ n, ·], and let G : 𝕃X′ → 𝕃X be an auxiliary special bijection. Then

X′ is isomorphic to X over RV^{n}.

(3) Let X, X′ ∈ RV[≤ n, ·], and let G : 𝕃X′ → 𝕃X be an elementary bijection. Then X′ is strongly isomorphic to a blowup of X.

(4) Let X, X′ ∈ RV[≤ n, ·], and let G : 𝕃X′ → 𝕃X be a special bijection. Then X′ is strongly isomorphic to an iterated blowup of X.

(5) Assume T is effective. If Y → X is an RV-blowup, there exists Y′ strongly isomorphic to Y over X and an elementary bijection c : 𝕃Y′ → 𝕃Y lying over Y′ → Y.

Proof. (1) Clear, using Lemma 3.52.

(2) This is a special case of (1).

(3) Clear from the definitions.

(4) Clear from (1-3).

(5) It suffices to consider elementary blowups; we use the notation in the definition there. So f_{n}(x) ∈ acl(f_{1}(x), …, f_{n−1}(x)) for x ∈ ϕ(Y). By effectiveness and Lemma 6.2 there exists a definable function s(x, y_{1}, …, y_{n−1}) such that if rv(y_{i}) = f_{i}(x) for i = 1, …, n − 1, then rvs(x, y) = f_{n}(x). This s is the additional data needed for an elementary bijection. □

Lemma 7.20. Let X = (X, f), X′ = (X′, f′) ∈ RV[≤ n, ·], and let h : X → W ⊆ RV^{∗}, h′ : X′ → W be definable maps. Let X_{c} = h^{−1}(c), X_{c} = (X_{c}, f|X_{c}) and similarly X′_{c}. If (X_{c}, X′_{c}) ∈ I_{sp}(RV_{c}[n, ·]) then (X, X′) ∈ I_{sp}.

Proof. Lemma 2.3 applies to RV_{Γ−vol}-isomorphisms, and hence using Remark 7.13 also to blowups. It also applies to RV[≤ n, ·]-isomorphisms; hence to I_{sp}-equivalence. □

Lemma 7.21. If (X, Y) ∈ I_{sp} then 𝕃X ≃ 𝕃Y.

Proof. Clear, since 𝕃[1]_{1} is the unit open ball around 1, 𝕃([RV^{>0}]_{1} is the punctured unit open ball around 0, and 𝕃1_{0} = {0}. □

7.5. The kernel of 𝕃.

Definition 7.22. VFR[k, l, ·] is the set of pairs X = (X, f), with X ⊆ VF^{k} × RV^{∗}, f : X → RV_{∞}^{l}, and such that f factors through the projection pr_{RV}(X) of X to the RV-coordinates. I_{sp} is the equivalence relation on VFR[k, l, ·]:

(X, Y) ∈ I_{sp} ⟺ (X_{a}, Y_{a}) ∈ I_{sp}(T_{a}) for each a ∈ VF^{k}

K_{+}VFR is the set of equivalence classes.

By the usual compactness argument, if (X, Y) ∈ I_{sp} then there are uniform formulas demonstrating this. The relative versions of Lemmas 7.14, 7.18 follow.

If U = (U, f) ∈ VFR[k, l, ·], and for u ∈ U we are uniformly given V_{u} = (V_{u}, g_{u}) ∈ VFR[k′, l′, ·] we can define a sum ∑_{u∈U} V_{u} ∈ VFR[k + k′, l + l′, ·]: it is the set ∪^{.}_{u∈U}V_{u}, with the function (u, v) ↦ (f(u), g_{u}(v)). When necessary, we denote this operation ∑^{(k,l;k′,l′)}. The special case k = l = 0 is understood as the default case.

By Proposition 7.6, the inverse of 𝕃 : RV[1, ·] → VF[1, ·] induces an isomorphism I_{1}^{1} : K_{+}VF[1, ·] → K_{+}RV[1, ·]/I_{sp}.

I([X]) = [Y]/I_{sp} ⟺ [𝕃Y] = [X]

Let J be a finite set of k elements. For j ∈ J, let π^{j} : VF^{k} × RV^{∗} → VF^{J−{j}} × RV^{∗} be the projection forgetting the j’th VF coordinate. We will write VF^{k}, VF^{k−1} for VF^{J}, VF^{J−{j}} respectively when the identity of the indices is not important.

Let X = (X, f) ∈ VFR[k, l, ·]. By assumption, f factors through π^{j}. We view the image (π^{j}X, f) as an element of VFR[k − 1, l, ·]. Note that each fiber of π^{j} is in VF[1, ·].

Relativizing I_{1}^{1} to π^{j}, we obtain a map

I^{j} = I_{k,l}^{j} : VFR[k, l, ·] → K_{+}VFR[k − 1, l + 1, ·]/I_{sp}

Lemma 7.23. Let X = (X, f), X′ = (X′, f′) ∈ VFR[k, l, ·].

(1) I^{j} commutes with maps into RV: if h : X → W ⊆ RV^{∗} is definable, X_{c} = h^{−1}(c), then

I^{j}(X) = ∑_{c∈W} I^{j}(X_{c})
(2) If ([X], [X′]) ∈ I_{sp} then (I^{j}(X), I^{j}(X′)) ∈ I_{sp}.
(3) I^{j} induces a map K_{+}VFR[k, l, ·]/I_{sp} → K_{+}VFR[k − 1, l + 1, ·]/I_{sp}.

Proof. (1) This reduces to the case of I_{1}^{1}, where it is an immediate consequence of the uniquness, and the fact that 𝕃 commutes with maps into RV in the same sense.

(2) All equivalences here are relative to the k − 1 coordinates of VF other than j, so we may assume k = 1.

For a ∈ VF, ([X_{a}], [X′_{a}]) ∈ I_{sp}(T_{a}). By stable embeddedness of RV, there exists α = α(a) ∈ RV^{∗} such that X_{a}, X′_{a} are T_{α}-definable and ([X]_{a}, [X′]_{a}) ∈ I_{sp}(T_{α}). Fibering over the map α we may assume by (1) and Lemma 7.20 that α is constant; so for some W ∈ VF[1], Y, Y′ ∈ RV[l, ·], we have X = W × Y, X′ = W × Y′, and ([Y], [Y′]) ∈ I_{sp}. Then I^{j}(X) = I^{j}(W) × Y, I^{j}(X′) = I^{j}(W) × Y′ and the conclusion is clear.

(3) by (2).

□

Lemma 7.24. Let X = (X, f), X ⊆ VF^{J} × RV^{∞}, f : X → RV^{l}. If j ≠ j′ ∈ J then I^{j}I^{j′} = I^{j′}I^{j} : K_{+}VFR[k, l, ·]/I_{sp} → K_{+}VFR[k − 2, l + 2, ·]/I_{sp}.

Proof. We may assume S = {1, 2}, j = 1, j′ = 2, since all is relative to VF^{S\{j,j′}}. By Lemma 7.23 (1) it suffices to prove the statement for each fiber of a given definable map into RV.

Hence we may assume X ⊆ VF^{2} and f is constant; and by Lemma 5.10, we can assume X is a basic 2-cell:

X = {(x, y) : x ∈ X_{1}, rv(y − G(x)) = α_{1}} X_{1} = rv^{−1}(δ_{1}) + c_{1}

The case where G is constant is easy, since then X is a finite union of rectangles. Otherwise G is invertible, and by niceness of G we can also write:

X = {(x, y) : y ∈ X_{2}, rv(x − G^{−1}(y)) = β}, X_{2} = rv^{−1}(δ_{2}) + c_{2}

We immediately compute:

I_{2}I_{1}(X) = (δ_{1}, α_{1}), I_{1}I_{2}(X) = (α_{2}, δ_{2})

Clearly [(δ_{1}, α_{1})]_{2} = [(α_{2}, δ_{2})]_{2}.

□

Proposition 7.25. Let X, Y ∈ RV[≤ n, ·]. If 𝕃X, 𝕃Y are isomorphic, then ([X], [Y]) ∈ I_{sp}.

Proof. define I = I_{1} … I_{n} : VF[n, ·] = VFR[n, 0, ·] → VFR[0, n, ·] = RV[≤ n, ·]. Let V ∈ VF[n, ·]. Claim 1. If σ ∈ Sym(n) then I = I_{σ(1)} … I_{σ(n)}.

Proof. We may assume σ just permutes two adjacent coordinates, say 2, 3 out of 1, 2, 3, 4. Then I = I_{1}I_{2}I_{3}I_{4} = I_{1}I_{3}I_{2}I_{4} by Lemma 7.24. Claim 2. When F : V → F (V) is a relatively unary bijection, we have I(V) = I(F (V))

Proof. By Claim 1 we may assume F is relatively unary with respect to pr^{n}. So F (V_{a}) = F (V)_{a} where V_{a}, F (V)_{a} are the pr^{n}-fibers. By definition of I_{1}^{1} we have I_{1}^{1}(V_{a}) = I_{1}^{1}(F (V)_{a}) ∈ RV[1, ·](T_{a}); but by definition of I^{n} , I_{n}(V)_{a} = I_{1}^{1}(V_{a}). So I^{n}(V) = I^{n}(F (V)) and thus I(V) = I(F (V)). Claim 3. When F : V → F (V) is any definable bijection, I(V) = I(F (V)).

Proof. Immediate from Claim 2 and Lemma 7.11.

Now turning to the statement of the Proposition, assume 𝕃X, 𝕃Y are isomorphic. We compute inductively that 𝕃(X) = [X]. By Claim 3, [X] = I(𝕃X) = I(𝕃Y) = [Y]. □

Notation 7.26. Let 𝕃^{∗} : K_{+}(VF) → K_{+}(RV[∗])/Isp be the inverse map to 𝕃.

Remark 7.27. When T is rv- effective, one can restate the conclusion of Proposition 7.25 as follows: if X, Y ∈ VF[n, ·] are ∼_{rv}-invariant and F : X → Y is a definable bijection, then there exist special bijections X′ → X and Y′ → Y and an ∼_{rv}-invariant -definable bijection G : X′ → Y′. (This follows from Proposition 7.25, Proposition 6.1, Lemma 7.18, and Lemma 7.19. The effectiveness hypothesis is actually unnecessary here, as will be seen in the proof of Proposition 8.26. Perhaps Question 7.9 can be answered simply by tracing the connection between F and G through the proof.

8. Definable sets over VF and RV: the main theorems

In stating the theorems we restrict attention to VF[n], i.e. to definable subsets of varieties, though the proof was given more generally for VF[n, ·] (definable subsets of VF^{n} × RV^{∗}.)

8.1. Definable subsets of varieties. Let T be V-minimal. We will look at the category of definable subsets of varieties, and definable maps between them. The results will be stated for VF[n]; analogous statements for VF[n, ·] are true with the same proofs.

We define three variants of the sets of objects. VF′′[n] is the category of ≤ n-dimensional definable sets over VF, i.e. of definable subsets of n-dimensional varieties. Let VF[n] be the category of definable subsets X ⊆ VF^{n} × RV^{∗} such that the projection X → VF^{n} has finite fibers. VF′[n] is the category of definable subsets X of V × RV^{∗}, where V ranges over all VF(A)-definable sets of dimension n, m ∈ ℕ, such that the projection X → V is finite-to-one. VF, VF′, VF′′ are the unions over all n. In all cases, the morphisms Mor (X, Y) are the definable functions X → Y. Lemma 8.1. The natural inclusion of VF[n] in VF′[n] is an equivalence. If T is effective, so is the inclusion of VF′′[n] in VF′[n].

Proof. We will omit the index ≤ n. The inclusion is fully faithful by definition, and we have to show that it hits every VF′ -isomorphism type; in other words, that any definable X ⊆ (V × RV^{m}) is definably isomorphic to some X′ ⊆ VF^{n} × RV^{m+l}, for some l (with n = dim(V).) Definable isomorphisms can be glued on pieces, so we may assume V is affine, and admits a finite-to-one map h : V → VF^{m}. By Lemma 3.9, each fiber h^{−1}(a) is A(a)-definably isomorphic to some F (a) ⊆ RV^{l}. By compactness, F can be chosen uniformly definable, F (a) = {y ∈ RV^{l} : (a, y) ∈ F } for some definable F ⊆ VF^{m} × RV^{l}; and there exists a definable isomorphism β : V → F, over VF^{m}. Let α(v, t) = (β(v), t), X′ = α(X).

Now assume T is effective. Let X ∈ Ob VF′; X ⊆ V × RV^{m}, V ⊆ VF^{n}, such that the projection X → V has finite fibers. Then by effectivity, for any v ∈ V (over any extension field), if (v, c_{1}, …, c_{m}) ∈ X then each c_{i}, viewed as a ball, has a point defined over A(v). Hence the partial map V × VF^{m} → X, (v, x_{1}, …, x_{m}) ↦ (v, rv(x_{1}), …, rv(x_{m})) has an A-definable section; the image of this section is a subset S of V × VF^{m}, definably isomorphic to X; and the Zariski closure V′ of S in V × VF^{m} has dimension ≤ dim(V). □

The following definition and proposition apply both to the category of definable sets, and to the definable sets with volume forms. Definition 8.2. X, Y are effectively isomorphic if

for any effective A, X, Y are definably isomorphic in T_{A}. If K_{+}^{eff}(VF) is the semi-ring of effective isomorphic classes of definable sets. K(VF) is the correpsponding ring. Similarly K_{+}^{eff}(VF[n]), etc.

Over an effective base, in particular if T is effective over any field-generated base, effectively isomorphic is the same as isomorphic. But Example 4.7 shows that this is not so in general. Proposition 8.3. Let T be V-minimal, or a finitely generated extension of a V-minimal theory. The following conditions are equivalent: Let X, Y ∈ VF[n].

(1) [𝕃^{∗}X] = [𝕃^{∗}Y] in K_{+}(RV[≤ n])/I_{sp}[≤ n]

(2) There exists a definable family ℱ of definable bijections X → Y, such that for any effective structure A, F (A) ≠ ∅.

(3) X, Y are effectively isomorphic.

(4) X, Y are definably isomorphic over any A such that VF^{∗}(A) → RV(A) is surjective.

(5) For some finite A_{0} ⊆ RV(< ∅ >), X, Y are definably isomorphic over any A such that A_{0} ⊆ rv(VF^{∗}(A)).

Proof. (1) implies (5): By Proposition 6.1 (6.3 in the measured case), the given isomorphism [𝕃^{∗}X] → [𝕃^{∗}Y] lifts to an isomorphism 𝕃𝕃^{∗}X → 𝕃𝕃^{∗}Y; since T_{A} ⊇ ACVF_{A}, this is also a T_{A} isomorphism; it can be composed with the isomorphisms X → 𝕃𝕃^{∗}X, Y → 𝕃𝕃^{∗}Y.

(2) implies (3), (5) implies (4) implies (3) trivially.

(3) implies (1),(2): Let E_{eff} be as in Proposition 3.51. By (3), X, Y are E_{eff}-isomorphic. By Proposition 7.25, [𝕃^{∗}X] = [𝕃^{∗}Y] in K_{+}(RV_{E_{eff}} [∗])/I_{sp}. But RV(E_{eff}), Γ(E_{eff}) ⊆ dcl(∅), so every E_{eff}-definable relation on RV is definable; i.e. RV_{E_{eff}} , RV are the same structure. So (1) holds.

Now by assumption, there exists an E_{eff}-definable bijection f′ : X → Y. f′ is an E_{eff}- definable element of a definable family ℱ of definable bijections X → Y. Since this family has an E_{eff}-point, and E_{eff} embeds into any effective B, it has a B point too. Thus (3) implies (2).

□

8.2. Invariants of all definable maps. Let [X] denote the class of X in K_{+}^{eff}(VF[n]). Proposition 8.4. Let T be V-minimal. There exists a canonical isomorphism of Grothendieck semigroups

D ∫ : K_{+}^{eff}(VF[n]) → K_{+}(RV[≤ n])/I_{sp}[≤ n]

satisfying

D ∫ [X] = W/I_{sp}[≤ n] ⟺ [X] = [𝕃W] ∈ K_{+}^{eff}(VF[n])

Proof. Recall Definition 4.8. Given X = (X, f) ∈ Ob RV[k] we have 𝕃X ∈ Ob VF[k] ⊆ Ob VF[n]. If X, X′ are isomorphic,then by Proposition 6.1, 𝕃X, 𝕃X′ are effectively isomorphic. Direct sums are clearly respected, so we have a semigroup homomorphism 𝕃 : K_{+}(RV[≤ n]) → K_{+}^{eff}(VF[n]). It is surjective by Proposition 4.5. By Proposition 8.3, the kernel is precisely I_{sp}[≤ n]. Inverting, we obtain ^{D} ∫. □

Definition 8.5. Let K_{+}VF[n]/(dim < n) be the Grothendieck ring of the category of definable subsets of n-dimensional varieties, and essential bijections between them. Let I_{sp}′[n] be the congruence on RV [n] generated by pairs (X, X × RV^{>0}) (where X ⊆ RV^{∗} is definable, of dimension < n.) Corollary 8.6. ^{D} ∫ induces an isomorphism K_{+}^{eff}(VF[n])/(dim < n) → RV[n]/I_{sp}′[n] □ Corollary 8.7. Let A, B ∈ RV[≤ n]. Let n′ > n, and let A_{N}, B_{N} be their images in RV[≤ N]. If (A_{N}, B_{N}) ∈ I_{sp}[≤ N] then (A, B) ∈ I_{sp}[≤ n].

Proof. By Proposition 8.4, (A, B) ∈ I_{sp}[ ≤ n] iff 𝕃A, 𝕃B are definably isomorphic; this latter condition does not depend on n.

□

Putting Proposition 8.4 together for all n we obtain:

Theorem 8.8. Let T be V-minimal. There exists a canonical isomorphism of filtered semirings

D ∫ : K_{+}(VF) → K_{+}(RV[∗])/I_{sp}

Let [X] denote the class of X in K_{+}(VF). Then

D ∫ [X] = W/I_{sp} ⟺ [X] = [𝕃W] ∈ K_{+}^{eff}(VF)
□

On the other hand, using the Grothendieck group isomorphisms of Proposition 8.4 and passing to the limit:

Corollary 8.9. Let T be V-minimal. The isomorphisms of Proposition 8.4 inducing an isomorphism of Grothendieck groups:

∫ ^{K} : K^{eff}(VF[n]) → K(RV[n])

The isomorphism ^{D} ∫ of Theorem 8.8 induces an injective ring homomorphism

∫ ^{K} : K^{eff}(VF) → K(RV)[J^{−1}]

where J = {1}_{1} − [RV^{>0}]_{1} ∈ K(RV).

Proof. We may work over an effective base. With subtraction allowed, the generating relation of I_{sp} can be read as: [{1}]_{0} = {1}_{1}−[RV^{>0}]_{1} := J; so that the groupfication of K_{+}(RV[≤ n])/I_{sp}[≤ n] is isomorphic to K(RV[n]), via the embedding of K_{+}(RV[n]) as a direct factor in K_{+}(RV[≤ n]). Thus the groupification of the homomorphism of Theorem 8.8 is a homomorphism

∫ ^{K} : K(VF) → lim_{n→∞} K(RV[n])

where the direct limit system maps are given by [X]_{d} ↦ ([X]_{d+1} − ([X]_{d} × (RV^{>0}))) = [X_{d}]J. This direct limit embeds into K(RV)[J^{−1}] by mapping X ∈ K(RV[n]) to XJ^{−n}. □

8.3. Definable volume forms: VF. We will now define the category μVF[n] of “n-dimensional T_{A}-definable sets with definable volume forms, up to RV- equivalence” and the same up to Γ-equivalence. We will represent the forms as functions to RV, that transform in the way volume forms do.

By way of motivation: in a local field with an absolute value, a top differential form ω induces a measure |dω|. For a regular isomorphism f : V → V′ have ω = hf^{∗}ω′ for a unique h, and f is measure preserving between V, |ω|) and (V′, |ω′|) iff |h| = 1.

We do not work with an absolute value into the reals, but instead define the analogue using the map rv or, a coarser version, the map val into Γ. When Γ = ℤ, the latter is the is the usual practice in Denef-style motivic integration. Using rv leaves room for considering an absolute value on the residue field, and iterating the integration functorially when places are composed, for instance ℂ((x))((y)) → ℂ((x)) → ℂ. This functoriality will be described in a sequel.

In the definition below, the words “almost every y ∈ Y ” will mean: for all y outside a set of VF-dimension < dim_{VF}(Y). Definition 8.10. Ob μVF[n, ·] consists of pairs (Y, ω), where Y is a definable subset of VF^{n} × RV^{∗}, and ω : Y → RV is a definable map. A morphism (Y, ω) → (Y′, ω′) is a definable essential bijection F such that for almost every y ∈ Y,

ω(y) = ω′(F (y)) · rv(JcbF (y))

(We will say: “F : (Y, ω) → (Y′, ω′) is measure preserving”.)

μ_{Γ}VF[n, ·] is the category of pairs (Y, ω) with ω : Y → Γ a definable function A morphism (Y, ω) → (Y′, ω′) is a definable essential bijection F : Y → Y′ such that for almost every y ∈ Y,

ω(y) = ω′(F (y)) + val(JcbF (y))

(“F : (Y, ω) → (Y′, ω′) is Γ-measure preserving”.)

μVF[n], μ_{Γ}VF[n] are the full subcategories of μVF[n, ·], μ_{Γ}VF[n, ·] (respectively) whose objects admit a finite-to-one map to VF^{n}.

In this definition, let t_{1}(y), …, t_{n}(y) be the VF-coordinates of y ∈ Y. One can think of the form as ω(y)dt_{1} · . . . · dt_{n}.

Note that VF_{vol} of Definition 5.19 is isomorphic to the full subcategory of μVF whose objects are pairs (Y, 1). Remark 8.11. When T is V-minimal and effective, the data ω of an object (Y, ω) of μVF[n] can be written as rv ∘ Φ for some Φ : Y → VF. (Write ω = ω̄ ∘ rv ∘ F for some F, and use Proposition 6.1 to lift ω̄ to some G, so that ω = rv ∘ G ∘ F.) It is thus possible to view ω as the RV-image (resp. Γ-image) of a definable volume form on Y. One could equivalently take ω to be a definable section of Λ^{n}T Y/(1 + ℳ), where T Y is the (appropriately defined) tangent bundle, Λ^{n} the n-th exterior power with n = dim(Y).

For VF_{Γ} the category we take is slightly more flexible than taking varieties with absolute values of volume forms, even if T is V-minimal and effective, in that expressions such as ∫ |\sqrt{x}|dx are allowed.

In either of these categories, one could restrict the objects to bounded ones: Definition 8.12. Let μVF_{bdd}[n] be the full subcategory of μVF[n] whose objects are bounded definable sets, with bounded definable forms ω. Similarly one defines μVF_{Γ;bdd}.

Here bounded means that there is a lower bound on the valuation of any coordinate of any element of the set. A similar definition applies in RV and μRV.

Note that if an object of μVF[n] is μVF[n]-isomorphic to an object of μVF_{bdd}[n], it must lie in μVF_{bdd}[n].

8.4. Definable volume forms: RV. We will define a category μRV[n] of definable subsets of (RV)^{m}, with additional data that can be viewed as a volume form. Unlike μVF[n], in μRV[n] subsets of dimension < n are not ignored: a point of RV^{n} corresponds to an open polydisc of VF^{n}, with nonzero n-dimensional volume.

In particular, the Jacobian of a morphism needs to be defined at every point, not just away from a smaller-dimensional set. However, in accord with Lemma 6.3, it may be modified by k^{∗}-multiplication on a smaller-dimensional set. Definition 8.13. The objects of μRV[n] are definable triples (X, f, ω), X ⊆ RV^{n+m}, f : X → RV^{n} finite-to-one, and ω : X → RV.

We define a multiplication μRV[n] × μRV[n′] → μRV[n + n′] by (X, f, ω) × (X′, f′, ω′) = (X × X′, f × f′, ω · ω′). Here ω · ω′(x, x′) = ω(x)ω′(x′).

Given X = (X, f, ω), we define an object 𝕃X of VF[n]; namely (𝕃X, 𝕃f, 𝕃ω) where 𝕃X = X ×_{f,rv} (VF^{×})^{n}, 𝕃f(a, b) = f(a, rv(b)), 𝕃ω(a, b) = ω(a, rv(b)). (Sometimes we will write f, ω for 𝕃f, 𝕃ω.)

A morphism α : X = (X, f, ω) → X′ = (X′, f′, ω′) is a definable bijection α : X → X′, such that

ω(y) = ω′(α(y)) · rv(Jcb_{RV}(α)(y)) for almost all y

where “almost all” means: away from a set Y with dim_{RV}(f(Y)) < n; and

val_{rv}ω(y) + ∑_{i=1}^{n} val_{rv}f_{i}(y) = val_{rv}ω′(α(y)) + ∑_{i=1}^{n} val_{rv}f′_{i}(αy) for all y

The objects of μ_{Γ}RV[n] are triples (X, f, ω), with f : X → RV^{n}, ω : X → Γ. A morphism α : (X, f, ω) → (X′, f′, ω′) is a definable bijection α : X → X′ such that val_{rv}ω(y) + ∑_{i=1}^{n} val_{rv}f_{i}(y) = val_{rv}ω′(α(y)) + ∑_{i=1}^{n} val_{rv}f′_{i}(αy) for all y. Disjoint sums and products are defined as for μRV.

μ_{Γ}RES[n] is the full subcategory of μ_{Γ}RV[n] with objects (X, f, ω), such that val_{rv}(X) is finite. In this case, ω takes finitely many values too.

K_{+}^{eff}μRV[n] is the Grothendieck semigroup of μRV[n] with respect to effective isomorphism. K_{+}^{eff}μRV is the direct sum ⊕_{n}K_{+}^{eff}μRV[n]; it clearly inherits a semiring structure from Cartesian multplication, (X, f, ω) × (X′, f′, ω′) = (X × X′, (f, f′), ω · ω′).

The morphisms of μ_{Γ}RV[n] are called Γ- measure preserving.

The category RV_{vol}[n, ·] of Definition 5.21 is isomorphic to the full subcategory whose objects have ω = 1.

Remark The semiring K_{+}^{eff}RV_{vol} is naturally a subsemiring of K_{+}^{eff}μRV. The latter is obtained by inverting [{a}]_{1} for a ∈ RV and taking the 0’th graded component. This process is needed in order to identify integrals of functions in n variables with volumes in n + 1 variables. Thus as semirings they are closely related. But if the dimension grading is taken into account, the subsemiring of RV-volumes contains finer information connected to integrability of forms.

8.5. The kernel of 𝕃 in the measured case. The description of the kernel of 𝕃 on the semigroups of definable sets with volume forms is essentially the same as for definable sets. We will now run through the proof, indicating the modficiations. The principal change is the introduction of a category with fewer morphisms, defined not only with reference to RV but also to VF. For effective bases, the category is identical to μRV, so it will be invisible in the statements of the main theorems; but during the induction in the proof, bases will not in general be effective and the mixed category introduced here has better properties.

Both the introduction of the various intermediate categories and the repetition of the proof would be unnecessary if we had a positive answer to Question 7.9. In this case the proof of Lemma 8.23 would immediately lift to higher dimensions. Indeed the characterization of the kernel of the map 𝕃 on Grothendieck groups would be uniformized not only for the categories we consider, but for a range of categories carrying more structure.

The integer n will be fixed in this subsection. Lemma 8.14. Let (X, ω) ∈ Ob μVF[n, ·], Y ∈ Ob VF[n, ·], and let F : Y → X be a definable bijection.

(1) There exists ψ : Y → RV such that F : (Y, ψ) → (X, ω) is measure preserving.

(2) ψ is essentially unique in the sense that if ψ′ meets the same condition, then ψ, ψ′ are equal away from a subset of X of smaller dimension.

(3) Dually, given F, X, Y, ψ there exists an essentially unique ω such that F : (Y, ψ) → (X, ω) is measure preserving.

(4) Lemma 7.11 applies to μVF[n, ·] and to μVF[n]

Proof. (1,2) Let ψ(y) = ω(α(y)) · rv(Jcb_{RV}(α)(y)). By definition of μVF this works, and is the only choice “almost everywhere”. (3) follows from the case of F^{−1}. (4) Now let X, Y ∈ Ob μVF[n] and let F ∈ Mor_{μVF[n]}(X, Y). We have X = (X, ω_{X}), Y = (Y, ω_{Y}) for some X, Y ∈ Ob VF[n] and ω_{X} : X → RV, ω_{Y} : Y → RV. By Lemma 7.11 there exist X = X_{1}, …, X_{n} = Y ∈ Ob VF[n] and essentially unary F_{i} : X_{i} → X_{i+1} with F = F_{n−1} ∘ ⋯ ∘ F_{1}. Let ω_{1} = ω_{X}, and inductively let ω_{i+1} be such that F_{i} ∈ Mor_{μVF[n]}((X_{i}, ω_{i}), (X_{i+1}, ω_{i+1}). Then F ∈ Mor_{μVF[n]}((X, ω), (Y, ω_{n})). By the uniqueness it follows that ω_{Y} , ω_{n} are essentially equal. □

Definition 8.15. Given X, Y ∈ Ob μRV[n, ·] call a definable bijection h : X → Y liftable if there exists F ∈ Mor_{μVF[n,·]}(𝕃X, 𝕃Y) with ρ_{Y} F = hρ_{X}.

Let 𝒞 = μ_{l}RV[n, ·] be the subcategory of μRV[n, ·] consisting of all objects and liftable morphisms.

By Proposition 5.22, liftable morphisms must preserve the volume forms, so 𝒞 is a subcategory of μRV[n, ·].

Over an effective base, 𝒞 = μRV[n, ·] ( Lemma 6.3), and the condition of existence of s in Definition 8.16 (1) below is equivalent to: f_{n}(y) ∈ acl(f_{1}(y), …, f_{n−1}(y)). Definition 8.16. (1) Let Y = (Y, f, ω) ∈ Ob μRV[n, ·] be such that there exists

s : Y ×_{f_{1},...,f_{n−1}} VF^{n−1} → VF with rv(s(y, u_{1}, …, u_{n−1})) = f_{n}(y). Let Y′ = Y ×RV^{>0}. For (y, t) ∈ Y′, define f′ = (f′_{1}, …, f′_{n}) by: f′_{i}(y, t) = f_{i}(y) for i < n, f′_{n}(y, t) = tf_{n}(y). Let ω′(y, t) = ω(y). Then Ỹ = (Y′, f′, ω′) is an elementary blowup of Y. It comes with the projection map Y′ → Y.

(2) Let X = (X, g, ω) ∈ Ob μRV[n, ·], X = X′ ∪^{.} X′′, g′ = g|X′, g′′ = g|X′′, ω′ =

ω|X′, ω′′ = ω|X′′, and let ϕ : Y → (X′, g′, ω′) be a μ_{l}RV[n, ·]-isomorphism. Then

the RV- blowup X̃_{ϕ} is defined to be Ỹ + (X′′, g′′, ω′′) = (Y′ ∪^{.} X′′, f′ ∪^{.} g′′, ω′ ∪^{.} ω′′). It

comes with b : Y′ ∪^{.} X′′ → X, defined to be the identity on X′′, and the projection on Y′. X′ is called the blowup locus of b : X̃_{ϕ} → X.

An iterated RV-blowup is obtained by finitely many iterations of RV- blowups. Definition 8.17. Let μI_{sp}[n] be the set of pairs (X_{1}, X_{2}) ∈ Ob μRV[n, ·] such that there exist iterated blowups b_{i} : X̃_{i} → X_{i} and a μ_{l}RV[n, ·]- isomorphism F : X̃_{1} → X̃_{2}.

When n is fixed, we will simply write μI_{sp}. On the other hand we will need to make explicit the dependence on the theory; we write μI_{sp}(A) for the congruence μI_{sp} of the theory T_{A}.

When X = (X, f, ω) ∈ Ob μRV[n, ·], h : X → W is a definable map, and c ∈ W, define X_{c} = (h^{−1}(c), f|h^{−1}(c), ω|h^{−1}(c)).

Let X_{1}, X_{2} ∈ Ob μRV[n, ·], and let f_{i} : X_{i} → Y be a definable map, with Y ⊆ RV^{∗}. In this situation the existence of μRV[n, ·](< a >)-isomorphisms between each pair of fibers X_{1}(a), X_{2}(a) (a ∈ Y) does not necessarily imply that X_{1} ≃_{μRV[≤n,·]} X_{2}, because of the explicit reference to dimension in the definition of morphisms; the dimension of the allowed exceptional sets may accumulate over Y. The definition of morphisms for μVF[n] also allows a smaller dimensional exceptional set; but this does not create a problem when fibered over W ⊆ RV^{∗}, since by Lemma 3.56 max_{c∈W} dim_{VF}(Z_{c}) = dim_{VF}(Z). Thus an RV-disjoint union of μVF[n]- isomorphisms is again a μVF[n]-isomorphism, and it follows that the same is true for μ_{l}RV[n, ·]. We thus have: Lemma 8.18. Let X = (X, f, ω), X′ = (X′, f′, ω) ∈ μRV[n, ·], and let h : X → W ⊆ RV^{∗}, h′ : X′ → W be definable maps. If for each c ∈ W, (X_{c}, X′_{c}) ∈ μI_{sp}(< c >) then (X, X′) ∈ μI_{sp}.

Proof. Lemma 2.3 applies to RV_{vol}-isomorphisms, and hence using Remark 7.13 also to blowups. It also applies to μ_{l}RV[n, ·]-isomorphisms by the discussion above, and hence to μI_{sp}-equivalence. □

In other words, there exists a well-defined direct sum operation on μRV[n, ·]/μI_{sp}, with respect to RV-indexed systems. Lemma 8.19. (1) Let Y′ be an elementary blowup of Y. Y′ is 𝒞- isomorphic to Y′′ = (Y′′, f′′, ω′), with

Y′′ = {(y, t) ∈ Y × RV_{∞} : val_{rv}(t) > f_{n}(y)}
f′′(y, t) = (f_{1}(y), …, f_{n−1}(y), t,), ω′(y, t) = ω(y)

(3) Up to isomorphism, the blowup depends only on blowup locus. In other words if X, X′, g, g′, ω, ω′ are as in Definition 8.16, and ϕ_{i} : Y_{i} → (X′, g′, ω′) (i = 1, 2) are μ_{l}RV[n, ·]- isomorphisms, then X̃_{ϕ_{1}}, X̃_{ϕ_{2}} are X-isomorphic in μ_{l}RV[n, ·].

Proof. (1) The isomorphism is given by h((y, t)) = (y, tf_{n}(y)); since f_{n} always lifts to a function F_{n} : 𝕃Y → VF (a coordinate projection), h can be lifted to H defined by H((y, t)) = (y, tF_{n}(y)).

(3) By assumption, ϕ_{1}, ϕ_{2} lift to measure preserving maps Φ_{i} : 𝕃Y_{i} → 𝕃X′. On the other hand, by the assumption on existence of a section s of f_{n}, we have measure preserving isomorphisms α_{1} : 𝕃Y_{1} → 𝕃Ỹ_{1}, (y, u_{1}, …, u_{n}) ↦ (y, u_{1}, …, u_{n−1}, (u_{n} − s)/s). Similarly we have α_{2} : 𝕃Y_{2} → 𝕃Ỹ_{2}. Composing we obtain α_{2}Φ_{2}^{−1}Φ_{1}α_{1}^{−1} : 𝕃Ỹ_{1} → 𝕃Ỹ_{2}; it is easy to check that this is ∼_{rv}-invariant and shows that 𝕃Ỹ_{1}, 𝕃Ỹ_{2} are Y-isomorphic in μ_{l}RV[n, ·]. Taking disjoint sum with the complement X′′ of X′ we obtain the result. □

Remark There is also a parallel of Lemma 7.15: Let Y = (Y, g) ∈ Ob RV_{∞}[n, ·], with dim(g(Y)) < n; let f : Y → RV^{n−1} be isogenous to g. Let h : Y → RV be definable, with h(y) ∈ acl(g(y)) for y ∈ Y, and with ∑(g) = ∑(f) + val_{rv}(h). Let Y′ = Y × RV_{∞}^{>0}, and f′(y, t) = (f(y), th(y)). Then for appropriate ω′, Y′ = (Y′, f′, ω′) with the projection map to Y is a blowup. This follows from Lemma 7.15, and Lemma 8.14 (3).

Notation For X ∈ RV[n, ·], [X] = [(X, 1)] denotes the corresponding object of μRV[n, ·] with form 1. Lemma 8.20. Lemma 7.18 (1-5) holds for μ_{l}RV[n, ·]. We also have:

(6) As a semi-ring congruence on K_{+}μ_{l}RV[n, ·], μI_{sp} is generated by ([[1_{k}]_{1}], [[RV^{>0}]_{1}) (with the forms 1)

Proof. (1)-(5) go through with the same proof. For (6), Let ∼ be the congruence generated by this element. By blowing up a point one sees immediately that ([[1]_{1}], [[RV^{>0}]_{1}) ∈ μI_{sp}, so ∼≤ μI_{sp}. For the converse direction we have to show that (Ỹ, Y) ∈∼ whenever Ỹ is a blowup of Y; the elementary case suffices, since the μ_{l}RV[n, ·]- isomorphisms of Definition 8.16 (2) are already accounted for in the semigroup K_{+}μ_{l}RV[n, ·]. Now Y = (Y, f, ω) with f_{n}(y) ∈ RV. Since dim(Y) < n, we have Y ≃ (Y, f′, ω′) where f′_{i} = f_{i} for i < n, f′_{n} = 1, and ω′ = f_{n}ω. So we may assume f_{n} = 1. In this case, as in the proof of Lemma 7.18 (6), (Ỹ, Y) ∈∼. □

Definition 8.21. Let J be a k-element set of natural numbers. VFR_{μ}[J, l, ·] is the set of triples X = (X, f, ω), with X ⊆ VF^{J} × RV^{∗}, f : X → RV_{∞}^{l}, ω : X → RV, and such that f and ω factor through the projection pr_{RV}(X) of X to the RV-coordinates. μI_{sp} is the equivalence relation on VFR_{μ}[J, l, ·]:

(X, Y) ∈ μI_{sp} ⟺ (X_{a}, Y_{a}) ∈ μI_{sp}(< a >) for each a ∈ VF^{J}

K_{+}VFR_{μ} is the set of equivalence classes.

For j ∈ J, let π^{j} : VF^{k} × RV^{∗} → VF^{J−{j}} × RV^{∗} be the projection forgetting the j’th VF coordinate. We will write VFR_{μ}[k, l, ·], VF^{k}, VF^{k−1} for VFR_{μ}[J, l, ·] VF^{J}, VF^{J−{j}} respectively when the identity of the indices is not important.

The map 𝕃 : Ob μRV[n, ·] → Ob μVF[n] induces, by Lemma 6.3, a homomorphism 𝕃 : K_{+}μRV[n, ·] → K_{+}μVF[n]. By Proposition 4.5 it is surjective. Lemma 8.22. Let X, X′ ∈ μRV[n, ·], and let G : 𝕃X′ → 𝕃X be a special bijection. Then X′ is isomorphic to an iterated blowup of X.

Proof. Clear from Lemma 7.19, since strong isomorphisms are also μ_{l}RV[n, ·]-isomorphisms. □

Lemma 8.23. The homomorphism 𝕃 : K_{+}μRV[1, ·] → K_{+}μVF[1, ·] is surjective, with kernel equal to μI_{sp}[1]. The image of K_{+}RV_{vol}[1, ·] is K_{+}VF_{vol}[1, ·]

Proof. Let X, Y ∈ μRV[1, ·], and let F : 𝕃X → 𝕃Y be a definable measure preserving bijection. We have X = (X, f, ω), Y = (Y, g, ω) with (X, f), (Y, g) ∈ RV[1, ·]. By Lemma 7.6 there exist special bijections b_{X} : 𝕃X′ → 𝕃X, b_{Y} : 𝕃Y′ → 𝕃Y and an ∼_{rv}-invariant definable bijection F′ : 𝕃X′ → 𝕃Y′ such that b_{Y} F′ = F b_{X}. We used here that any ∼_{rv}-invariant object can be written as 𝕃X′ for some X′. Since F, b_{X}, b_{Y} are measure preserving bijections, so is F′. By Lemma 8.22, X′ → X and Y′ → Y are blowups; and F′ descends to a definable bijection between them. This bijection is measure preserving by Lemma 5.22. Hence by definition (X, Y) ∈ μI_{sp}. □

By Proposition 8.23, the inverse of 𝕃 : RV[1, ·] → VF[1, ·] induces an isomorphism I_{1}^{vol} : K_{+}VF_{vol}[1, ·] → K_{+}RV_{vol}[1, ·]/μI_{sp}.

I_{1}^{vol}([X]) = [Y]/μI_{sp} ⟺ [𝕃Y] = [X]

Let X = (X, f, ω) ∈ VFR_{μ}[k, l, ·]. By assumption, f, ω factor through π^{j}, so that they can be viewed as functions on π^{j}X. We view the image (π^{j}X, f, ω) as an element of VFR_{μ}[k − 1, l, ·]. Each fiber of π^{j} is a subset of VF; it can be viewed as an element of VF_{vol}[1] ⊆ μVF[1] ⊆ μVF[1, ·]. Claim Relative μI_{sp}-equivalence implies μI_{sp}-equivalence, in the following sense. Let X_{i} ⊆ RV^{∗} (i = 1, 2); h_{i} : X_{i} → W ⊆ RV^{∗}; f_{W} : W → RV^{l}, ω : W → RV, and f_{i} : X → RV^{k} be definable sets and functions. Let X_{i} = (X_{i}, (f_{W} ∘ h_{i}, f_{i}), ω ∘ h_{i}). Let X_{i}(w) = (X_{i}(w), f_{i}|X_{i}(w), ω ∘ h_{i}|X_{i}(w)) where X_{i}(w) = h_{i}^{−1}(w). If X_{1}(w), X_{2}(w) ∈ I_{sp}(< w >) for each w ∈ W, then (X_{1}, X_{2}) ∈ μI_{sp}.

Proof. Clear, using Lemma 8.18. □

The Claim allows us to relativize I_{1}^{vol} to π^{j}. We obtain a map

I^{j} = I_{k,l}^{j} : VFR_{μ}[k, l, ·] → K_{+}VFR_{μ}[k − 1, l + 1, ·]/μI_{sp}

Lemma 8.24. Let X = (X, f, ω), X′ = (X′, f′, ω′) ∈ VFR_{μ}[k, l, ·].

(1) I^{j} commutes with maps into RV: if h : X → W ⊆ RV^{∗} is definable, X_{c} = h^{−1}(c), then

I^{j}(X) = ∑_{c∈W} I^{j}(X_{c})
(2) If ([X], [X′]) ∈ μI_{sp} then (I^{j}(X), I^{j}(X′)) ∈ μI_{sp}.
(3) I^{j} induces a map K_{+}VFR_{μ}[k, l, ·]/μI_{sp} → K_{+}VFR_{μ}[k − 1, l + 1, ·]/μI_{sp}.

Proof. (1) This reduces to the case of I_{1}^{vol}, where it is an immediate consequence of the uniquness, and the fact that 𝕃 commutes with maps into RV in the same sense.

(2) All equivalences here are relative to the k − 1 coordinates of VF other than j, so we may assume k = 1, and write I for I^{j}.

For a ∈ VF, ([X_{a}], [X′_{a}]) ∈ μI_{sp}(< a >). By stable embeddedness of RV, there exists α = α(a) ∈ RV^{∗} such that X_{a}, X′_{a} are < α >-definable there are < α >-definable blowups X̃_{a}, X̃′_{a} and an < α >-definable isomorphism between them, lifting to an a-definable isomorphism. Using (1) and Lemma 8.18 we may assume that α is constant. So for some W ∈ Ob VF[1], Y, Y′ ∈ μRV[l + 1, ·], we have X = W × Y, X′ = W × Y′, Ỹ, Ỹ′ are blowups of Y, Y′ respectively, ϕ : Y → Y′ is a bijection, and for any w ∈ W there exists a measure preserving F_{w} : 𝕃Ỹ → 𝕃Ỹ lifting ϕ. Then I(X) = I(W) × Y, I(X′) = I(W) × Y′ and the bijection Id_{I(W)} × ϕ is lifted by the measure preserving bijection (w, y) ↦ (w, F_{w}(y)).

(3) by (2).

□

Lemma 8.25. Let X = (X, f, ω) ∈ Ob VFR_{μ}[J, l, ·]. If j ≠ j′ ∈ J then I^{j}I^{j′} = I^{j′}I^{j} : K_{+}VFR_{μ}[J, l, ·]/μI_{sp} → K_{+}VFR_{μ}[J \ {j, j′}, l + 2, ·]/μI_{sp}.

Proof. We may assume S = {1, 2}, j = 1, j′ = 2, since all is relative to VF^{S\{j,j′}}. By Lemma 7.23 (1) and Lemma 8.18 it suffices to prove the statement for each fiber of a given map into RV[l]. Hence we may assume X ⊆ VF^{2} so that f is constant; and by Lemma 5.10, we can assume X is a basic 2-cell:

X = {(x, y) : x ∈ X_{1}, rv(y − G(x)) = α_{1}} X_{1} = rv^{−1}(δ_{1}) + c_{1}

The case where G is constant is easy, since then X is a finite union of rectangles. Otherwise G is invertible, and by niceness of G we can also write:

X = {(x, y) : y ∈ X_{2}, rv(x − G^{−1}(y)) = β}, X_{2} = rv^{−1}(δ_{2}) + c_{2}

We immediately compute:

I_{2}I_{1}(X) = (δ_{1}, α_{1}), I_{1}I_{2}(X) = (α_{2}, δ_{2})

and necessarily val_{rv}δ_{1} + val_{rv}α_{1} = val_{rv}α_{2} + val_{rv}δ_{2}. ( Lemma 5.4). We have bijections F_{j} : X → 𝕃I_{j}(X). The map F_{1}F_{2}F_{1}^{−1}F_{2}^{−1} : 𝕃I_{2}I_{1}(X) → 𝕃I_{1}I_{2}(X) lifts the unique bijection between the singleton sets {(δ_{1}, α_{1})}, {(α_{2}, δ_{2})}, and shows that [(δ_{1}, α_{1})]_{2} = [(α_{2}, δ_{2})]_{2}. □

Proposition 8.26. Let X, Y ∈ μRV[≤ n, ·]. If 𝕃X, 𝕃Y are isomorphic, then ([X], [Y]) ∈ μI_{sp}.

Proof. Identical to the proof of Proposition 7.25, only quoting Lemma 8.25 in place of Lemma 7.24, and Lemma 8.14 to enable using Lemma 7.11. □

Proposition 8.27. Proposition 8.3 is valid for μVF[n], μRV[n], μI_{sp}[n].

Proof. Same as the proof of Proposition 8.3, but using Proposition 6.3 in place of 6.1 and Proposition 8.26 in place of 7.25. □

8.6. Invariants of measure preserving maps, and some induced isomorphisms. Theorem 8.28. Let T be V-minimal. There exists a canonical isomorphism of Grothendieck semigroups

∫ : K_{+}^{eff}μVF[n, ·] → K_{+}(μRV[n, ·])/μI_{sp}[n]

Let [X] denote the class of X in K_{+}^{eff}(μVF[n]). Then

∫ [X] = W/μI_{sp}[n] ⟺ [X] = [𝕃W] ∈ K_{+}^{eff}(μVF[n])

Proof. Given X = (X, f, ω) ∈ Ob μRV[n] we have 𝕃X ∈ Ob μVF[n].

If X, X′ are isomorphic, then by Lemma 6.3, 𝕃X, 𝕃X′ are effectively isomorphic.

Direct sums are clearly respected, so we have a semigroup homomorphism 𝕃 : K_{+}(μRV[n]) → K_{+}^{eff}(μVF[n]).

It is surjective by Proposition 4.5, injective by Proposition 8.3.

Inverting, we obtain I. □

Let μI_{sp}′ be the semigroup congruence on RV_{vol}[n] generated by ((Y, f), (Y × RV^{>0}, f′)) where Y, f, f′ are as in Definition 7.12. Let μ_{Γ}I_{sp} be the congruence on K_{+}μ_{Γ}RV[n] generated by ([[1_{k}]_{1}], [[RV^{>0}]_{1}), with the constant Γ-form 0 ∈ Γ.

Assume given a distinguished subgroup N_{1} of the multiplicative group of the residue field k. For example, N_{1} may be the group of elements of norm one, with respect to some absolute value |, |on k. With this example in mind, write |x| = 1 for x ∈ N_{1}. Let |μ|VF[n] be the subcategory of VF[n] with the same objects, and such that F ∈ Mor_{|μ|VF[n]} iff F ∈ Mor_{μ_{Γ}VF[n]} and |JRV (F)| = 1 almost everywhere. Similarly define |μ^{.}|RV[n].

Theorem 8.29. The isomorphism ∫ of Theorem 8.28 induces isomorphisms:

(9)  K_{+}^{eff}VF_{vol}[n] → K_{+}RV_{vol}[n]/μI_{sp}′[n]
(10)  K_{+}^{eff}VF_{vol}^{bdd}[n] → K_{+}RV_{vol}^{bdd}[n]/μI_{sp}[n]
(11)  K_{+}^{eff}μVF^{bdd}[n] → K_{+}μRV^{bdd}[n]/μI_{sp}[n]
(12)  K_{+}^{eff}|μ|VF[n] → K_{+}|μ^{.}|RV[n]/μI_{sp}[n]
(13)  K_{+}^{eff}μ_{Γ}VF[n] → K_{+}μ_{Γ}RV[n]/μ_{Γ}I_{sp}[n]

(14)

Proof. Since Proposition 4.5 uses measure preserving maps, Proposition 6.1 do not go out of the subcategory VF_{vol}, and RV_{vol}[n] is a full subcategory of μRV[n], we have (9). It is similarly easy to see that “dimension < n” and boundedness are preserved, hence (10)-(11).

We have K_{+}^{eff}|μ|VF = K_{+}^{eff}μVF/N_{VF}, where N_{VF} = {([X, ω], [X, gω]) : g : X → RV, |g| = 1}. Similarly for K_{+}^{eff}|μ^{.}|RV. Thus for (12) it suffices to show that (∫ (X), ∫ (Y)) ∈ N_{RV} ⟺ (X, Y) ∈ N_{VF}. For X ∈ Ob μVF[n] or X ∈ Ob μRV[n] with RV-volume form ω, given g : X → RV, let ^{g}X denote the same object but with volume form gω. In one direction we have to show: (𝕃X, 𝕃Y) ∈ N_{VF} if (X, Y) ∈ N_{RV}. This is clear since 𝕃(^{g}X) =^{g} (𝕃X). Conversely we have to show that (∫ [^{g}Z], ∫ [Z]) ∈ N_{RV}. Since ∫ commutes with RV-sums, we may assume g is constant, with value a. But then 𝕃(^{a}X) =^{a} (𝕃X) implies ∫ (^{a}Z) =^{a} ∫ Z as required. This gives (12); (13) is a special case. □

9. The Grothendieck semirings of Γ

Let T = DOAG_{A} be the theory of divisible ordered Abelian groups Γ, with distinguished constants for elements of a subgroup A. Let DOAG_{A}[∗] be the category of all DOAG_{A} definable sets and bijections. Our primary concern is not with DOAG_{A}, but rather a proper subcategory Γ[∗], having the same objects but only piecewise integral morphisms (Definition 9.1.) Our interest in Γ[∗] derives from this: the morphisms of Γ[∗] are precisely those that lift to morphisms of RV[∗], and it is K_{+}[Γ[∗]] that forms a part of K_{+}[RV[∗]]. (cf. §3.3.) This category depends on A, but will nevertheless be denoted Γ[∗] when A is fixed and understood.

We will first describe K(Γ^{fin}[∗]), the subring of classes of finite definable sets. Next we will analyze K(DOAG_{A}), obtaining two Euler characteristics. This repeats earlier work by Maříková. We retain our proofs as they give a rapid path to the Euler characteristics, but [26] includes a complete analysis of the semiring K(DOAG_{A}), that may well be useful in future applications.

At the level of Grothendieck rings, the categories Γ[∗]_{A} and DOAG_{A} may be rather close; see Lemma 9.8 and Question 9.9. But the semiring homomorphism K_{+}(Γ[∗]_{A}) → K(DOAG_{A}) is far from being an isomorphism, and it remains important to give a good description of K_{+}(Γ[∗]_{A}). We believe that further invariants can be found by mapping K_{+}[Γ[∗]] into the Grothendieck semirings of other completions of the universal theory of ordered Abelian groups over A, as well as DOAG, in the manner of Proposition 9.2; it is possible that all invariants appear in this way.

A description of K_{+}(Γ[∗]_{A}) would include information about the Grothendieck group of subcategories, such as the category of bounded definable sets. We will only sample one bit of the information available there, in the form of a “volume” map on bounded subsets of K_{+}[Γ[∗]] into the rationals, and a discrete analogue.

Definition 9.1. An object of Γ[n] is a finite disjoint union of subsets of Γ^{n} defined by linear equalities and inequalities with ℤ-coefficients and parameters in A. Given X, Y ∈ Ob Γ[n], f ∈ Mor_{Γ}(X, Y) iff f is a bijection, and there exists a partition X = ∪_{i=1}^{n}X_{i}, M_{i} ∈ GL_{n}(ℤ), a_{i} ∈ A^{n}, such that for x ∈ X_{i},

f(x) = M_{i}x + a_{i}

Γ[∗] is the category of definable subsets of Γ^{n} for any n, with the same morphisms. Since there are no morphisms between different dimensions, it is simply the direct sum of the categories Γ[n], and the Grothendieck semi-ring K_{+}[Γ] of Γ[∗] is the graded direct sum of the semigroups K_{+}(Γ[n]). We will write K[Γ] for the corresponding group.

Let Γ^{bdd}[∗] be the full subcategory of Γ[∗] consisting of bounded sets, i.e. an element of Ob Γ^{bdd}[n] is a definable subset of [−γ, γ]^{n} for some γ ∈ Γ.

Γ_{A} is a subcategory of Γ_{ℚ⊗A} (a category with the same objects, but more morphisms, generated by additional translations) and this in turn is a subcategory of DOAG_{ℚ⊗A}.

There is therefore always a natural morphism from K_{+}(Γ_{A}[∗]) to the simpler semigroup K_{+}(DOAG_{ℚ⊗A}). We will exhibit two independent Euler characteristics on DOAG_{ℚ⊗A} and show that they define an isomorphism K(DOAG_{ℚ⊗A}) → ℤ^{2}. Taking the dimension grading into account, this will give rise to two families of Euler characteristics on K(Γ_{A}), with ℤ[T]- coefficients.

9.1. Finite sets. Let Γ^{fin}[n] be the full subcategory of Γ_{A}[n] consisting of finite sets. The Grothendieck semi-ring of Γ^{fin}[∗] embeds into the semi-rings of both Γ_{A} and RES, within the Grothendieck semi-ring of RV_{A}, and we will see that K_{+}(RV_{A}) is freely generated by them over K_{+}(Γ^{fin}[∗]. We proceed to analyze K_{+}(Γ^{fin}[∗]) in detail.

Let τ = [0]_{1} ∈ K_{+}(Γ^{fin}[1]) be the class of the singleton {0}.

The unit element of K(Γ) is the class of Γ^{0}. Note that the bijection between τ and Γ^{0} is not a morphism in Γ[∗]; in fact 1, τ, τ^{2}, … are distinct and ℚ-linearly independent in K(Γ). The motivation for this choice of category becomes clear if one thinks of the lift to RV: the inverse image of τ^{n} in RV (also denoted τ^{n}) has dimension n, and cannot be a union of isomorphic copies of τ^{m} for smaller m.

Let K(Γ^{fin})[τ^{−1}] be the localization. This ring is naturally ℤ-graded ring; let H_{fin} be the 0-dimensional component.

Let Ξ_{A} be the space of subgroups of (ℚ⊗A)/A, or equivalently of subgroups of ℚ⊗A containing A. View it as a closed subspace of the Tychonoff space 2^{(ℚ⊗A)/A}, via the characteristic function 1_{s} of a subgroup s ∈ Ξ_{A}. Let C(Ξ_{A}, ℤ) be the ring of continuous functions Ξ_{A} → ℤ (where ℤ is discrete.)

A cancellation semigroup is a semigroup where a + b = a + c implies b = c; in other words, a subsemigroup of an Abelian group. Proposition 9.2. K_{+}(Γ^{fin}[n]) is a cancellation semigroup. As a semiring, K_{+}(Γ^{fin}[∗]) is generated by K_{+}(Γ^{fin}[1]). We have:

K(Γ^{fin})[τ^{−1}] = H_{fin}[τ, τ^{−1}]
H_{fin} ≃ C(Ξ_{A}, ℤ)

Proof. Since Γ is ordered, any finite definable subset of Γ^{n} is a union of definable singletons. Thus the semi-group K_{+}(Γ^{fin}[n]) is freely generated by the isomorphism classes of singletons a ∈ Γ^{n}, and in particular is a cancellation semigroup. The displayed equality is thus clear; we proceed to prove the isomorphism.

A definable singleton of Γ^{n} has the form (a_{1}, …, a_{n}), where for some N ∈ ℕ, Na_{1}, …, Na_{n} ∈ A. So [(a_{1}, …, a_{n})] = [(a_{1})] ⋯ [(a_{n})].

For any commutative ring R, let Idem(R) be the Boolean algebra of idempotent elements of a commutative ring R with the operations 1, 0, xy, x + y − xy. Note that the elements [(a_{1}, …, a_{n})]τ^{−n} ∈ H_{fin} belong to Idem(H_{fin}): in K_{+}(Γ^{fin}): for any a ∈ Γ we have the relation: [a]^{2} = [a]τ. Let B be the Boolean subalgebra of Idem(H_{fin}) generated by the elements [(a_{1}, …, a_{n})]τ^{−n}. For a maximal ideal M of B, let I_{M} be the ideal of H_{fin} generated by M. Note H_{fin} = ℤB. Hence we have to show:

(1) The Stone space of B is Ξ_{A}

(2) For any maximal ideal M of B, H_{fin}/I_{M} ≃ ℤ naturally.

For any commutative ring R, a finitely generated Boolean ideal of Idem(R) is generated by a single element b; if b ≠ 1, then bR ≠ R since b(1 − b) = 0. Thus if M is a proper ideal of Idem(R), then MR is a proper ideal of R. Applying this to B, viewed as a Boolean subalgebra of Idem(ℚ⊗H_{fin}), we see that I_{M} ∩ ℤ = (0) for any maximal ideal M of B. Thus the composition ℤ → H_{fin} → H_{fin}/I_{M} is injective. On the other hand, H_{fin} is generated over ℤ by the elements [a]/τ, and each of them equals 0 or 1 modulo I_{M}, so the map is surjective too. This proves the second point.

To prove the first, we define a map Φ : Ξ_{A} → Stone(B).

Let t = T/A, T ≤ ℚ⊗A. If [(a_{1}, …, a_{n})] = [(b_{1}, …, b_{n})], then some element of GL_{n}(ℤ)⋉A^{n} takes (a_{1}, …, a_{n}) to (b_{1}, …, b_{n}); in this case, if a_{i} ∈ T for each i then b_{i} ∈ T for each i; so Π_{i=1}^{n}1_{t}(a_{i} + A) = Π_{i=1}^{n}1_{t}(b_{i} + A). Thus, given t ∈ Ξ_{A}, we can define a homomorphism h_{t} : H_{fin} → ℤ by:

\frac{[(a_{1}, …, a_{n})]}{τ^{n}} ↦ Π_{i=1}^{n}1_{t}(a_{i} + A)

Let M(t) = ker(h_{t}) ∩ B.

The map Φ : t ↦ M(t) is clearly continuous. If t, t′ are distinct subgroups, let a ∈ t, a ∉ t′ (say); then [a]/τ ∈ M(t), [a]/τ ∉ M(t′). So Φ is injective. If P is a maximal filter of B, let t_{P} = {a + A : [a]/τ ∈ P }. Claim t_{P} is a subgroup.

Proof:

Suppose a + A, b + A ∈ t_{P} and let c = a + b. Then we have the relation

[a][b]τ = [a][b][c]

in K_{+}(Γ^{fin}), arising from the map

(x, y, z) ↦ (x, y, xyz)

Thus ([a]/τ)([b]/τ)(1 − [c]/τ) = 0. As ([a]/τ), ([b]/τ) ∈ P we have (1 − [c]/τ) ∉ P, so [c]/τ ∈ P. Clearly P = M(t_{P}). Thus Φ is surjective, and so a homeomorphism. □ Example We always have a homomorphism K(Γ^{fin}) → ℤ (by counting points of a finite set

in the divisible hull); when A is divisible, this identifies K(Γ^{fin}) with ℤ[τ]. In general, we have the surjective morphism K(Γ^{fin})) → K(Γ_{ℚ⊗A}^{fin}) = ℤ[τ].

Lemma 9.3. Let Y be an A-definable subset of Γ^{n}, of dimension < n. Then Y is a finite union of GL_{n}(ℤ)-conjugates of sets Y_{i} ⊆ {c_{i}} × Γ^{n−1}, with c_{i} ∈ ℚ⊗A.

Proof. Y can be divided into finitely many A-definable pieces, each contained in some Adefinable hyperplane of Γ^{n}. So we may assume Y itself is contained in some such hyperplane, i.e. ∑ r_{i}y_{i} = c for some c ∈ ℚ⊗val_{rv}(A). We may assume r_{i} ∈ ℤ and (r_{1}, …, r_{n}) have no common divisor. In this case ℤ^{n}/ℤ(r_{1}, …, r_{n}) is torsion free, hence free, so ℤ(r_{1}, …, r_{n}) is a direct summand of ℤ^{n}. Thus after effecting a transformation of GL_{n}(ℤ), we may assume (r_{1}, …, r_{n}) = (1, 0, …, 0), i.e. Y lies in the hyperplane y_{1} = c. Let Z be the projection of Y to the coordinates (2, …, n). Then Y = {c} × Z. □

9.2. Euler characteristics of DOAG. We describe two independent Euler characteristics on A- definable subsets of Γ, i.e. additive, multiplicative ℤ[τ]-valued functions invariant under all definable bijections. The values are in ℤ[τ] rather than ℤ because Γ[∗] = ⊕_{n}Γ[n] is graded by ambient dimension. Proposition 9.4 - Lemma 9.6 were obtained earlier in [26], and independently in [20].

In fact these two Euler characteristics come from Euler characteristics of DOAG_{ℚ⊗A}. There, they are the only ones:

Proposition 9.4. Let A be a divisible ordered Abelian group. Then K(DOAG_{A}) ≃ ℤ^{2}

Proof. We begin by noting that there are at most two possibilities.

In DOAG, all definable singletons are isomorphic. The identity element of the ring K(DOAG) is the class of any singleton. Thus the image of K(Γ^{fin}[∗]) in K(DOAG_{A}) is isomorphic to ℤ. Claim The image of K(Γ_{A}^{bdd}) in K(DOAG_{A}) equals the image of K(Γ^{fin}[∗]) there.

Translation by a gives an equality of classes in K(Γ), [(0, ∞) = [(a, ∞)] so

[(0, a)] + [{pt}] = [(0, a] = 0

Thus bounded segments are equivalent to linear combinations of points. This can be seen directly by induction on dimension and on ambient dimension: consider the class of a bounded set Y ⊂ Γ^{n+1}. Y is a Boolean combination of sets of the form {(x, y) : x ∈ X, f(x) < y < g(x)}. This is DOAG_{A}-isomorphic to Y′ = {(x, y) : x ∈ X, 0 < y < h(x)}, where h(x) = g(x) − f(x). Let Z = {(x, y) : x ∈ X, y > 0}, Z′ = {(x, y) : x ∈ X, y > h(x)}. Then the map (x, y) ↦ (x, y + h(x)) shows that [Z] = [Z′]. On the other hand Z′ is the disjoint union of Z, Y and a smaller-dimensional set W. Thus [Z] = [Z′] = [Z′]+[Y]+[W] so [Y] = −[W], and by induction [Y] lies in the image of K(Γ^{fin}[∗]).

Now consider t = [(0, ∞)] ∈ K(Γ_{A}). We have a homomorphism K(Γ_{A}^{bdd})[t] → K(Γ). To see that it is surjective, again by induction it suffices to look at sets such as {(x, y) : x ∈ X, f(x) < y} or {(x, y) : x ∈ X, f(x) < y < g(x)}. The latter is equivalent to a smaller-dimensional set, by induction, as above. The former is equivalent to {(x, y) : x ∈ X, 0 < y} so that it has the class [X] × t and is thus in the image of K(Γ_{A}^{bdd})[t].

Let T = {(x, y) : 0 < y ≤ x}. The map (x, y) ↦ (x, y + x) takes T to {(x, y) : 0 < x < y ≤ 2x}, so 2[T] = [{(x, y) : 0 < y ≤ 2x}]. The same map shows that t^{2} − [T] = t^{2} − 2[T] so [T] = 0. But then [{(x, y) : 0 < x ≤ y}] = 0, and adding we obtain: 0+0 = t^{2}+[{(x, x) : 0 < x}] = t^{2}+t. Thus K(DOAG_{A}) is a homomorphic image of ℤ[t]/(t^{2}+t) ≃ ℤ^{2}. To see that the homomorphism is bijective, it remains to exhibit a homomorphism K(DOAG_{A}) → ℤ with t ↦ 0 and another with t ↦ −1. The two lemmas below show this, in a form suitable also for a dimension- graded version.

□

Lemma 9.5. There exists a ring homomorphism χ_{O} : K(Γ) → ℤ[τ], with χ_{O}((0, ∞)) = τ. It is invariant under GL_{n}(ℚ) acting on Γ^{n}.

Proof. Let RCF be the theory of real closed fields. See [36] for the existence and definability of an Euler characteristic map χ : K(RCF) → ℤ. For any definable X, P, f : X → P of RCF, there exists m ∈ ℕ and a definable partition P = ∪_{−m≤i≤m}P_{i}, such that for any i, any M ⊧ RCF and b ∈ P_{i}(M), χ(X_{b}) = i. Here X_{b} = f^{−1}(b), and χ(X_{b}) = i iff there exists an M-definable partition of X_{b} into definable cells C_{j}, with ∑_{j}(−1)^{dim(C_{j})} = i.

The language of Γ (the language of ordered Abelian groups) is contained in the language of RCF. So if X, P, f : X → P are definable in the language of ordered Abelian groups, they are RCF-definable. Therefore, the above result specializes, and we obtain an Euler characteristic map χ : K(Γ_{A}[n]) → ℤ, valid for any divisible group A. This χ is invariant under all definable bijections (not only the morphisms of Γ[∗]), and is additive and multiplicative. We have χ_{O}({0}) = 1, χ_{O}((a, b)) = −1 for a < b, and χ_{O}(0, ∞) = −1 too (though (0, 1) and (0, ∞) are not definably isomorphic in the linear structure.) Now let χ_{O}(X) = χ(X)τ^{n} for X ⊆ Γ^{n}, and extend to Γ[∗] by additivity. □

Remark The Euler characteristic constructed in this proof appears to depend on an embedding of A into the additive group of a model of RCF. But by the uniqueness shown above it does not. In fact, as pointed out to us by Van den Dries, Ealy and Maříková, an Euler characteristic with the requisite properties is defined in [36] directly for any O-minimal structure; moreover the use of RCF in the lemma below can also be replaced by a direct inductive argument, and some simple facts about Fourier-Motzkin elimination.

Another Euler characteristic can be obtained as follows: given a definable set Y ⊂ Γ^{n}, let

χ′(Y) = lim_{r→∞} χ(Y ∩ C_{r})

where C_{r} is the bounded closed cube [−r, r]^{n}. By O-minimality, the value of χ(Y ∩ C_{r}) is eventually constant.

Note that χ′ is not invariant under semi-algebraic bijections, since the bounded and unbounded open intervals are given different measures. Still,

Lemma 9.6. χ′ induces a group homomorphism K(Γ[n]) → ℤ; and yields a ring homomorphism K(Γ[∗]) → ℤ[τ]. Moreoever χ′ is invariant under piecewise GL_{n}(ℚ)-transformations.

Proof. χ′ is clearly additive and multiplicative. Isomorphism invariance can be checked as follows: first, Claim If X ≠ ∅ is defined by a finite number of weak (≤) affine equalities and inequalities, then χ′(X) = 1.

Proof. It suffices to show that this is true in (ℝ, +); since then it is true in any model of the theory of divisible ordered Abelian groups. Now we may compute the Euler characteristic χ of the bounded sets X ∩ C_{r} in (ℝ, +, ·). Let p ∈ X. For large enough r, p ∈ X ∩ C_{r} there is a definable retraction of the closed bounded set X ∩C_{r} to p (along lines through p.) Thus X ∩C_{r} has the same homology groups as a point, and so Euler characteristic 1. □

To prove the lemma we must show that if ϕ : X → Y is a definable bijection, X, Y ⊆ Γ^{n}, then χ′(X) = χ′(Y). We use induction on dim(X). By additivity, if X is a Boolean combination of finitely many pieces, it suffices to prove the lemma for each piece. We may therefore assume that ϕ is linear (rather than only piecewise linear) on X. Let ϕ′ be a linear automorphism extending ϕ. Expressing X as a union of basic pieces, we may assume X is defined by some inequalities ∑ α_{i}x_{i} ≤ c, as well as some equalities and strict inequalities. So X is convex. We have to show that χ′(X) = χ′(ϕ′X). Let X̄ be the closure of X (defined by the corresponding weak inequalities.) Then X̄ \ X has dimension < dim(X), so by induction χ′(ϕ′(X̄ \ X) = χ′(X̄ \ X). But X̄ is closed and convex, so χ_{O′}(X̄) = 1 = χ_{O′}(ϕ′X̄). Subtracting, χ′(ϕ′(X̄)) = χ′(X̄).

Once again, using the ambient dimension grading, we can define χ′_{O} : Γ[∗] → ℤ[τ] with χ′_{O}(x) = χ′(x)τ^{n} for x ∈ Γ[n]. □

In the following lemma, all classes are taken in K(Γ_{A})[∗]. Let e_{a} be the class in K(Γ_{A})[1] of the singleton {a}, and τ_{a} the class of the segment (0, a).

Lemma 9.7. Let a ∈ ℚ⊗A, b ∈ A.

(1) τ_{a} = τ_{a+b}, e_{a} = e_{a+b}.

(2) If b < c ∈ A then [(b, c)] = −e_{0}.

(3) e_{a}e_{0} = e_{a}^{2}
(4) τ_{a}(τ_{a} + e_{0}) = 0
(5) If 2a ∈ A then 2τ_{a} + e_{a} = −e_{0}, and e_{0}(e_{a} − e_{0}) = 0

Proof. (1) τ_{a} = [(0, a)] = [(0, ∞)] − [(a, ∞)] − e_{a} and similarly τ_{a+b}. The map x ↦ x + b shows that [(a, ∞) = [(a + b, ∞)] and e_{a} = e_{a+b}, hence also τ_{a} = τ_{a+b}.

(2) [(b, c)] = [(b, ∞)] − [(c, ∞)] − e_{c} = −e_{0} by (1), since c − b ∈ A.

(3) The map (x, y) ↦ (x, y + x) is an SL_{2}(ℤ)-bijection between {(a, 0)} and (a, a).

(4) Let

D = {(x, y) : 0 < x < a, 0 < y ≤ x}
D′ = {(x, y) : 0 < y < a, 0 < x ≤ y}
D_{1} = {(x, y) : 0 < x < a, y > 0}
T (x, y) = (x, y + x)

Then T (D_{1}) = D_{1} \ D. Since [T (D_{1})] = [D_{1}], [D] = 0. Similarly [D′] = 0. Note also

T ((0, a) × {0}) = {(x, x) : 0 < x < a}

Thus

0 = [D] + [D′] = [(0, a)^{2}] + [{(x, x) : 0 < x < a}] = τ_{a}^{2} + τ_{a}e_{0}

(5) Let 0 < 2a ∈ A. Then [(0, a)] = [(a, 2a)] using the map x ↦ 2a − x. So 2τ_{a} + e_{a} = [(0, a) ∪ {a} ∪ (a, 2a)] = [(0, 2a)] = −e_{0} (by (2)). So (−e_{0} − e_{a})(e_{0} − e_{a}) = (2τ_{a})(2τ_{a} + 2e_{0}) = 0 by (1). Thus e_{a}e_{0} = e_{a}^{2} = e_{0}^{2}. □

The next lemma will not be used, except as a partial indication towards the question that follows, regarding the difference at the level of Grothendieck groups between GL_{n}(ℤ) and GL_{n}(ℚ) transformations. Let Ann(e_{0}) be the annihilator ideal of e_{0}; it is a graded ideal. Let R = K(Γ_{A})[∗]/Ann(e_{0}), the image of K(Γ_{A})[∗] in the localization K(Γ_{A})[∗](e_{0}^{−1}). In the next lemma, the classes of definable sets are taken in R, viewed as a subring of K(Γ_{A})[∗](e_{0}^{−1}). Let e_{a} = e_{a}/e_{0}, t_{a} = τ_{a}/e_{0}.

Lemma 9.8. Let A′ = {a ∈ ℚ⊗A : e_{a} = 1}.

(1) If X ⊆ Γ^{n} is definable by linear inequalities over A, and T ∈ GL_{n}(ℤ) ⋉ (A′)^{n}, then [T X] = [X] ∈ R.

(2) A′ is a subgroup of ℚ⊗A.

(3) e_{a}^{2} = e_{a}, t_{a}(t_{a} + 1) = 0

(4) A′ is 2-divisible.

Proof. (1) It suffices to show this when T is a translation by an element a ∈ (A′)^{n}. The map (x, y) ↦ (x + y, y) is in SL_{2n}(ℤ), hence [X × {a}] = [T X × {a}] in K(Γ_{A})[2n]. Since a ∈ (A′)^{n},

[a] = e_{0}^{n}. So [X]e_{0}^{n} = [T X]e_{0}^{n}, and upon dividing by e_{0}^{n} the statement follows.

(2) Clear from (1). For the following clauses, note that by (1),(2), Lemma 9.7 applies with A replaced by A′.

(3) By Lemma 9.7 (3),(4) divided by e_{0}^{2}.

(4) By Lemma 9.7(5) applied to A′, if 2a ∈ A′ then e_{0}(e_{a} − e_{0}) = 0; so e_{a} − 1 = 0, i.e. a ∈ A′. □

Question 9.9. Is it true that K(Γ_{A}[∗])/Ann(e_{0}) = K(DOAG_{A}[∗])/Ann(e_{0}) ?

A positive answer would follow from an extension of (4) to odd primes, over arbitrary A; by an inductive argument, or by integration by parts.

9.3. Bounded sets: volume homomorphism. Let Ā = ℚ⊗A. Recall Γ^{bdd}[n] is the category of bounded A-definable subsets of Γ^{n}, with piecewise GL_{n}(ℤ) ⋉ A - bijections for morphisms. Let Sym(Ā) be the symmetric algebra on A.

Proposition 9.10. There exists a natural “volume” ring homomorphism K(Γ^{bdd}[∗]) → Sym(Ā)

Proof. We first work with DOAG without parameters, defining a polynomial associated with a family of definable sets.

Let C(x, u) = C(x_{1}, …, x_{n}; u_{1}, …, u_{m}) be a formula of DOAG. Write C_{b} = {x : C(x, b)}; this is a definable family of definable sets. Assume the sets C_{b} are uniformly bounded: equivalently, as one easily sees, for some q ∈ ℕ, for each i, C(x, u) implies |x_{i}| ≤ q ∑_{j} |u_{j}|. For b ∈ ℝ^{m}, let v(b) = volC_{b}(ℝ^{n}). Here vol is the Lebesgue measure.

By a constructible function into ℚ, we mean a ℚ-linear combination of characteristic functions of definable sets of DOAG. Let R be the ℚ-algebra of constructible functions into ℚ. Claim 1 There exists a polynomial P_{C}(u) ∈ R[u] such that for all b ∈ ℝ^{m}, volC_{b}(ℝ^{n}) = P_{C}(b).

In other words, the volume of a rational polytope is piecewise polynomial in the parameters, with linear pieces. The proof of the claim is standard, using iterated integration. For each C, fix such a polynomial P_{C}.

At this point we re-introduce A. Any A-definable bounded subset of Γ^{n} has the form C_{b} for some C as above and some b ∈ Ā^{m}. Claim 2 If C_{b} = C′_{b′} then P_{C}(b) = P_{C′}(b′).

Proof. (See also below for a more algebraic proof). Fix the formulas C, C′. Write b = Ne, b′ = N′e where e ∈ Ā^{l} is a vector of ℚ-linearly independent elements of Ā, and N, N′ are rational matrices. Write P_{C} = ∑ a_{ν}(u)u^{ν} where a_{ν} is a constructible function into ℚ; similarly P_{C′}.

Note now that any formula ψ(x_{1}, …, x_{l}) of DOAG of dimension l has a solution in ℝ^{l} whose entries are algebraically independent. Use this to find algebraically independent ẽ ∈ ℝ^{l} such that C_{Nẽ} = C′_{N′ẽ}, and a_{ν}(Nẽ) = a_{ν}(b), a_{ν}(N′ẽ) = a′_{ν}(b′) for each multi-index ν of degree d.

By definition of P_{C} we have P_{C}(Nẽ) = P_{C′}(N′ẽ). So ∑ a_{ν}(b)(Nẽ)^{ν} = ∑ a′_{ν}(b′)(N′ẽ)^{ν}. By algebraic independence, ∑ a_{ν}(b)(Nv)^{ν} = ∑ a′_{ν}(b′)(N′v)^{ν} as ℚ-polynomials. So P_{C}(Ne) = P_{C′}(N′e). □

Thus we can define: v(C_{b}) = P_{C}(b). Let us show that v defines a ring homomorphism.

Given C, C′ one can find C′′ such that C′′_{b,b′} = C_{b}∪C_{b′}, and similarly C′′′ with C′′′_{b,b′} = C_{b}∩C_{b′}. Then P_{C} + P_{C′} = P_{C′′} + P_{C′′′}. It follows that v is additive. Similarly v is multiplicative, and translation invariant. Since | det(M)| = 1 for M ∈ GL_{n}(ℤ), if ϕ^{M}(x, u) = ϕ(Mx, u) then P_{ϕ^{M}} = P_{ϕ}.

□

Van den Dries, Ealy, and Maříková pointed out that Claim 2 can also be reduced to the following statement: if Q ∈ R[u], B is any 0-definable set of , and Q vanishes on B(ℝ), then Q vanishes on B(Γ). They prove it as follows: let B̄ be the Zariski closure of B; B̄ is clearly a finite union of linear subspaces, and by intersecting B with each of these, we may assume B̄ is linear, so it is cut out by homogeneous linear polynomials Q_{1}, …, Q_{m}. Each Q_{i} vanishes on B(ℝ) and hence on B(Γ). So Q lies in the (radical) ideal generated by Q_{1}, …, Q_{m}, hence vanishes on B(Γ).

9.3.1. The counting homomorphism in the discrete case. Suppose A has a least positive element 1, and assume given a homomorphism h_{p} : A → ℤ_{p} for each p. Then A embeds into a ℤ-group Ã, i.e. an ordered Abelian group whose theory is the theory T h(ℤ) of (ℤ, <, +). (We have Ã ∩ (ℚ⊗A) = {a/n ∈ ℚ⊗A : (∀p)(n|h_{p}(a)).) We have a homomorphism [X] ↦ [X(Ã)] from K_{+}(Γ[∗]) to K_{+}(T h(ℤ)_{A}). On the other hand the polynomial formula for the number of integral points in a polytope defined by linear equations over ℤ yields a homomorphism K(T h(ℤ)^{bdd}[∗]) → ℚ[A]. By composing we obtain a homomorphism K(Γ^{bdd}[∗]) → ℚ[A].

Remark Using integration by parts one can see that the homomorphism K(T h(ℤ)^{bdd}[∗]) → ℚ[A] above is actually an isomorphism.

9.4. The measured case. We repeat the definition of μΓ from the introduction, along with two related categories. The category volΓ corresponds to integrable volume forms, i.e. those that can be transformed by a definable change of variable to the standard form on a definable subsets of affine n-space. By Lemma 3.26, the liftability condition in (2) is equivalent to being piecewise in GL_{n}(ℤ) ⋉ A^{n}, A^{n} being the group of definable points.

Definition 9.11. (1) For c = (c_{1}, …, c_{n}) ∈ Γ^{n}, let ∑(c) = ∑_{i=1}^{n} c_{i}.

(2) For n ≥ 0 let μΓ[n] be the category whose objects are pairs (X, ω), with X ∈ Ob Γ[n] and ω : X → Γ a definable map. A morphism (X, ω) → (X′, ω′) is a definable bijection f : X → X′ liftable to a definable bijection val_{rv}^{−1}X → val_{rv}^{−1}X′, such that ∑(x) + ω(x) = ∑(x′) + ω′(x′) for x ∈ X, x′ = f(x).

(4) Let μΓ^{bdd}[n] be the full subcategory of μΓ[n] with objects X ⊆ [γ, ∞)^{n} for some γ ∈ Γ.

(3) Let Ob volΓ[n] be the set of finite disjoint unions of definable subsets of Γ^{n}. Given X, Y ∈ Ob volΓ[n], f ∈ Mor_{volΓ[n]}(X, Y) iff f ∈ Mor_{Γ[n]} and ∑(x) = ∑(f(x)) for x ∈ X.

(5) μΓ[∗] is the direct sum of the μΓ[n], and similarly for the related categories.

Recall the Grothendieck rings of functions from §2.2. F n(Γ, K_{+}(Γ)) is a semigroup with pointwise addition. We also have a convolution product: if f is represented by a definable F ⊆ Γ × Γ^{m}, in the sense that f(γ) = [F (γ)], and g by a definable G ⊆ Γ × Γ^{n}, let

f ∗ g(γ) = [{(α, b, c) : α ∈ Γ, b ∈ F (α), c ∈ G(γ − α)}]

The coordinate α in the definition is needed in order to make the union disjoint. In general, it yields an element represented by a subset of Γ × Γ^{m+n+1} rather than m + n. But let F n(Γ, K_{+}(Γ))[n] be the set of [F] ∈ F n(Γ, K_{+}(Γ[n])) such that dim(F (a)) < n for all but finitely many a ∈ Γ. If f ∈ F n(Γ, K_{+}(Γ))[m] and g ∈ F n(Γ, K_{+}(Γ))[n] then f ∗ g ∈ F n(Γ, K_{+}(Γ))[m + n]. Let F n(Γ, K_{+}(Γ))[∗] = ⊕_{m}F n(Γ, K_{+}(Γ))[m], a graded semiring.

Lemma 9.12. (1) K_{+}(μΓ)[n] ≃ F n(Γ, K_{+}(Γ))[n] (2) K_{+}μΓ^{bdd}[n] ≃ {f ∈ F n(Γ, K_{+}(Γ^{bdd}))[n] : (∃γ_{0})(∀γ < γ_{0})(f(γ) = 0)}

(3) K_{+}volΓ[n] ≃ F n(Γ, K_{+}(Γ[n − 1]))

Proof. (1) Let (X, ω) ∈ Ob μΓ[n], with X ⊆ Γ^{n} and ω : X → Γ. Let d(x) = ω(x) + ∑(x). For a ∈ Γ, let X_{a} = {x ∈ X : d(x) = a}. This determines an element F (X, ω) ∈ F n(Γ, K_{+}(Γ[n])), namely a ↦ [X_{a}]. It is clear from additivity of dimension that dim(X_{a}) < n for all but finitely many a; so F (X, ω) ∈ F n(Γ, K_{+}(Γ))[n]. If h ∈ Mor_{μΓ[n]}(X, Y), then by definition of μΓ we have h(X_{a}) = Y_{a}; so [X_{a}] = [Y_{a}] in K_{+}(Γ)[n]. Conversely if for all a ∈ Γ we have [X_{a}] = [Y_{a}] in K_{+}(Γ)[n], then val_{rv}^{−1}(X_{a}), val_{rv}^{−1}(Y_{a}) are a-definably isomorphic. By Lemma 2.3 there exists a definable H : val_{rv}^{−1}(X) → val_{rv}^{−1}(Y) such that for any x ∈ val_{rv}^{−1}(X), H(x) = h_{a}(x) where a = ∑ val_{rv}(x). Clearly H descends to H̄ : X → Y; by construction H̄ lifts to RV, and preserves ∑ +ω, so H̄ ∈ Mor_{μΓ[n]}(X, Y). We have thus shown that [X] ↦ [F (X)] is injective. It is clearly a semiring homomorphism.

For surjectivity, let g ∈ F n(Γ, K_{+}(Γ))[n] be represented by G ⊆ Γ × Γ^{n}. It suffices to consider either g with singleton support {γ_{0}}, or g such that dim(G(a)) < n for all a ∈ Γ. In the first case, g = F (X, ω) where X = G(γ_{0}) and ω(x) = γ_{0} − ∑(x). In the second: after effecting a partition and a permuation of the variables, we may assume G(a) ⊆ Γ^{n−1} × {ψ(a)} for some definable function ψ(a). With another partition of Γ, we may assume g is supported on S ⊆ Γ, i.e. g(x) = 0 for x ∉ S, and ψ is either injective or constant on S. In fact we may assume ψ is injective on S: if ψ is constant on S, let G′ = {(a, (b_{1}, …, b_{n−1}, b_{n} + a)) : (a, (b_{1}, …, b_{n})) ∈ G, a ∈ S}. Then G′ also represents g, and for G′ the function ψ is injective. Now let X = ∪_{a∈S}G(a), and let ω(x) = − ∑(x) + ψ^{−1}(x_{n}). Then F (X, ω) = g.

(2) follows from (1) by restricting the isomorphism.

(3) is proved in a similar manner to (1) though more simply and we omit the details. The key point is that GL_{n}(ℤ) acts transitively on ℙ^{n}(ℚ); this can be seen as a consequence of the fact that finitely generated torsion free Abelian groups are free. More specifically the co-vector (1, …, 1) is GL_{n}(ℤ)-conjugate to (1, 0, …, 0). Thus the catgegory volΓ[n] is equivalent to the one defined using the weighting x_{1} in place of ∑(x_{i}). For this category the assertion is clear.

□

This lemma reduces the study of K_{+}(μΓ) to that of K_{+}(Γ).

10. The Grothendieck semirings of RV

10.1. Decomposition to Γ, RES. Recall that RV is a structure with an exact sequence

0 → k^{∗} → RV →_{val_{rv}} Γ → 0

We study here the Grothendieck semiring of RV in a theory T_{RV} satisfying the assumptions of Lemma 3.26. The intended case is the structure induced from ACV F_{A} for some RV, Γ-generated base structure A.)

We show that the Grothendieck ring of RV decomposes into a tensor product of those of RES, and of Γ.

The category Γ[∗] was described in §9. We used GL_{n}(ℤ) rather than GL_{n}(ℚ) morphisms. The reason is given by: Lemma 10.1. The morphisms of Γ[n] are precisely those definable maps that lift to morphisms of RV[n]. The map X ↦ val_{rv}^{−1}(X) therefore induces a functor Γ[n] → RV[n], yielding an embedding of Grothendieck semirings K_{+}[Γ[n]] → K_{+}[RV[n]].

Proof. Any morphism of Γ[∗] obviously lifts to RV, since GL_{n}(ℤ) acts on C^{n} for any group C. The converse is a consequence of Lemma 3.28. □

We also have an inclusion morphism K_{+}(RES) → K_{+}(RV).

Observe that K_{+}(Γ^{fin}) forms a part of both K_{+}(RES[∗]) and K_{+}(Γ[∗]): the embedding of K_{+}(Γ[∗]) into K_{+}(RV[∗]) takes K_{+}(Γ^{fin}) to a subring of K_{+}(RES[∗]), namely the subring generated by the pullbacks val_{rv}(γ), γ ∈ Γ a definable point.

Given two semirings R_{1}, R_{2} and a homomorphism f_{i} : S → R_{i}, define R_{1}⊗_{S}R_{2} by the universal property for triples (R, h_{1}, h_{2}), with R a semi-ring and h_{i} : R_{i} → R a semiring homomorphism, satisfying h_{1}f_{1} = h_{2}f_{2}.

We have a natural map K_{+}(RES)⊗K_{+}(Γ[∗]) → K_{+}(RV), [X]⊗[Y] ↦ [X × val_{rv}^{−1}(Y)]. By the universal property it induces a map on K_{+}(RES)⊗_{K_{+}(Γ^{fin})}K_{+}(Γ[∗]). A typical element of the image is represented by a definable set of the form ∪^{.} (X_{i} × val_{rv}^{−1}(Y_{i})), with X_{i} ⊆ RES^{∗}, Y_{i} ⊆ Γ^{∗}.

Proposition 10.2. The natural map K_{+}(RES)⊗_{K_{+}(Γ^{fin})}K_{+}(Γ[∗]) → K_{+}(RV) is an isomorphism.

Proof. Surjectivity is Corollary 3.25. We will prove injectivity. In this proof, X⊗Y will always denote an element of K_{+}(RES)⊗_{K_{+}(Γ^{fin})}K_{+}(Γ[∗]). Claim 1 Any element of K_{+}(Γ[∗]) can be expressed as ∑_{j=1}^{l}[Y_{j}] × {p_{j}}, for some Y_{j} ⊆ Γ^{m_{j}}, dim(Y_{j}) = m_{j}, and p_{j} ∈ Γ^{l_{j}}.

Proof. Let Y ⊆ Γ^{m} be definable. If dim(Y) < m, then Y can be partitioned into finitely many sets Y_{j}, each of which lies in some definable affine hypersurface ∑_{i=1}^{m} α_{i}x_{i} = c, with α_{i} ∈ ℚ, not all 0. In other words x ↦ α · x is constant on Y_{j}, where α = (α_{1}, …, α_{m}). We may assume that each α_{i} ∈ ℤ and that they are relatively prime. Then (α) is the first row of a matrix M ∈ GL_{m}(ℤ). The map x ↦ Mx takes Y_{j} to a set of the form Y′_{j} × {c}, Y′_{j} ⊆ Γ^{m−1}. Since [MY_{j}] = [Y_{j}] in K_{+}(Γ[∗]), the Claim follows by induction. □

Claim 2 Any element of K_{+}(RES)⊗_{K_{+}(Γ^{fin})}K_{+}(Γ[∗]) can be represented as ∑_{i=1}^{k} X_{i}⊗val_{rv}^{−1}Y_{i}, where X_{i} ⊆ RES^{n_{i}} and Y_{i} ⊆ Γ^{m_{i}} are definable sets, and m_{i} = dim Y_{i}.

Proof. By definition of K_{+}(RES)⊗_{K_{+}(Γ^{fin})}K_{+}(Γ[∗]) and by Claim 1, any element is a sum of tensors X⊗val_{rv}^{−1}(Y × {p}); using the ⊗_{K_{+}(Γ^{fin})}-relation, X⊗val_{rv}^{−1}(Y × {p}) = (X × val_{rv}^{−1}(p))⊗Y. □

Now let X_{i}, X′_{i} ⊆ RES^{∗}, Y_{i}, Y′_{i} ⊆ Γ^{∗} be definable sets, and let

F : ∪^{.} (X_{i} × val_{rv}^{−1}(Y_{i})) → ∪^{.} (X′_{i′} × val_{rv}^{−1}(Y′_{i′}))

be a definable isomorphism. Let m be the maximal dimension m of any Y_{i} or Y′_{i′}. Assume (by Claim (2)):

(*) for each i′, Y′_{i′} ⊆ Γ^{dim(Y′_{i′})} and similarly for the Y_{i}. Claim 3

Let P be a complete type of Y_{i} of dimension m, and Q a complete type of X_{i}. Then F (Q × val_{rv}^{−1}P) = Q′ × val_{rv}^{−1}P′ where Q′ is a complete type of some X′_{i′}, and P′ a complete type type of Y′_{i′}.

Moreover there exist definable sets P̃, Q̃, P̃′, Q̃′ containing P, Q, P′, Q′ respectively, such that (1) F restricts to a bijection Q̃ × val_{rv}^{−1}P̃ → Q̃′ × val_{rv}^{−1}P̃′

(2) there exist definable bijections f : P̃ → P̃′ and g : Q̃ → Q̃′.

(3) For any x ∈ Q̃, y ∈ P̃, F restricts to a bijection {x} × val_{rv}^{−1}(y) → {f(x)} × val_{rv}^{−1}(g(y)).

Proof. By Lemma 3.17, val_{rv}^{−1}(P) is a complete type; by the same lemma, Q × val_{rv}^{−1}(P) is complete; hence so is F (Q × val_{rv}^{−1}(P)). We have F (Q × val_{rv}^{−1}(P)) ⊆ (X′_{i′} × val_{rv}^{−1}(Y′_{i′})) for some i′. Let Q′ = pr_{1}(F (Q × val_{rv}^{−1}(P))), V′ = pr_{2}(F (Q × val_{rv}^{−1}(P))), P′ = val_{rv}(V′) ⊆ Y′_{i′}. where pr_{1} : X′_{i′} × val_{rv}^{−1}(Y′_{i′}) → X_{i} ⊆ RES, pr_{2} : X′_{i′} × val_{rv}^{−1}(Y′_{i′}) → val_{rv}^{−1}(Y′_{i′}) are the projections. Then Q′, V′, P′ are complete types. We have m = dim(P′) ≥ dim(Y′_{i′}) so by maximality of m, equality holds. We thus have P′ ⊆ Γ^{dim(P′)}. By Lemma 3.17, Q′×val_{rv}^{−1}(P′) is also complete type. So F (Q × P) = Q′ × val_{rv}^{−1}P′.

By one more use of Lemma 3.17, the function f_{y} : x ↦ pr_{1}F (x, y), whose graph is a subset of the stable set Q × Q′, cannot depend on y ∈ P. Thus f_{y} = f, i.e. F (x, y) = (f(x), pr_{2}F (x, y)).

Since Q×val_{rv}^{−1}(y) is stable, val_{rv}pr_{2}F must be constant on it; so val_{rv}pr_{2}F (x, y) = g(y) on P ×Q. This shows that (3) of the “moreover” holds on P ×Q. By compactness, it holds on some definable Q̃ × P̃ (and we may take f injective on Q̃, and g on P̃.) Let Q̃′ = f(Q̃), P̃′ = g(P̃). Then (1,2) hold also. □

Claim 4 Assume (*) holds. Then there exist finitely many definable Y_{i}^{j} (j = 0, …, N_{i}) and X_{i}^{j} such that dim(Y_{i}^{0}) < m, and the conclusion of Claim 3 holds on each X_{i}^{j} × val_{rv}^{−1}Y_{i}^{j} for j ≥ 1. Moreover we may take the Y_{i}^{j}, X_{i}^{j} pairwise disjoint.

Proof. This follows from Claim 3 by compactness; the disjointness can be achieved by noting that if Claim 3 (3) holds for P̃, Q̃ then it holds for their definable subsets too. □

We now show that if ∪^{.} (X_{i} × val_{rv}^{−1}(Y_{i})) and ∪^{.} (X′_{i′} × val_{rv}^{−1}(Y′_{i′})) are definably isomorphic then ∑_{i′}[X_{i′}]⊗[Y_{i′}] = ∑_{i}[X_{i}⊗Y_{i}]. We use induction on the maximal dimension m of any Y_{i} or Y′_{i′}, and also on the number of indices i such that dim(Y_{i}) = m. Say dim(Y_{1}) = m.

By Claim 2, without changing ∑_{i′} X′_{i′}⊗val_{rv}^{−1}(Y′_{i′})) as an element of K_{+}(RES)⊗_{K_{+}(Γ^{fin})}K_{+}(Γ[∗]), we can arrange that dim(Y_{i′}) = m_{i′}, i.e. (*) holds. So Claims 3,4 apply.

The Y_{1}^{j} for j ≥ 1 may be removed from Y_{1}, if their images are correspondingly excised from the appropriate Y′_{j}, since [Q̃]⊗_{K_{+}(Γ^{fin})}[P̃] = [f(Q̃)]⊗_{K_{+}(Γ^{fin})}[g(P̃)]. What is left in Y_{1} has Γ-dimension < m, and so by induction the classes are equal.

The injectivity and the Proposition follow.

□

For applications to VF, we need a version of Proposition 10.2 keeping track of dimensions. Below, the tensor product is in the category of graded semirings. Corollary 10.3. The natural map K_{+}(RES[∗])⊗_{K_{+}(Γ^{fin})}K_{+}(Γ[∗]) → K_{+}(RV[∗]) is an isomorphism.

Proof. For each n we have a surjective homomorphism

⊕_{k=1}^{n}K_{+}(RES[k])⊗K_{+}(Γ[n − k]) → K_{+}(RV[n])

K_{+}RV[n] can be identified with a subset of the semiring K_{+}RV, namely {[X] : dim(X) ≤ n}. The proof of Proposition 10.2 shows that the kernel is generated by relations of the form

(X × val_{rv}^{−1}(Y))⊗Z = X⊗(Y ⊗Z)

when Y ∈ K_{+}(Γ^{fin}) and dim(X) + dim(val_{rv}^{−1}(Y)) + dim(val_{rv}^{−1}(Z)) = n. These relations are taken into account in the ring K_{+}(RES[∗])⊗_{K_{+}(Γ^{fin})}K_{+}(Γ[∗]), so that the natural map K_{+}(RES[∗])⊗_{K_{+}(Γ^{fin})}K_{+}(Γ[∗]) → K_{+}(RV[∗]) is injective, hence an isomorphism. □

Recall the classes e_{a} = [{a}])_{1} in K(Γ[1]), defined for a ∈ Γ(< ∅ >) They are in K_{+}(Γ^{fin}), hence identified with classes in K(RES[1]), namely e_{a} = [val_{rv}^{−1}(a)]. When denoting classes of varieties V over the residue field, we will write [V] for [V (k)], when no confusion can arise. Definition 10.4. Let I! be the ideal of K(RES[∗]) generated by all differences e_{a} − e_{0}, where a ∈ Γ(< ∅ >). Let !K(RES[∗]) = K(RES[∗])/I!.

By Lemma 9.7 (3), the natural homomorphism K(RES[∗]) into the localization of K(RV[∗]) by all classes e_{a} factors through !K(RES[∗]).

Since I! is a homogeneous ideal, !K(RES[∗]) is a graded ring.

The theorem that follows, when combined with the canonical isomorphisms K(VF[n]) → K(RV[≤ n])/I_{sp} and K(VF) → K(RV[∗])/I_{sp}, yields homomorphisms

R ∫ : K(VF) → !K(RES)[[𝔸_{1}(k)]^{−1}]

R ∫ ′ : K(VF) → !K(RES)

Theorem 10.5. (1) There exists a group homomorphism

ℰ_{n} : K(RV[≤ n])/I_{sp} → !K(RES[n])

with:

[RV^{>0}]_{1} ↦ −[𝔸^{n−1} × G_{m}]_{n}

and

[X]_{k} ↦ [X × 𝔸^{n−k}]_{n}

for X ∈ RES[k]

(2) There exists a ring homomorphism ℰ : K(RV[∗])/I_{sp} → !K(RES)[[𝔸_{1}]^{−1}] with ℰ([X]_{k}) = [X]_{k}/𝔸^{k} for X ∈ RES[k].

(3) There exists a group homomorphism

ℰ′_{n} : K(RV[≤ n])/I_{sp} → !K(RES[n])

with [RV^{>0}]_{1} ↦ 0, and [X]_{k} ↦ [X]_{n} for X ∈ RES[k].

(4) There exists a ring homomorphism ℰ′ : K(RV[∗])/I_{sp} → !K(RES) with ℰ([X]_{k}) = [X]_{k} for X ∈ RES[k].

Proof. (1) We first define a homomorphism χ[m] : K(RV[m]) → !K(RES[m]). By Corollary 10.3,

K(RV[m]) = ⊕_{l=1}^{m}K(RES[m − l])⊗_{K_{+}(Γ^{fin})}K(Γ[l])

Let χ_{0} = Id_{KRES[m]}. For l ≥ 1 recall the homomorphism χ : K(Γ[l]) → ℤ of Lemma 9.5. It induces χ_{l} : K(RES[k])⊗_{K_{+}(Γ^{fin})}K(Γ[l]) → !K(RES[k]) by a⊗b ↦ χ(b) · [G_{m}]^{l} · a.

Define a group homomorphism

χ[m] : K(RV[m]) → K(RES[m]), χ[m] = ⊕_{l}χ_{l}

We have

χ[m_{1} + m_{2}](ab) = χ[m_{1}](a)χ[m_{2}](b)

when a ∈ K(RV[m_{1}]), b ∈ K(RV[m_{2}]). This can be checked on homogeneous elements with respect to the grading ⊕_{l}K_{+}(RES[m − l])⊗K_{+}(Γ[l]).

We compute: χ[1]([RV^{>0}]_{1}) = χ_{1}(1⊗[Γ^{>0}]_{1}) = −[G_{m}] ∈ K(RES[1]).

Next define a group homomorphism β_{m} : !K(RES[m]) → !K(RES[n]) by β_{m}([X]) = [X × 𝔸^{n−m}].

Define γ : ⊕_{m≤n}K(RV[m]) → !K(RES[n]) by γ = ∑_{m} β_{m} ∘ χ[m].

Then γ is a group homomorphism, and γ(a)γ(b) = γ(ab) × [𝔸^{n}] for a ∈ K(RV[m_{1}]), b ∈ K(RV[m_{2}]), m_{1} + m_{2} ≤ n. Again this is easy to verify on homogeneous elements.

Finally we compute γ on the standard generator J = [RV^{>0}]_{1} + [1]_{0} − [1]_{1} of I_{sp}. Since χ[1]([RV^{>0}]_{1}) = −[G_{m}], we have

γ([RV^{>0}]_{1}) = β_{1}(−[G_{m}]) = −[G_{m} × 𝔸^{n−1}]_{1}

On the other hand

γ([1]_{0}) = β_{0}([1]_{0}) = [𝔸^{n}]_{n}
γ([1]_{1}) = β_{1}([1]_{1}) = [𝔸^{n−1}]_{n}

So γ(J) = [𝔸^{n−1}]_{n−1} × (−[G_{m}]_{1} + [𝔸^{1}]_{1} − [1]_{1}) = 0. A homomorphism K(RV[≤ n])/I_{sp} → K(RES[n]) is thus induced.

(2) For a ∈ K(RV[m]), let ℰ(a) = β_{m}(a)/[𝔸^{m}]. For any large enough n we have ℰ(a) = ℰ_{n}(a)/[𝔸^{n}]. The formulas in (1) prove that ℰ is a ring homomorphism.

(3,4) The proof is similar, using χ′ from Lemma 9.6 in place of χ of Lemma 9.5, and the identity in place of β_{m}. □

Corollary 10.6. The natural morphism K(RES[n]) → K(RV[≤ n])/I_{sp} has kernel contained in !I. □

Lemma 10.7. Let T = ACVF_{F ((t))} or T = ACVF^{R}_{F ((t))}, F a field of characteristic 0, with val(F) = (0), val(F ((t))) = ℤ, and val(t) = 1 ∈ ℤ. Then there exists a retraction ρ_{t} : K_{+}(RES) → K_{+}(V ar_{F}). It induces a retraction !K(RES) → K(Var_{F})

Proof. Let t_{n} ∈ F ((t))^{alg} be such that t_{1} = t and t_{nm}^{n} = t_{m}. For α = m/n ∈ ℚ, with m ∈ ℤ, n ∈ ℕ, let t_{α} = t_{n}^{m}. So α → t_{a} is a homomorphism ℚ → G_{m}(F ((t))^{alg}).

Let V (α) = val_{rv}^{−1}(α). Let t_{α} = rv(t_{α}). Then t_{a} ∈ V (α).

Let X ∈ RES[n]. Then for some α_{1}, …, α_{n} ∈ ℚ we have X ⊆ Π_{i=1}^{n}V (α_{i}), where V (α_{i}) = val_{rv}^{−1}(α_{i}). Define f(x_{1}, …, x_{n}) = (x_{1}/t_{α_{1}}, …, x_{n}/t_{α_{n}}). Then f is F ((t^{1/m}))-definable for some m, but not in general definable. Nevertheless, F (X) =: Y is definable. This is because the Galois group G = Aut(F^{a}((t^{1/m}))/F^{a}((t))) extends to a group of valued field automorphisms Aut(k((t^{1/m}))/k((t))) fixing the entire residue field k; while Y ⊆ k; thus G fixes Y pointwise and hence setwise.

The map X ↦ Y of definable sets described above clearly respects disjoint unions. It also respects definable bijections: if h : X → X′ is a definable bijection, Y = f(X), Y′ = F (Y′), then fhf^{−1} is an F ((t^{1/∞}))-definable bijection Y → Y′; by the Galois argument above, it is in fact definable.

The definable subsets of k are just the F -constructible sets. Thus we have an induced homomorphism ρ_{t} : K_{+}(RES) → K_{+}(V ar_{F}); it is clearly the identity on K_{+}(RES). It induces a homomorphism K(RES) → K(Var_{F}).

Finally ρ_{t}(val_{rv}^{−1}(α)) = [G_{m}] for any α ∈ ℚ; so a homomorphism on !K(RES) is induced. □

This example can be generalized as follows. Let L be a valued field with residue field F of characteristic 0, T = ACVF_{L} or ACVF^{R}_{L}. Let A = res(L), A = ℚ⊗A, and let t : A → G_{m}(L^{a}) be a monomorphism, with t(A) ⊆ G_{m}(L). Then there exists a retraction ρ_{t} : K_{+}(RES) → K_{+}(V ar_{F}).

From Theorem 10.5 and Lemma 10.7 we obtain the example discussed in the introduction:

Proposition 10.8. Let T = ACVF^{R}_{F ((t))}, F a field of characteristic 0, with val(F) = (0) and val(t) = 1 ∈ ℤ. Then there exists a ring homomorphism ℰ_{t} : K(VF) → K(Var_{F})[[𝔸^{1}]^{−1}], with [ℳ] ↦ −[G_{m}]/[G_{a}], 𝕃([X]_{k}) ↦ [X]_{k}/[𝔸^{k}] for X ∈ Var_{F} [k]. There is also a ring homomorphism ℰ′_{t} : K(VF) → K(Var_{F}) with 𝕃([X]_{k}) ↦ [X]_{k}.

10.2. Decomposition of μRV. An analogous decomposition is valid for the measured Grothendieck semiring μ_{Γ}RV (Definition 8.13).

Lemma 10.9. There exists a homomorphism K_{+}μΓ[n] → K_{+}μ_{Γ}RV[n] with [(X, ω)] ↦ [(val_{rv}^{−1}(X), Id, ω ∘ val_{rv})].

Proof. We have to show that an μΓ[n]-isomorphism X → Y lifts to an μ_{Γ}RV[n]-isomorphism. This follows immediately from the definitions. □

Recall μ_{Γ}RES from Definition 8.13. Along the lines of Lemma 9.12, we can also describe K_{+}μ_{Γ}RES[n] as the semigroup of functions with finite support Γ → K_{+}(RES[n]). We also have the inclusion K_{+}μ_{Γ}RES[n] → K_{+}μ_{Γ}RV[n], [(X, f)] ↦ [(X, f, 1)].

Let μΓ^{fin}[n] be full subcategory of μΓ[n] whose objects are finite. We have a homomorphism K_{+}(μΓ^{fin})[n] → μ_{Γ}RES[n], (X, ω) ↦ (val_{rv}^{−1}(X), Id, ω ∘ val_{rv}). As before we obtain a homomorphism K_{+}μ_{Γ}RES[∗]⊗_{K_{+}(μΓ^{fin})}K_{+}(μΓ[∗]) → K_{+}(RV[∗]).

Let RES_{Γ−vol} be the full subcategory of RV_{Γ−vol} whose objects are in RES; this is the same as RV except that morphisms must respect ∑ val_{rv}. Let volΓ^{fin} be the subcategory of finite objects of volΓ. Proposition 10.10. (1) The natural map K_{+}(μ_{Γ}RES[∗])⊗_{K_{+}(μΓ^{fin})}K_{+}(μΓ[∗]) → K_{+}(μ_{Γ}RV[∗]) is an isomorphism.

(2) So is K_{+}(RES_{Γ−vol}[∗])⊗_{K_{+}(volΓ^{fin}[∗])}K_{+}(volΓ[∗]) → K_{+}(RV_{Γ−vol}[∗]).

(3) The decompositions of this section preserve the subsemirings of bounded sets.

Proof. We first prove surjectivity in (1). By the surjectivity in Corollary 10.3, it suffices to consider a class c = [(X × val_{rv}^{−1}(Y), f, ω)] with X ∈ RES[k], Y ⊆ Γ^{l}, f(x, y) = (f_{0}(x), y), and ω : X × (valr^{−1}(Y)) → RV. In fact as in Proposition 10.2 we may take dim(Y) = l, and inductively we may assume that any class [(X′ × Y′, f′, ω′)] with dim(Y′) < l is in the image. Since we may remove a subset of Y of smaller dimension, applying Lemma 3.17 to ω : X × val_{rv}^{−1}(Y) → Γ, we may assume ω(x, y) = ω′(γ) when val_{rv}(y) = γ. Now c = [(X, f_{0}, 1)]⊗[(Y, ω′)].

The proof of surjectivity in (2) is similar.

The proof of injectivity in (1,2) is the same as of Proposition 10.2 and Corollary 10.3. (3) is clear by inspection of the homomorphisms. □

We now deduce Theorem 1.3. For a finite extension L of ℚ_{p}, write vol_{L}(U) for vol_{L}(U(L)). Let r be the ramification degree, i.e. val(L^{∗}) = (1/r)ℤ. Let Q = q^{r}. The normalization is such that ℳ has volume 1; so an open ball of valuative radius γ has volume q^{rγ} = Q^{γ}. Thus the volume of val_{rv}^{−1}(γ) is (q − 1)Q^{γ}. Also the norm satisfies |y| = Q^{val(y)}.

Proof of Theorem 1.3. For a ∈ Γ^{k} let Z(a) = {x ∈ 𝒪_{L}^{n} : val(f_{1}(x)) = a_{1} … val(f_{k}(x)) = a_{k}}. Then

∫_{𝒪_{L}^{n}} |f|^{s} = ∑_{a∈(Γ^{≥0})^{k}} Q^{s·a}vol_{L}(Z(a))

According to Proposition 4.5 and Proposition 10.10 we can write

Z(a) ∼ ∪^{.} _{i=1}^{ν}𝕃X_{i} × 𝕃Δ_{i}(a)

where Δ_{i} is a definable subset of Γ^{k+n_{2}(i)}, h^{i} : Δ_{i} → Γ^{k} the projection to the first k coordinates, Δ_{i}(a) = {d ∈ Γ^{n_{2}(i)} : h^{i}(d) = a}, X_{i} = (X_{i}, f_{i}) ∈ RES[n_{1}(i)], and ∼ denotes equivalence up to an admissible transformation. Thus

vol_{L}(Z(a)) = vol_{L}( ∪^{.} _{i=1}^{ν}𝕃X_{i} × 𝕃Δ_{i}(a)) = ∑_{i=1}^{ν} vol_{L}(𝕃X_{i})vol_{L}(𝕃Δ_{i}(a))

If b = (b_{1}, …, b_{k+n_{2}(i)}) ∈ Δ_{i}, let h_{0}^{i}(b) be the sum of the last n_{2}(i) coordinates.

Since val_{rv} takes only finitely many values on a definable subset of RES, we may assume ∑ val_{rv}(f(x)) = γ(i) is constant on x ∈ X_{i}. Then vol_{L}(𝕃X_{i}(L)) = Q^{γ(i)}|X_{i}(L)|. Thus

(15)  ∫_{𝒪_{L}^{n}} |f|^{s} = ∑_{i} |X_{i}(L)|Q^{γ(i)} ∑_{a∈(Γ^{≥0})^{k}} Q^{s·a}vol_{L}(𝕃Δ_{i}(a))

Now vol_{L}(𝕃Δ_{i}(a)) = ∑_{b∈Δ_{i},h(b)=a}(q − 1)^{n_{2}(i)}Q^{h_{0}(b)}. So (16)

∑_{a∈(Γ^{≥0})^{k}} Q^{s·a}vol_{L}(𝕃Δ_{i}(a)) = ∑_{b∈Δ_{i}} Q^{s_{1}h_{1}^{i}(b)+...+s_{k}h_{k}^{i}(b)}(q −1)^{n_{2}(i)}Q^{h_{0}(b)} = (q −1)^{n_{2}(i)}ev_{h^{i},s,Q}(Δ_{i})

The theorem follows from Equations (15),(16). □

Let A be the set of definable points of Γ. Recall that for X ⊆ RV, [X]_{1} denotes the class [(X, Id_{X})] ∈ RV[1] of X with the identity map to RV, and the constant form 1. For a ∈ A, let ẽ_{a} = [(val_{rv}^{−1}(0), Id, a)] ∈ RES[1], f_{a} = [{1}_{k}, Id, a] ∈ RES[1] where a in the third coordinate is the constant function with value a. If a lifts to a definable point d of RV, multiplication by d shows that ẽ_{a} = [val_{rv}^{−1}(a), Id, 0], f_{a} = [{d}, Id, 0]. Note ẽ_{a}ẽ_{b} = ẽ_{a+b}ẽ_{0}; and ẽ_{0} = [G_{m}]. Let τ_{a} ∈ RES[1] be the class of (val_{rv}^{−1}((a, ∞)), Id, 0). The generating relation of μ_{Γ}I_{sp} is thus (τ_{0}, f_{0}) ( Lemma 8.20 (6)). Let 𝔥 be the class of [(RV^{>0}, Id, x^{−1})].

Let !I_{μ}^{0} be the ideal of K(μ_{Γ}RES[∗]) generated by the relations ẽ_{a+b} = [(val_{rv}^{−1}(a), Id, b)], where a, b ∈ A, b denoting the constant function b. Let !I_{μ} be the ideal generated by !I_{μ}^{0} as well as the element [𝔸_{1}]_{1}.

Theorem 10.11. There exist two graded ring homomorphisms

e ∫ , e ∫ ′ : K^{eff}(μ_{Γ}VF[∗]) = K(μ_{Γ}RV[∗])/μI_{sp} → K(μ_{Γ}RES[∗])/!I_{μ}

such that the composition K(μ_{Γ}RES[∗]) → K(μ_{Γ}RV[∗])/μI_{sp} → K(μ_{Γ}RES[∗])/!I_{μ} equals the natural projection π : K(μ_{Γ}RES[∗]) → K(μ_{Γ}RES[∗])/!I_{μ}; with

e ∫ 𝔥 = −[{0_{k}}]_{1}, e ∫ ′ 𝔥 = 0

Proof. The identification K^{eff}(μ_{Γ}VF[∗]) = K(μ_{Γ}RV[∗])/μI_{sp} is given by Theorem 8.28, and we work with K(μ_{Γ}RV[∗])/μI_{sp}.

According to Proposition 10.10, we can identify

K(μ_{Γ}RV[∗]) = K(μ_{Γ}RES[∗])⊗_{K_{+}(μΓ^{fin})}K(μΓ[∗])

We first construct two homomorphisms of graded rings R, R′ : K(μ_{Γ}RV[∗]) → K(μ_{Γ}RES[∗])/!I_{μ}. This amounts to finding graded ring homomorphisms K(μΓ[∗]) → K(μ_{Γ}RES[∗])/!I_{μ}, agreeing with π on the graded ring K_{+}(μΓ^{fin}). It will be simpler to work with R, R′ together, i.e. construct

R′′ = (R, R′) : K(μΓ[n]) → (K(μ_{Γ}RES[n])/!I_{μ})^{2}

Recall from Lemma 9.12 the isomorphism

ϕ : K(μΓ[n]) → F n(Γ, K(Γ))[n]

Let χ′′ : K(Γ[n]) → ℤ^{2} be the Euler characteristic of Proposition 9.4; so that χ′′ = (χ, χ′), cf. Lemmas 9.5,9.6. We obtain by composition a map E′′_{n} = (E_{n}, E′_{n}) : F n(Γ, K(Γ[n])) → F n(Γ, ℤ)^{2}. Here F n(Γ, ℤ) is the group of functions g : Γ → ℤ such that g(Γ) is finite and g^{−1}(z) is a definable subset of Γ (a finite union of definable intervals and points.) Thus F n(Γ, ℤ) is freely generated as an Abelian group by {p_{a}, q_{a}, r} where r is the constant function 1, and for a ∈ A, p_{a},q_{a} are the characteristic functions of {a}, {(a, ∞)}, respectively. Define ψ_{n} : F n(Γ, ℤ) → K(μ_{Γ}RES[∗]):

ψ_{m}(p_{a}) = [G_{m}]^{n−1}ẽ_{a} = [G_{m}]^{n}f_{a}, ψ_{n}(q_{a}) = −[G_{m}]^{n}f_{a}, ψ_{n}(r) = 0

For u ∈ K(μΓ[n]), let R′′(u) = ψ_{n}(E′′_{n}(ϕ(u))).

Claim R′′ : K(μΓ[∗]) → K(μ_{Γ}RES[m])^{2} is a graded ring homomorphism.

Proof. : We have already seen that ϕ is a ring homorphism, so it remains to show this for ψ_{∗}∘E′′_{∗}. Now by Proposition 9.4, χ′′(Y) = χ′′(Y′) iff [Y] = [Y′] in the Grothendieck group of DOAG. Hence given a families Y_{t}, Y_{t′} of pairwise disjoint sets with χ′′(Y_{t}) = χ′′(Y′_{t}), by Lemma 2.3 we have χ′′(∪_{t}Y_{t}) = χ′′(∪_{t}Y′_{t}). From this and the definition of multiplication in F n(Γ, K(Γ))[∗], and the multiplicativity of E′′_{n}, it follows that if E′′_{n}(f) = E′′_{n}(f′) and E′′_{m}(g) = E′′_{m}(g′) then E′′_{n+m}(fg) = E′′_{n+m}(fg). In other words, E′′_{∗} is a graded homomorphism from into (F n(Γ, ℤ)^{2}, ⋆) for some uniquely determined multplication ⋆ on F n(Γ, ℤ)^{2}. Clearly (a, b)⋆(c, d) = (a∗_{1}c, b∗_{2}d) for two operations ∗_{1}, ∗_{2} on F n(Γ, ℤ).

Now we can compute these operations explicitly on the generators:

p_{a} ∗ p_{b} = p_{a+b}, p_{a} ∗ q_{b} = q_{a+b}, q_{a} ∗ q_{b} = −q_{a+b}

for both ∗_{1} and ∗_{2}, and

r∗_{1}ẽ_{a} = r, r∗_{1}q_{a} = −r, r∗_{1}r = r
r∗_{2}ẽ_{a} = −r, r∗_{2}q_{a} = 0, r∗_{2}r = −r

composing with ψ, we see that R′′ is indeed a graded ring homomorphism. □

Let R, R′ be the components of R′′. Claim R, R′, π agree on K_{+}(μΓ^{fin}). R(τ_{0}) = R′(τ_{0}) = −ẽ_{0}.

This is a direct computation. It folllows that R, R′ induce homomorphisms K(μ_{Γ}RV[∗]) → K(μ_{Γ}RES[∗])/. Since ẽ_{0} + f_{0} = [(𝔸_{1}, Id, 0)], modulo !I_{μ} both R, R′ equalize μ_{Γ}I_{sp}, and hence induce homomorphisms on K(μ_{Γ}RV[∗])/μ_{Γ}I_{sp} → K(μ_{Γ}RES[∗])/!I_{μ}. □

Remark

The construction is heavily, perhaps completely constrained. The value of ψ_{m}(p_{a}) is determined by the tensor relation over K_{+}(μΓ^{fin}). The value of ψ_{m}(q_{a}) is determined by the relation I_{sp}. The choice ψ(r) = 0 is not forced, but the multiplicative relation shows that either r or −r is idempotent, so one has a product of two rings, with ψ(r) = 0 and with ψ(r) = ±1. In the latter case we obtain the isomorphisms of Theorem 10.5. Thus the only choice involved is to factor the fibers of an element of F n(Γ, K(Γ))[n] through χ′′, i.e. through K(DOAG). It is possible that K(Γ[n]) = K(DOAG[n]) (cf. Question 9.9). In this case, ^{e} ∫ ,^{e} ∫′,^{R} ∫ ,^{R} ∫′ are injective as a quadruple, and determine K(μVF[∗]) completely, at least when localized by the volume of a unit ball.

11. Integration with an additive character

Let Ω = VF/ℳ. Let ψ : VF → Ω be the canonical map.

Motivation: for any p, Ω(ℚ_{p}) can be identified with the p-th power roots of unity via an additive character on ℚ_{p}. For other local fields, the universal ψ we use is tantamount to integration with respect to all additive characters of conductor ℳ at once. Thus Ω is our motivic analog of the roots of unity, and the natural map VF → VF/ℳ an analogue of a generic additive character.

Throughout this paper we have been able to avoid subtractions and work with semi-groups, but here it appears to be essential to work with a group or at least a cancellation semigroup. The reason is that we will introduce, as the essential feature of integration with an additive character, an identification of the integral of a function f with f + g if g is 𝒪-invariant. This corresponds to the rule that the sum over a subgroup of a nontrivial character vanishes. Now for any h : Ω → K_{+}(μVF), it is easy to construct h′ : Ω → K_{+}(μVF) such that h + h′ is 𝒪-invariant. Thus if f + h = f′ + h for some h, then f = f + h + h′ = f′ + h + h′ = f′. So cancellation appears to come of itself.

If we allow all definable sets and volume forms, a great deal of collapsing is caused by the cancellation rule. We thus use the classical remedy and work with bounded sets and volume forms. The setting is flexible and can be compatible with stricter notions of boundedness. This is only a partial remedy in the case of higher dimensional local fields, cf. Example 12.12.

The theory can be carried out for any of the settings we considered. Let ℛ be one of these groups or rings, with 𝒟 the corresponding data. For instance: 𝒟 the set of pairs (X, ϕ) with X a bounded definable subset of VF^{n} × RV^{∗}, and ϕ : X → RV a bounded definable function; ℛ the corresponding Grothendieck ring. Similarly we can take Γ-volumes, or pure isomorphism invariants without volume forms. In this last case there is no point restricting to bounded sets. As we saw, two Euler characteristics into the Grothendieck group of varieties over RES do survive.

In each case, we we think of ℛ as a Grothendieck ring of associated RV-data, modulo a canonical ideal.

Everything can be graded by dimension, but for the moment we have no need to keep track of it, so in the volume case we can take the direct sum over all n or fix one n and omit it from the notation.

The corresponding group for the theory T_{A} or T_{<a>} will be denoted ℛ_{A}, ℛ_{a}, etc. When V is a definable set, we let 𝒟_{V}, ℛ_{V} denote the corresponding objects over V. For instance in the case of bounded RV-volumes, 𝒟_{V} is the set of pairs (X ⊆ V × W, ϕ : X → RV^{∗}) such that for any a ∈ V, (X_{a}, ϕ|X_{a}) with X_{a} bounded.

If ℛ is our definable analog of the real numbers (as recipients of values of p-adic integration), the group ring 𝒞 = ℛ[Ω] will take the role of the complex numbers. We have a canonical group homomorphism (VF, +) → Ω ⊆ G_{m}(𝒞), corresponding to a generic additive character.

Integration with an additive character can be presented in two ways: in terms of definable functions f : X → Ω (Riemann style), where we wish to evaluate expressions such as ∫_{X} f(x)ϕ(x); classically f usually has the form ψ(h(x)), where h is a regular function and ψ is the additive character. Or we can treat definable functions F : Ω → ℛ (Lebesgue style), and evaluation ∫_{ω∈Ω} F (ω). We will work with the latter. Given this, to reconstruct a Riemann style integral, given f : X → Ω, and an ℛ-valued volume form ϕ on X, let

F (ω) = ∫_{f^{−1}(ω)} ϕ(x)

Then we can define

∫_{X} f(x)ϕ(x) = ∫_{ω∈Ω} ωF (ω)

It thus suffices to define the integral of a definable function on Ω. Such a function can be interpreted as an ℳ-invariant function on VF. We impose one rule (cancellation): the integral of a function that is constant on each 𝒪-class equals zero. The integral is a homomorphism on the group of ℳ-invariant functions VF → ℛ, vanishing on the 𝒪-invariant ones. We give a full description of the quotient group, showing that the universal homomorphism of this type factors through a similar group on the residue field.

Recall the group F n(V, ℛ) of §2.2.1. We will not need to refer to the dimension grading explicitly.

If V is a definable group, V acts on on F n(V, ℛ) by translation. Definition 11.1. For a definable subgroup W of V, let F n(V, ℛ)^{W} be the set of W -invariant elements of F n(V, ℛ): they are represented by a definable X, such that if t ∈ W and a ∈ V then X[a], X[a + t] represent the same class in K(μVF_{a,t})[n].

Lemma 11.2. An element of F n(VF, ℛ)^{ℳ} can be represented by an ℳ-invariant X ⊆ (VF×∗).

Proof. Let Y ∈ 𝒟_{VF}^{RV} represent an element of F n(VF, ℛ)^{ℳ} . So each fiber Y_{a} ∈ 𝒟^{RV}. By Lemma 3.52, for a ∈ VF/ℳ one can find Y′_{a} ∈ 𝒟^{RV} such that for some a ∈ VF with a + ℳ = a, Y_{a} = Y′_{a}. As in Lemma 2.3 there exists Y′ ∈ ℛ_{VF/ℳ} such that Y′_{a} to be the fiber of Y′ over a. Pulling back to VF gives the required ℳ-invariant representative. □

Since the equivalence is defined in terms of effective isomorphism, Definition 8.2, it is clear that two elements of 𝒟_{Ω} are equivalent iff the corresponding pullbacks to F n(VF, ℛ)^{ℳ} are equivalent.

The groups F n(VF, ℛ)^{ℳ} and F n(VF/ℳ, ℛ) can thus be identified.

Note that effective isomorphism agrees with pointwise isomorphism for F n(VF, ℛ)^{ℳ} , but not for F n(VF/ℳ, ℛ).

The group we seek to describe is 𝒜 = 𝒜_{T} = F n(VF, ℛ)^{ℳ} /F n(VF, ℛ)^{𝒪} . The quotient corresponds to the cancellation rule discussed earlier.

Let F n(k, ℛ) be the Grothendieck group of functions k → ℛ, with addition induced from ℛ. Let 𝒞 = ℛ[Ω] be the ring of definable functions Ω → ℛ with finite support, convolution

product.

Remark 𝒞 embeds into the Galois-invariant elements of the abstract group ring ℛT̃[ΩT̃], where T̃ = T_{acl(∅)}.

The additive group k = 𝒪/ℳ is a subgroup of Ω = VF/ℳ, and so acts on Ω by translation. It also acts naturally on F n(k, ℛ). This gives two actions on F n(k, 𝒞) = F n(k, ℛ)[Ω]. Let F n(k, 𝒞)_{k} denote the coinvariants with respect to the anti-diagonal action, i.e. the largest quotient on which the two actions coincide.

In general, the upper index denotes invariants, the lower index co-invariants.

F n(VF, ℛ) is the ring of definable functions from VF to ℛ. F n(k, ℛ) is the ring of definable functions from k to ℛ. F n(k, 𝒞) is the ring of definable functions from k to 𝒞; equivalently, it is the set of Galois-invariant elements of the group ring F n(k, ℛ)[Ω].

The action of k on F n(k, 𝒞) is by translation on k, and negative translation on Ω and hence on 𝒞. The term (Const) refers to the image of the constant functions of F n(k, 𝒞) in F n(k, 𝒞)_{k} (it is isomorphic to (𝒞/k).)

Theorem 11.3. There exists a canonical isomorphism F n(k, 𝒞)_{k}/(Const) ≅^{→} F n(VF, ℛ)^{ℳ} /F n(VF, ℛ)^{𝒪}

Proof. Let 𝒜_{fin} be the subring of F n(VF, ℛ)^{ℳ} consisting of functions represented by elements of F n(VF, 𝒟)^{ℳ} whose support projects to a finite subset of VF/𝒪.

A definable function on k can be viewed as an ℳ-invariant function on 𝒪; this gives

(17)  F n(k, ℛ) ≅^{→} F n(𝒪, ℛ)^{ℳ}

On the other hand we can define a homomorphism

(18)  F n(𝒪, ℛ)^{ℳ} [Ω] → 𝒜_{fin} : ∑_{ω∈W} a(ω)ω ↦ ∑_{ω∈W} a(ω)_{ω}

where W is a finite A-definable subset of Ω, a : W → F n(𝒪, ℛ)^{ℳ} is an A-definable function, (so that ∑_{a∈W} a(ω)ω is a typical element of the group ring F n(𝒪, ℛ)^{ℳ} [Ω]), and b_{ω} is the translation of b by ω, i.e. b_{ω}(x) = b(x − ω).

(18) is surjective: let f ∈ 𝒜_{fin} be represented by F, with support Z, a finite union of translates of 𝒪. By Lemma 3.39 there exists a finite definable set W, meeting each ball of Z in a unique point. Define a : W → F n(𝒪, ℛ)^{ℳ} by

a(ω) = (f|ω + 𝒪)_{−ω}

Then (18) maps ∑ a(ω)ω to f.

The kernel of (18) is the equalizer of the two actions of k. Composing with (17) we obtain an isomorphism (F n(k, ℛ)[Ω])_{k} ≅^{→} 𝒜_{fin}, or equivalently

(19)  F n(k, 𝒞)_{k} ≅^{→} 𝒜_{fin}

The last ingredient is the homomorphism

(20)  𝒜_{fin} → 𝒜

We need to show that it is surjective, and to describe the kernel.

Using the representation 𝒟 of elements of ℛ by RV-data, An element of 𝒜 is is represented by an ℳ-invariant definable W ⊂ VF × RV^{∗}.

By Lemma 3.37, for each coset C of 𝒪 in VF apart from a finite number, W ∩ (C × RV^{n+l}) is invariant under translation of the first coordinate by elements of 𝒪. Thus W is the disjoint sum of an 𝒪-invariant set W′ and a set W′′ ⊂ VF × RV^{∗} projecting to a finite union Z of cosets of 𝒪 in VF, i.e. representing a function in 𝒜_{fin}.

Clearly W′ ×_{RV}^{n} VF^{n} lies in F n(VF, ℛ)^{𝒪} .

Thus (20) is surjective; the kernel is 𝒜_{fin}^{𝒪}. Composing (19),(20) we obtain an isomorphism

𝒜 ≅^{→} (F n(k, ℛ)[Ω])_{k}/(Const)

Using the identification F n(k, ℛ)[Ω] ≃ F n(k, 𝒞), the theorem follows.

□

Note that F n(k, 𝒞)^{k} ≃ 𝒞, via F n(k, 𝒞) ≃ F n(k × Ω, ℛ)_{fin}.

11.1. Definable distributions. ℛ is graded by dimension (VF-presentation) or ambient dimension (RV-presentation.) Write ℛ = ⊕_{n≥0}ℛ[n].

Let ℛ_{df} be the dimension-free version: first form the localization ℛ[[0]_{1}^{−1}], where [0]_{1} is the class of the point 1 ∈ RV, as an element of RV[1]. Equivalently, [0]_{1}^{n} is the volume of the open n-dimensional polydisc 𝒪^{n}. Let ℛ_{df} be the zero-dimensional component of this localization. Similarly define 𝒞_{df}; so that 𝒞_{df} = ℛ_{df}[Ω]. We can also define K_{+}(𝒟)_{df}, and check that the groupification is ℛ_{df}.

Given a = (a_{1}, …, a_{n}) ∈ VF^{n} and γ = (γ_{1}, …, γ_{n}) ∈ Γ^{n}, let B(a, γ) = Π_{i=1}^{n}B(a_{i}, γ_{i}), where B(a_{i}, γ_{i}) = {c ∈ VF : val(c − a_{i}) > γ_{i}}. Call B(a, γ) an open poly-disc of dimensions γ. If γ ∈ Γ, let B(a, γ) = B(a, (γ, …, γ)) (the open cube of side γ.)

Note that [B(0, γ)] is invertible in ℛ_{df}, in each dimension. In particular in dimension 1, [B(0, γ)][B(0, −γ)] = [0]_{1}^{2}. Note also: [B(a, γ)] =_{a} [B(0, γ)].

We proceed to define integrals of definable functions.

Let U be a bounded definable subset of VF^{n}. A definable function f : U → K_{+}(𝒟)_{df} has the form [0]_{1}^{−m}F, where F : U → K_{+}𝒟[m] is a definable function, represented by some F̄ ∈ 𝒟[m + n]_{U}. In case F̄ can be taken bounded, define

∫_{U} f = [0]_{1}^{−m+n}[F]_{n+m}

We say that f is boundedly represented in this case.

In particular vol(U) = ∫_{U} 1 = [0]_{1}^{−m}[U]_{m} is treated as a pure number now, without dimension units. (Check independence of the choices.)

This extends by linearity to ∫_{U} f for f : U → ℛ_{df}, provided f can be expressed as the difference of two boundedly represented functions U → K_{+}(𝒟)_{df}.

We now note that averaging twice, with appropriate weighting, is the same as doing it once. The function γ′ in the lemmas below corresponds to a partition of U into cubes; γ′(u) is the side of the cube around u ∈ U. Lemma 11.4. Let U be a bounded open subset of VF^{n}, f a boundedly represented function on U. Let γ′ : U → Γ be a definable function such that if u ∈ U and u′ ∈ B(u, γ′(u)) then u′ ∈ U and γ′(u′) = γ′(u). Then

∫_{U} f = ∫_{U}[vol(B(u, γ′(u)))^{−1} ∫_{B(u,γ′(u))} f]

Proof. Let f = [0]_{1}^{−m}F, where F : U → K_{+}𝒟[m] is bounded. We have vol(B(u, γ′)) = [0]_{1}^{−n}[γ′(u)]^{n} so

vol(B(u, γ′))^{−1} = [0]_{1}^{n}[γ′(u)]^{−n} = [0]_{1}^{−n}[−γ′(u)]^{n}

Thus, multiplying by [0]_{1}^{3n+m}, we have to show

[0]_{1}^{2n}[F] = [−γ′(u)]^{n}][{(u, u′, z) : u ∈ U, u′ ∈ B(u, γ′(u)), (u′, z) ∈ F }]

Now u′ ∈ B(u, γ′(u)) iff u ∈ B(u′, γ′(u′)). Applying the measure preserving bijection (u, u′, z) ↦ (u − u′, u′, z′) we see that the [{(u, u′, z) : u ∈ U, u′ ∈ B(u, γ′(u)), (u′, z) ∈ F }] = [γ]_{1}^{n}[{(u′, z) : (u′, z) ∈ F] so the equality is clear. □

We now define the integral of definable functions into 𝒞_{df}. By definition, such a function is a finite sum of products fg, with f ∈ F n(U, ℛ_{df}) and g ∈ F n(U, Ω). Define

∫_{U} fg = ∫_{ω∈Ω} ω ∫_{g^{−1}(ω)} f

and extend by linearity.

Note that this is defined as soon as g is boundedly represented. (Again, check independence of choices.) Definition 11.5. A definable distribution on an open U ⊆ VF^{n} is a definable function 𝔡 : U × Γ → 𝒞_{df}, such that 𝔡(a, γ) = 𝔡(a′, γ) if B(a, γ) = B(a′, γ), and whenever γ′ > γ in each coordinate,

𝔡(b, γ) = ∫_{u∈B(b,γ)} vol(B(0, γ′))^{−1}𝔡(u, γ′)

As in Lemma 11.2, the invariance condition means that 𝔡 can be viewed as a function on open polydiscs, and we will view it this way below.

If 𝔡 takes values in ℛ_{df}, we say it is ℛ_{df}-valued. By definition, 𝔡 can be written as a finite sum ∑ ω_{i}𝔡_{i}, where 𝔡_{i} is an ℛ_{df}-valued function; in fact 𝔡_{i} is an ℛ_{df}-valued distribution.

We wish to strengthen the definition of a distribution so as to apply to sub-polydiscs of variable size. For this we need a preliminary lemma. Lemma 11.6. Let U = B(a, γ) be a polydisc. Let γ′ : B(a, γ) → Γ be a definable function such that γ′(u′) = γ′(u) for u′ ∈ B(u, γ′(u)). Then γ′ is bounded on U.

Proof. suppose for contradiction that γ′ is not bounded on B(a, γ), i.e.

(∀δ ∈ Γ)(∃u ∈ B(a, γ))(γ′(u) > δ)

This will not change if we add a generic element of Γ to the base, so we may assume Γ(dcl(∅)) ≠ (0). By Lemma 3.51, there exists a resolved structure with the same RV-part as < ∅ >; hence we may assume T is resolved. By §6.0.1 any VF-generated structure is resolved. By Lemma 3.49, for any M ⊧ T and c ∈ VF(M), acl(c) is an elementary submodel of M. Consider c with val(c) ⊧ p_{0}, where p_{0} is the generic type at ∞ of elements of Γ, i.e. p_{0}|A = {x > δ : δ ∈ Γ(A)}. Since

acl(c) ⊧ (∀δ ∈ Γ)(∃u ∈ B(a, γ))(γ′(u) > δ)

there exists e ∈ acl(c) with e ∈ B(a, γ) and γ′(e) > val(c). By Lemma 5.12, there exists e_{0} ∈ acl(∅) such that (c, e) → (0, e_{0}). In particular e_{0} ∈ B(a, γ). But then since e → e_{0} and γ′(e_{0}) ∈ Γ(acl(∅)) we have e ∈ B(e_{0}, γ′(e_{0})). So γ′(e) = γ′(e_{0}). But then γ′(e_{0}) > val(c), contradicting the choice of c. □

Lemma 11.7. (1) Let 𝔡 : U × Γ → 𝒞_{df} be a definable distribution. Let γ′ : U → Γ be a definable function with γ′(u) > γ, such that γ′(u′) = γ′(u) for u′ ∈ B(u, γ′(u)). Then

(21)  𝔡(b, γ) = ∫_{u∈B(b,γ)} vol(B(0, γ′(u))^{−1}𝔡(u, γ′(u))

(2) Let 𝔡_{1}, 𝔡_{2} be definable distributions on U such that for any x ∈ U, for all large enough γ ∈ Γ, for any y ∈ B(x, γ) and any γ′ > γ, 𝔡_{1}(B(y, γ′)) = 𝔡_{2}(B(y, γ′)). Then 𝔡_{1} = 𝔡_{2}.

Proof. (1) To prove (21), fix b, γ. We may assume U = B(b, γ). Using Lemma 11.6, pick a constant γ′′ with γ′′ > γ′(u) for all u ∈ B(b, γ). Use the definition of a distribution with respect to γ′′ to compute both 𝔡(B(b, γ)) and for each u 𝔡(u, γ′(u)), and compare the integrals using Lemma 11.4.

(3) Define γ′(u) to be the smallest γ′ such that for all γ′′ > γ′ and all y ∈ B(u, γ), 𝔡_{1}(B(y, γ′′)) = 𝔡_{2}(B(y, γ′′)). It is clear that γ′(u′) = γ′(u) for u′ ∈ B(u, γ′(u)). (21) gives the same integral formula for 𝔡_{1}(b, γ) and 𝔡_{2}(b, γ). □

Let 𝔡 be a definable distribution, and U an arbitrary bounded open set. We can define 𝔡(U) as follows. For any x ∈ U, let ρ(x, U) be the smallest ρ ∈ Γ such that B(x, ρ) ⊆ U. Let B(x, U) = B(x, ρ(x, U)); this is the largest open cube around x contained in U. Note that two such cubes B(x, U), B(x′, U) are disjoint or equal. Define

𝔡(U) = ∫_{x∈U} vol(B(x, U))^{−1}𝔡(x, ρ(x, U))

More generally, if h is a locally constant function on VF^{n} into ℛ_{df} with bounded support, we can define

(22)  𝔡(h) = ∫_{x∈VF^{n}} h(x)[B(x, h)]^{−1}𝔡(x, ρ(x, U))

where now B(x, h) = B(x, ρ(x, U)) is the largest open cube around x on which h is constant.

Proposition 11.8. Let 𝔡 be a definable distribution. Then there exists a definable open set G ⊆ VF^{n} whose complement Z has dimension < n, and a definable function g : G → 𝒞_{df} such that for any polydisc U ⊆ G

𝔡(U) = ∫_{U} g

Proof. Since 𝔡 is a finite sum of ℛ_{df}-valued distributions, we may assume it is ℛ_{df}-valued. Given a ∈ VF^{n}, we have a function α_{a} : Γ → ℛ_{df} defined by α_{a}(ρ) = 𝔡(B(a, ρ)). Using the RV-description of ℛ, and the stable embeddedness of RV ∪ Γ, we see that α_{a} has a canonical code c(a) ∈ (RV ∪ Γ)^{∗}.

Let G be the union of all polydiscs W such that c is constant on W. Let Z = VF^{n} \ G. By Lemma 5.13, dim(Z) < n. Claim Let W be a polydisc such that c is constant on W. Then for some r ∈ ℛ_{df}, for any polydisc U = B(a, ρ) ⊆ W 𝔡(a, ρ) = rvol(U).

.

Proof. Since c is constant on W, for some function δ, all ρ and all b ∈ W with B(w, ρ) ⊆ W, we have 𝔡(B(w, ρ)) = δ(ρ). By definition of a distribution we have, for any a ∈ W

δ(ρ)vol(B(a, ρ′)) =_{a} volB(a, ρ)δ(ρ′)

Now vol(B(a, ρ)) =_{a} volB(0, ρ). So δ(ρ)vol(B(0, ρ′)) =_{a} volB(0, ρ)δ(ρ′). Since this holds for any a ∈ W, by Proposition 3.51 we have

δ(ρ)vol(B(0, ρ′))) = volB(0, ρ)δ(ρ′)

So δ(ρ)/volB(0, ρ) = r is constant. The Claim follows. □

The Proposition follows too using Lemma 11.7. □

11.2. Fourier transform. Let ψ be the tautological projection K → K/ℳ = Ω.

Let g : VF^{n} → 𝒞_{df} be a definable function, bounded on bounded subsets of VF^{n}. Define a function ℱ(g) by

ℱ(g)(U) = ∫_{y∈VF} g(y)(∫_{x∈U} ψ(x · y))

This makes sense since for a given U, (∫_{x∈U} ψ(x · y)) vanishes for y outside a certain polydisc (with sides inverse to U.) Moreover, Lemma 11.9. ℱ(g) is a definable distribution.

Proof. This follows from Fubini, Lemma 11.4, and chasing the definitions. □

Corollary 11.10. Fix integers n, d. For all local fields L of sufficiently large residue characteristic, for any polynomial G ∈ L[X_{1}, …, X_{n}] of degree ≤ d, there exists a proper variety V_{G} of L^{n} such that ℱ(|G|) agrees with a locally constant function outside V_{G}.

Proof. By Lemma 11.9 and Lemma 11.8. □

See [4] for the real case.

12. Expansions and rational points over Henselian fields

We have worked everywhere with the geometry of algebraically closed valued fields, or more generally of T, but at a geometric level; all objects and morphisms can be lifted to the algebraic closure, and quantifiers are interpreted there.

For many purposes, we believe this is the right framework. It includes for instance Igusa integrals ∫_{x∈X(F)} |f(x)|^{s}, and we will show in a sequel how to interpret in it some constructions of representation theory. See also [21].

In other situations, however, one wishes to integrate definable sets over Henselian fields rather than only constructible sets; and to have a change of variable formula for definable maps, as obtained by Denef-Loeser and Cluckers-Loeser (cf. [7]). It turns out that our formalism lends itself immediately to this generalization; we explain in this section how to recover it. The point is that an arbitrary definable set is an RV-union of constructible ones, and the integration theory commutes with RV-unions.

We will consider F that admit quantifier elimination in a language 𝕃^{+} obtained from the language of T by adding relations to RV only. For example, if F = T h(ℂ((X))), F has quantifier elimination in a language expanded with names D_{n} for subgroups of Γ (with D_{n}(F) = nΓ(F).)

There are two steps in moving from F^{alg} to F. We will try to clarify the situation by taking them one at a time. The two steps are to restrict the points to a smaller set (the F -rational points), and the enlarge the language to a larger one (with enough relation symbols for F - quantifier elimination.) We will take these steps in the reverse order. In §12.1 we show how to extend the results of this paper to expansions of the language in the RV sorts, and in §12.3 how to pass to sets of rational points over a Hensel field.

The reader who wishes to restrict attention to constructible integrals (still taking rational points) may skip §12.1, taking T^{+} = T in §12.3. In this case one still has a change of variable formula for constructible change of variable, but not for definable change of variable. An advantage is that the the target ring correspondingly involves the Grothendieck group of constructible sets and maps rather than definable ones, which sometimes has more faithful information; cf. Example 12.12.

12.1. Expansions of the RV sort. Let T be V-minimal.

Let T^{+} be an expansion of T obtained by adding relations to RV. We assume that every M ⊧ T embeds into the restriction to the language of T of some N ⊧ T^{+} (as T is complete, this is actually automatic.) By adding some more basic relations, without changing the class of definable relations, we may assume T^{+} eliminates RV-quantifiers. As T eliminates field quantifiers, and T^{+} has no new atomic formulas with VF variables, T^{+} eliminates VF- quantifiers too, hence all quantifiers.

For instance T+ may include a name for a subfield of the residue field (say, pseudo-finite) or the angular coefficients the the Denef-Pas language (where RV is split). Write +-definable for T^{+}-definable; similarly tp_{+} will denote the type in T^{+}¡ etc. The unqualified words formula, type, definable closure will refer to quantifier-free formulas of T. Lemma 12.1. Let M ⊧ T^{+}. Let A be a substructure of M, c ∈ M, B = A(c) ∩ RV.

(1) tp(c/A ∪ B) ∪ T^{+}_{A∪B} implies tp_{+}(c/A ∪ B).

(2) Assume c is T^{+}_{A}-definable. Then c ∈ dcl(A, b) for some b ∈ A(c) ∩ RV

Proof. (1) This follows immediately from the quantifier elimination for T^{+}. Indeed let ϕ(x) ∈ tp_{+}(c/A ∪ B). Then ϕ is a Boolean combination of atomic formulas, and it is sufficient to consider the case of ϕ atomic, or the negation of an atomic formula. Now since any basic function VF^{n} → VF is already in the language of T, every basic function of the language of T^{+} denoting a function V F^{n} → RV factors through a T-definable function into RV. Hence the same is true for all terms (compositions of basic functions). And any basic relation is either the equality relation on VF, or else a relation between variables of RV. If ϕ is an equality or inequality between f(x), g(x), it is already in tp(c/A). Now suppose ϕ is a relation R(f_{1}(x), …, f_{n}(x)) between elements of RV. Since B(c) ∩ RV ⊆ B, the formula f_{i}(x) = b_{i} lies in tp(c/A ∪ B) for some b_{i} ∈ B. On the other hand R(b_{1}, …, b_{n}) is part of T^{+}_{B}. These formulas together imply R(f_{1}(x), …, f_{n}(x)).

(2) We must show that c ∈ dcl(A ∪ B). Let p = tp(c/A ∪ B). By (1), p generates a complete type of T^{+}_{A∪B}. Since this is the type of c and c is T^{+}_{A}-definable, and since any model of T embeds into a model of T^{+}, p has a unique solution solution in any model of T. Thus c ∈ dcl(A ∪ B). □

We will now see that any T^{+}-definable bijection decomposes into T-bijections, and bijections of the form x ↦ (x, j(g(x))) where g is a T-definable map into RV^{m} and j is a T^{+}-definable map on RV.

Corollary 12.2. (1) Let P be a T^{+}-definable set. There exist T-definable f : P̃ → RV^{∗}

and a T^{+}-definable Q ⊆ RV^{∗} such that P = f^{−1}Q.

(2) Let P_{1}, P_{2} be T^{+}-definable sets, and let F : P_{1} → P_{2} be a T^{+}-definable bijection. Then there exist g_{i} : P̃_{i} → R_{i} ⊆ RV^{m}, R ⊆ RV^{m}, h_{i} : R → R_{i}, and a bijection H : P̃_{1} ×_{g_{1},h_{1}} R → P̃_{2} ×_{g_{2},h_{2}} R over R, all T-definable, and T^{+}-definable Q_{i} ⊆ R_{i}, Q ⊆ R, and j_{i} : Q_{i} → Q such that P_{i} = g_{i}^{−1}Q_{i}, h_{i}j_{i} = Id_{Q_{i}}, and for x ∈ P_{1}

(⋄) j_{1}g_{1}(x) = j_{2}g_{2}(F (x)) =: j(x), and H(x, j(x)) = (F (x), j(x))

Moreover if P_{i} ⊆ VF^{n} × RV^{m} projects finite-to-one to VF^{n}, then R → R_{i} is finiteto-one.

Proof. (1) Let ℱ be the family of all T-definable functions f : W → RV^{m} where W is a definable set. Claim If tp(c) = tp(d) and f(c) = f(d) for all f ∈ ℱ with c, d ∈ dom(f), then c ∈ P ⟺ d ∈ P.

Proof. We have tp(c, f(c)) = tp(d, f(d)) = tp(d, f(c)), so tp(c/f(c)) = tp(d/f(c)) for all f ∈ ℱ with c ∈ dom(f), and thus tp(c/B) = tp(d/B), where B = A(c) ∩ RV. It follows that tp_{+}(c) = tp_{+}(d), and in particular c ∈ P ⟺ d ∈ P.

By compactness, there are (f_{i}, W_{i})_{i=1}^{m} ∈ ℱ such that if c ∈ W_{i} ⟺ d ∈ W_{i} and f_{i}(c) = f_{i}(d) whenever c, d ∈ W_{i}, then c ∈ P ⟺ d ∈ P. Let P̃ = ∪_{i}W_{i}, and extend f_{i} to P̃ by f_{i}(x) = ∞ if x ∉ W_{i}. Let f(x) = (f_{1}(x), …, f_{m}(x)). Letting P̃ = ∪_{i}W_{i} and Q = f(P), (1) follows.

For (2), consider first a T^{+}- type p = tp_{+}(c_{1}), c_{1} ∈ P_{1}. Let c_{2} = F (c_{1}). Using Lemma 3.48, there exists g_{i}^{p} ∈ ℱ such that e_{i} = g_{i}^{p}(c_{i}) generates dcl(c_{i}) ∩ RV. It follows as in Lemma 12.1 (1) that e_{i} generates dcl_{+}(c_{i}) ∩ RV. Let e generate dcl(c_{1}, c_{2}) ∩ RV; we have e_{i} = h_{i}^{p}(e) for appropriate T-definable h_{i}^{p}. Note dcl_{+}(c_{1}) = dcl_{+}(c_{2}), and so e ∈ dcl_{+}(c_{i}). Now quantifier elimination for T^{+} implies the stable embeddedness of RV, in the same way as for ACVF (cf. §2.1.5.) By Lemma 2.9 tp_{+}(c_{i}/e_{i}) implies tp_{+}(c_{i}/RV); in particular since e ∈ dcl_{+}(c_{i}) e = j_{i}^{p}(e_{i}) for some T^{+}-definable j_{i}^{p}. By Lemma 12.1 (2) over dcl(c_{1}), c_{2} ∈ dcl(c_{1}, e); similarly c_{1} ∈ dcl(c_{2}, e); so there exists a T-definable invertible H^{p} with H^{p}(c_{1}, e) = (c_{2}, e). The equations (⋄) have been shown to hold on p. Now g_{i} extends to a T-definable function g_{i} : P̃_{i} → R_{i}. By compactness (⋄) holds on some definable neighborhood of p; and by (1) this neighborhood can be taken to have the form g_{1}^{−1}Q_{1} for some Q_{1}. Finitely many such neighborhoods cover P_{1}, and the data can be sewed together as in (1). We thus find P̃_{1}, R, R_{1}, R_{2}, g_{1}, g_{2}, h_{1}, h_{2}, H, Q_{1}, j_{1}, j_{2} such that h_{i}j_{i}(x) = x and (⋄) holds on g_{1}^{−1}Q_{1} = P_{1}. Let Q_{2} = h_{2}j_{1}Q_{1}; it follows that P_{2} = F (P_{1}) = g_{2}^{−1}Q_{2}.

To prove the last point, since c_{2} ∈ dcl(c_{1}, e) we have ( Lemma 3.41) c_{2} ∈ acl(c_{1}). But e ∈ dcl(c_{1}, c_{2}) so e ∈ acl(dcl(c_{1})); and as e ∈ RV^{m} for some m, e ∈ acl(dcl(e_{1})). □

Let VF^{+} be the category of +- definable subsets of varieties over VF∩dcl(∅), and +-definable maps. Define effective isomorphism as in Definition 8.2; let K_{+}^{eff} denote the Grothendieck group of effective isomorphism classes, and let [X] be the class of X.

Let RV^{+}[∗] be the category of pairs (Y, f), where Y is a +-definable subset of X for some (X, f) ∈ Ob RV[∗] (Definition 3.66). A morphism (Y, f) → (Y′, f′) is a definable bijection h : Y → Y′ such that f′(h(y)) ∈ acl(f(y)) for y ∈ Y.

Let K_{+}(RV^{+}[∗]) be the Grothendieck semi-group of isomrphisms classes of RV^{+}; let I_{sp} be the congruence generated by (J, 1_{1}) where J = {1}_{0} + [RV^{>0}]_{1}. Proposition 12.3. There exists a canonical surjective homomorphism of Grothendieck semigroups

D ∫ : K_{+}(VF^{+}[∗]) → K_{+}(RV^{+}[∗])/I_{sp}

determined by:

D ∫ [X] = [W]/I_{sp} ⟺ [X] = [𝕃W]

Proof. We have to show:

(i) any element of K_{+}(VF^{+}) is effectively isomorphic to one of the form [𝕃W]; (ii) if [𝕃W_{1}] = [𝕃W_{2}] then ([W_{1}], [W_{2}]) ∈ I_{sp}.

(i) By Corollary 12.2 (1), a typical element of K_{+}(VF^{+}) is represented by P = f^{−1}Q, where Q ⊆ RV^{∗}is T^{+}-definable, f : P̃ → RV^{∗} is T-definable. For any a ∈ RV^{∗}, f^{−1}(a) is T_{a}-definable, and [f^{−1}(a)] = [𝕃C_{a}] where [C_{a}] = [∫ f^{−1}(a)]. Since 𝕃 commutes with RV-disjoint unions, it follows that [P] = [𝕃W] where W = ∪^{.}_{a∈Q}C_{a}.

(ii) Assume [𝕃W_{1}] = [𝕃W_{2}]. By Proposition 3.51, the base can be enlarged so as to be made effective, without change to RV; thus to show that ([W_{1}], [W_{2}]) ∈ I_{sp} we may assume 𝕃W_{1}, 𝕃W_{2} are isomorphic. Let f : 𝕃W_{1} → 𝕃W_{2} be an isomorphism. Let P_{i} = 𝕃W_{i} and let P̃_{i}, R_{i}, g_{i}, h_{i}, R, H, Q, Q_{i}, j_{i} be as in Corollary 12.2 (2).

Since P_{i} = g_{i}^{−1}Q_{i} = 𝕃W_{i}, the maximal ∼_{rv}-invariant subset of P̃_{i} contains P_{i}, so we may as-

sume P̃_{i} is ∼_{rv}-invariant ; in other words P̃_{i} = 𝕃W̃_{i} for some T-definable W̃_{i} ∈ RV[∗, ·] containing W_{i}.

By Lemma 7.8, there exists a special bijection σ : 𝕃W̃_{i}^{∗} → 𝕃W̃_{i} such that g_{i} ∘ σ factors through ρ, i.e. for some e_{i} : W̃_{i}^{∗} → R_{i} we have g_{i} ∘ σ = e_{i} ∘ ρ on 𝕃W̃_{i}. Let W_{i}^{∗} be the pullback of W_{i} to W̃_{i}^{∗}, so that σ(𝕃W_{i}^{∗}) = 𝕃W_{i} = P_{i}. Then ([W_{i}], [W_{i}^{∗}]) ∈ I_{sp}, so it suffices to show that (W_{1}^{∗}, W_{2}^{∗}) ∈ I_{sp}. Since P_{i} = g_{i}^{−1}Q_{i}, we have W_{i}^{∗} = e_{i}^{−1}Q_{i}.

For c ∈ R, let P̃_{i}(c) = σ^{−1}g_{i}^{−1}(h_{i}(c)), W̃_{i}(c) = e_{i}^{−1}(h_{i}(c)). Then P̃_{i}(c) = 𝕃W̃_{i}(c). Now H induces a bijection P̃_{1}(c) → P̃_{2}(c). Thus by Proposition 7.25, (W̃_{1}(c), W̃_{2}(c)) ∈ I_{sp}. In particular this is true for c ∈ Q; now h_{i} : Q → Q_{i} is a bijection, and W_{i}^{∗} = ∪^{.}_{c∈Q}W̃_{i}(c). Thus ([W_{1}^{∗}], [W_{2}^{∗}]) ∈ I_{sp}. □

Remark Since the structure on RV in T^{+} is arbitrary, we cannot expect the homomorphism of Corollary 12.3 to be injective. We could make it so tautologically by modifying the category RV^{+}, taking only liftable morphisms, i.e. those that lift to VF; we then obtain an isomorphism. In specific cases it may be possible to check that all morphisms are liftable.

12.2. Transitivity. Motivation. Consider a tower of valued fields, such as ℂ ≤ 𝒞((s)) ≤ 𝒞((s))((t)). Given a definable set over 𝒞((s))((t)), we can integrate with respect to the tvaluation, obtaining data over 𝒞((s)) and the value group. The 𝒞((s)) can then be integrated with respect to the s-valuation. On the other hand, we can consider directly the ℤ^{2}-valued valuation of 𝒞((s))((t)), and integrate so as to obtain an answer involving the Grothendieck group of varieties over ℂ. We develop below the language for comparing these answers, and show that they coincide.

For simplicity we accept here a Denef-Pas splitting, i.e. we expand RV so as to split the sequence k^{∗} → RV^{∗} → Γ. Then rv splits into two maps, ac : VF^{∗} → k^{∗} and val : VF^{∗} → Γ.

This expansion of ACVF(0, 0) is denoted ACVF^{DP}. Note that this falls under the framework of §12.1, as will the furher expansions below.

Consider two expansions of ACVF^{DP}: (1) expand the residue field to have the structure of a valued field (itself a model of ACVF^{DP}.) (2) expand the value group to be a lexicographically ordered product of two ordered Abelian groups. Then (1),(2) yield bi-interpretable theories. In more detail:

First expansion: Rename the VF sort as VF_{21}, the residue field as VF_{1}, and the value group Γ_{1}. VF_{1} carries a field structure; expand it to a model of ACVF^{DP}, with residue field F_{0} and value group Γ_{0}. Let ac_{21}, val_{21} have their natural meanings.

Second expansion: Rename the VF-sort as VF_{20}, the residue field as F_{0} and the value group as Γ_{20}. Add a predicate Γ_{0} for a proper convex subgroup of Γ_{20}, and a predicate Γ_{1} for a complementary subgroup, so that Γ_{20} is identified with the lexicographically ordered Γ_{0} × Γ_{1}. Lemma 12.4. The two theories described above are bi-interpretable. A model of (1) can canonically be made into a model of (2) with the same class of definable relations; and vice versa.

Proof. Given (1), let VF_{20} = VF_{21} as fields. Define

(23)  ac_{20} = ac_{10} ∘ ac_{21}

Let Γ_{20} = Γ_{1} × Γ_{0}, and define val_{20} : VF_{21}^{∗} → Γ_{20} by

(24)  val_{20}(x) = (val_{21}(x), val_{10}(ac_{21}(x)))

Conversely given (2), let VF_{21} = VF_{20} as fields;

𝒪_{21} = {x ∈ VF_{21} : (∃t ∈ Γ_{0})(val_{20}(x) ≥ t)}
ℳ_{21} = {x ∈ VF_{21} : (∀t ∈ Γ_{0})(val_{20}(x) > t)}
VF_{1} = 𝒪_{21}/ℳ_{21}

Let VF_{21} have the valued field structure with residue field VF_{1}; note that the value group VF_{21}^{∗}/𝒪_{21}^{∗} can be identified with Γ_{1}. Note that ker ac_{20} ⊃ 1+ℳ_{21}, so that factors through VF_{1}^{∗}, and define ac_{10}, ac_{21} so as to make (23) hold. Then define val_{21}, val_{10} so that (24) holds. □

Let VF^{+}[∗] denote the category of definable subsets of VF_{21}, equivalently VF_{20}, in the expansions (1) or (2). According to Proposition 12.3, and Lemma 2.11, we have canonical maps K_{+}(VF^{+}[∗]) → K_{+}(RV_{1}^{+}[∗])/I_{sp} and K_{+}(VF^{+}[∗]) → K_{+}(RV_{2}^{+}[∗])/I_{sp}, where RV_{i}^{+}[∗] denotes the expansion of RV according to (1),(2) respectively.

By Proposition 8.4 we have canonical maps:

(25) K_{+}(VF^{+}[∗]) → K_{+}(VF_{1}[∗])⊗K_{+}(Γ_{21}[∗])/I_{sp} → (K_{+}(F_{0})⊗K_{+}(Γ_{10}))⊗K_{+}(Γ_{21})/I_{sp}_{1}

for a certain congruence I_{sp}_{1}. And on the other hand:

(26) K_{+}(VF^{+}[∗]) → K_{+}(F_{0}[∗])⊗K_{+}(Γ_{20}[∗])/I_{sp} = K_{+}(F_{0}[∗])⊗(K_{+}(Γ_{10}[∗])⊗K_{+}(Γ_{21}[∗]))/I_{sp}_{2}

For an appropriate I_{sp}_{2}. The tensor products here are over ℤ, in each dimension separately. Using transitivity of the tensor product we identify (K_{+}(F_{0})⊗K_{+}(Γ_{10}))⊗K_{+}(Γ_{21}) with

K_{+}(F_{0}[∗])⊗(K_{+}(Γ_{10}[∗])⊗K_{+}(Γ_{21}[∗])). Then Theorem 12.5. I_{sp}_{1}, I_{sp}_{2} are equal and the maps of (25),(26) coincide.

Proof. It suffices to show in the opposite direction that the compositions of maps induced by 𝕃

(27)  (K_{+}(F_{0}[∗])⊗K_{+}(Γ_{10}[∗])⊗K_{+}(Γ_{21}[∗]) → K_{+}(VF_{1}[∗])⊗K_{+}(Γ_{21}[∗]) → K_{+}(VF^{+}[∗])

(28) K_{+}(F_{0}[∗])⊗K_{+}(Γ_{10}[∗])⊗K_{+}(Γ_{21}[∗]) → K_{+}(F_{0}[∗])⊗K_{+}(Γ_{20}[∗]) → K_{+}(VF^{+}[∗])

coincide. But this reduces by RV-additivity to the case of points, and by multplicativity to the individual factors F_{0}, Γ_{21}, Γ_{10}, yielding to an obvious computation in each case.

□

12.3. Rational points over a Henselian subfield: constructible sets and morphisms. Let T be V-minimal, and T^{+} an expansion of T in the RV sorts.

Let F be an effective substructure of a model of T. Thus F = (F_{VF}, F_{RV}), with F_{VF} a field, and rv(F_{VF}) = F_{RV}; and , F closed under definable functions of T. For example, if T = T^{+} = ACV F (0, 0), this is the case iff F_{VF} is a Henselian field and F_{RV} = F/ℳ(F); any Hensel field of residue characteristic 0 can be viewed in this way. See Example 12.8.

By a +- constructible subset of F^{n}, we mean a set of the form X(F) = X ∩ F^{n}, with X a quantifier-free formula of T^{+}. Let VF^{+}(F) be the category of such sets, and +-constructible functions between them. The Grothendieck semiring K_{+}VF^{+}(F) is thus the quotient of K_{+}VF by the semiring congruence

I_{F} = {([X], [Y]) : X, Y ∈ Ob VF^{+}, X(F) = Y (F)}

(one can verify this is an ideal; in fact if X(F) = Y (F) and X ≃ X′, then there exists Y′ ≃ Y with X′(F) = Y′(F).)

Similarly we can define I_{F}^{RV} and form K_{+}RV(F) ≃ (K_{+}RV)/I_{F}^{RV}. As usual, let I_{sp} denote the congruence generated by ([1]_{0} + [RV^{>0}]_{1}, [1]_{1}), and I_{F}^{RV} + I_{sp} their sum. Claim If ([X], [X′]) ∈ I_{F} then (^{D} ∫ [X],^{D} ∫ [X′]) ∈ I_{F}^{RV} + I_{sp}.

Proof. We may assume, changing X within the VF-isomorphism class [X], that X(F) = X′(F). Then X(F) = (X ∪ X′)(F) = X′(F), and it suffices to show: (^{D} ∫ [X],^{D} ∫ [X ∪ X′]), (^{D} ∫ [X′],^{D} ∫ [X ∪ X′]) ∈ I_{F}^{RV}. Thus we may assume X ⊆ X′. Let Z = X′ \ X. Then Z(F) = ∅, and it suffices to show: (^{D} ∫ (Z), ∅) ∈ I_{F}^{RV}. Now ^{D} ∫ (Z) = [Y] for some Y with Z definably isomorphic to 𝕃Y. So 𝕃Y (F) = ∅, hence Y (F) = ∅. Thus ([Y], ∅) ∈ I_{F}^{RV}, as required. □

As an immediate consequence we have: Proposition 12.6. Assume F ≤ M ⊧ T, with F closed under definable functions of T. The homomorphism ^{D} ∫ of Theorem 8.8 induces a homomorphism

∫_{F} : K_{+}VF^{+}(F) → K_{+}RV^{+}(F)/I_{sp}
□

12.4. Quantifier elimination for Hensel fields. Let T be a V-minimal theory in a language L_{T}, with sorts (VF, RV) (cf. §2.1.) Assume T admits quantifier-elimination, and moreover that any definable function is given by a basic function symbol. This can be achieved by an expansion-by-definition of the language.

Let T_{h} = (T)_{∀} ∪ {(∀y ∈ RV)(∃x ∈ VF)(rv(x) = y).

A model of T_{h} is thus the same as a substructure A of a model of T, such that RV(A) = rv(VF(A)).

Lemma 12.7. Any formula of L_{T} is T-equivalent to a Boolean combination of formulas in VFvariables alone, and formula ψ(t(x), u) where t is a sequence of terms for functions VF^{n} → RV, u is a sequence of RV-variables, and ψ is a formula of RV variables only.

Proof. This follows from stable embeddedness of RV, Corollary 3.24, Lemma 2.8 and the fact ( Lemma 7.10) that definable functions into Γ factor through definable functions into RV. □

Example 12.8. If T = ACVF(0, 0), then T_{h} is an expansion-by-definition of the theory of Hensel fields of residue characteristic zero.

Proof. We must show that a Henselian valued field is definably closed in its algebraic closure, in the two sorts VF, RV.

Let K ⊧ T_{Hensel}, K ≤ M ⊧ ACVF. Let X ⊆ VF^{k} × RV^{l}, Y ⊆ VF^{k′} × RV^{l′} be ACVF_{K}- definable sets, and F : X → Y an ACVF_{K}-definable bijection. We have to show that F (X ∩ K^{k} × RV(K)^{l}) = Y ∩ K^{k′} × RV(K)^{l′}.

K^{alg} is an elementary submodel of M; we may assume K^{alg} = M. By one of the characterizations of Henselianity, the valuation on K extends uniquely to K^{alg}. Hence every field automorphism of M over K is a valued field automorphism. So K is the fixed field of Aut(M/K) (in the sense of valued fields), and hence K = dcl(K). Since ACVF_{K} is effective, any definable point of RV lifts to a definable point of VF; so dcl(K) ∩ RV = RV_{K}. Thus K is definably closed in M in both sorts. □

Let L ⊃ L_{T}; assume L\L_{T} consists of relations and functions on RV only. If A ≤ M ⊧ T, let L_{T}(A) be the languages enriched with constants for each element of A; let T_{h}(A) = T_{A} ∪ T_{h}, where T_{A} is the set of quantifier-free valued field formulas true of A. Proposition 12.9. T_{h} admits elimination of field quantifiers.

Proof. Let A be as above. Let Φ_{A} be the set of L(A)- formulas with no VF-quantifiers. Claim Let ϕ(x, y) ∈ Φ_{A} with x a free VF-variable. Then (∃x)ϕ(x, y) is T_{h}(A)-equivalent to a formula in Φ_{A}.

Proof. By the usual methods of compactness and absorbing the y-variables into A, it suffices to prove this when x is the only variable. Assume first that ϕ(x) is an L_{T}(A)-formula. By Lemma 4.2, there exists an ACVF-definable bijection between the definable set defined by ϕ(x), and a definable set of the form 𝕃ϕ′(x′, u), where ϕ′ is an L_{T}(A)-formula in RV-variables only (including a distinguished variable x′ on which 𝕃 acts.) By definition of T_{h}, in any model of T_{h}, ϕ has a solution iff 𝕃ϕ′(x′, u) has a solution. But clearly 𝕃ϕ′(x′, u) has a solution iff ϕ′(x′, u) does. Thus T_{h}(A) ⊧ (∃x)ϕ(x) ⟺ (∃x′, u)ϕ′(x′, u).

Now let ϕ(x) be an arbitrary Φ_{A} formula. Let Ψ be the set of formulas of L(A) involving RV-variables only. Let Θ be the set of conjunctions of formulas of L_{T}(A) in VF-variables only, and of formulas of the form ψ(t(x)), where ψ ∈ Ψ and t is a term of L_{T}(A).The set of disjunctions of formulas in Θ is then closed under Boolean combinations, and under existential RV-quantification. By Lemma 12.7 it includes all L_{T}-formulas, up to equivalence; and also all formulas in RV-variables only. Thus ϕ(x) is a disjunction of formulas in Θ, and we may assume ϕ(x) ∈ Θ. Say ϕ = ϕ_{0}(x) ∧ ψ(t(x)), with ϕ_{0} ∈ L_{T}(A) and ψ ∈ Ψ. By Claim 1, for some formula ρ(y) of Φ_{A}, we have T_{h}(A) ⊧ ρ(y) ⟺ (∃x)(t(x) = y&ϕ_{0}(x)). Hence (∃x)ϕ(x) ⟺ (∃y)(ψ(y)&ρ(y)). □

Quantifier elimination now follows by induction. □

Remark Since only field quantifiers are mentioned, this immediately extends to expansions in the field sort.

In particular, one can split the sequence 0 → k^{∗} → RV → Γ → 0 if one wishes. This yields the quantifier-elimination [29] in the Denef-Pas language.

The results of Ax-Kochen and Ershov, and the large literature that developed around them, appeared to require methods of “quasi-convergent sequences”. It is thus curious that they can also be obtained directly from Robinson’s earlier and purely “algebraic” quantifier elimination for ACVF. Note that in the case of ACVF, there is no need to expand the language to obtain QE; and then Lemma 12.7 requires no proof beyond inspection of the language.

12.5. Rational points: definable sets and morphisms. In this subsection we will work with completions T of T_{h} ∪ {(∃x ∈ Γ)(x > 0)}. . These are theories of valued fields of residue characteristic 0, possibly expanded, not necessarily algebraically closed. The language of T is thus the language of T^{+}. The words formula, type, definable closure will refer to quantifier-free formulas of T^{+}. Definable closure, types with respect to T are referred to explicitly as dcl^{T}, tp^{T} etc.

Let F ⊧ T. Since F ⊧ T_{∀}, F embeds into a model M′ of T^{+}. Since Γ(F) ≠ (0), by Proposition 3.51 and Lemma 3.49, there exists F′ ⊆ M′ containing F, with Γ(F′) = Γ(F), and M = acl(F′) an elementary submodel of M′. Hence F embeds into a model M of T^{+} with Γ(F) cofinal in Γ(M). Lemma 12.10. Let F ⊧ T, F ≤ M ⊧ T^{+}, Γ(F) cofinal in Γ(M). Let A be a substructure of M, c ∈ F, B = A(c) ∩ RV ∩ F,

(1) tp(c/B) ∪ T_{B} implies tp^{T} (c/B).

(2) Assume c is T_{A}-definable. Then c ∈ dcl(A, b) for some b ∈ B.

Proof. (1) This follows immediately from the quantifier elimination for T and Lemma 12.1 (1).

(2) We have B ⊆ dcl^{T} (A) ∩ RV. We must show that c ∈ dcl(A ∪ B). Let p = tp(c/A ∪ B). By (1), p generates a complete type of T_{A∪B}. Since this is the type of c and c is T_{A}-definable, some formula P in the language of T_{A∪B} with P ∈ p has a unique solution in F. Now the values of F are cofinal in the value group of F^{a}; so P cannot contain any ball around c (any such ball would have an additional point of F, obtained by adding to c some element of large valuation.) Let P′ be the set of isolated elements of P; then P′ is finite (as is the case for every definable P), T_{A}-definable, and c ∈ P′. By Lemma 3.9, there exists an T_{A}-definable bijection f : P′ → Q with Q ⊆ RV^{n}. Then f(c) ∈ dcl_{T} (A) = B, and c = f^{−1}(f(c)) ∈ dcl(A ∪ B). □

Corollary 12.11. Two definably isomorphic definable subsets of F have the same class in K_{+}VF^{+}(F).

Proof. T -definable bijections are restrictions of T^{+}-definable bijections. Hence Corollary 12.2 is true with T replacing T^{+}. □

Thus Proposition 12.6 includes a change-of-variable formalism for definable bijections.

12.6. Some specializations.

12.6.1. Tim Mellor’s Euler characteristic. Consider the theory RCVF of real closed valued fields. Let RV_{RCVF}, RES_{RCVF}, VAL_{RCVF} denote the categories of definable sets and maps that lift to bijections of RCVF (on RV and on the residue field, value group respectively. We do not need to use the sorts of RES other than the residue field here, say all structures A of interest have Γ_{A} divisible.) From Proposition 12.6 and Corollary 12.11 we obtain an isomorphism: K(RCVF) → K(RV_{RCVF})/([0]_{1} − [RV^{>0}]_{1} − [0]_{0}).

The residue field is a model of the theory RCF of real closed fields; K(RCF) = ℤ via the Euler characteristic (cf. [36]). Since the ambient dimension grading is respected here, K(RES_{RCVF}) = ℤ[t].

The value group is a model of DOAG, and moreover, any definable bijection on Γ[n] for fixed n lifts to RV and indeed to RCVF. This is because the multiplicative group of positive elements is uniquely divisible, and so SL_{n}(ℚ) acts the n’th power of this group. By Proposition 9.4, K(DOAG)[n] = ℤ^{2} for each n ≥ 1, and K(VAL_{RCVF}) = ℤ[s]^{(2)} := {(f, g) ∈ ℤ[s] : f(0) = g(0)}.

Thus K(RV_{RCVF} = ℤ[t]⊗ℤ[s]^{(2)} ≤ ℤ[t, s]^{2}; and J is identified with the class (1, 1)−(0, −s)− (t, t). Thus we obtain two homomorphisms K(RV_{RCVF})/J → ℤ[s] (one mapping t ↦ 1, the other with t ↦ 1 − s; and as a pair they are injective.

Equivalently, we have found two ring homomorphisms χ, χ′ : K(RCVF) → ℤ[t]. One of these was found in [27].

12.6.2. Cluckers-Haskell. Take the theory of the p-adics. By Proposition 12.6 and Corollary 12.11 we obtain an isomorphism: K(pCF) → K(RV_{pCF})/I_{sp}. However RV_{pCF} is a finite extension of ℤ, and evidently K(ℤ) = 0, since [[0, ∞)] = [[1, ∞)]. Thus K(pCF) = 0.

12.7. Higher dimensional local fields. We have seen that the Grothendieck group of definable sets with volume forms loses a great deal of information compared to the semi-group. Over fields with discrete value groups, restricting to bounded sets is helpful; in this the Grothendieck group retains information about volumes. In case of higher dimensional local fields, with value group is A = ℤ^{n}, simple boundedness is insufficient to save from collapse. We show that using a simple-minded notion of boundedness is only partly helpful, and loses much of the volume information (all but one ℤ factor.) Example 12.12. Let K_{μ}^{bdd}(T h(ℂ((s_{1}))((s_{2})))[n]) be the Grothendieck ring of definable bounded sets and measure preserving maps in ℂ((s_{1}))((s_{2})) (with val(s_{1}) << val(s_{2})). Let Q^{t} denote the class of the thin annulus of radius t. In particular Q^{0} is the volume of the units of the valuation ring. Then in K_{μ}^{bdd}(T h(ℂ((s_{1}))((s_{2})))[2]) we have for example (Q^{0})^{2} = 0. To see this directly let

Y = {(x, y) : val(x) = 0, val(y) = 0}, X = {(x, y) : 0 < 2val(x) < val(s_{2}), val(x) + val(y) = 0}

Then X is bounded. Let f(x, y) = (x/s_{1}, s_{1}y). Then f is a measure preserving bijection X → X′ = {(x, y) : 0 < 2(val(x) + val(s_{1})) < val(s_{2}), val(x) + val(y) = 0} But in ℂ((s_{1}))((s_{2})), 2val(x) < val(s_{2}) iff 2(val(x) + val(s_{1})) < val(s_{2}) so X′(ℂ((s_{1}))((s_{2}))) = X(ℂ((s_{1}))((s_{2})) ∪ Y (ℂ((s_{1}))((s_{2}))). Remark 12.13.

(2[[0, y/2]] − [[0, y]])(2[[0, y/2)] − [[0, y)]), is a class of the Grothendieck group of Γ that vanishes identically in the ℤ-evaluation, but not in the ℤ^{2}-evaluation.

13. The Grothendieck group of algebraic varieties

Let X, Y be smooth nonsingular curves in ℙ^{3}, or in some other smooth projective variety Z, and assume Z \ X, Z \ Y are biregularly isomorphic. Say X, Y, Z are defined over ℚ. Then for almost all p, |X(𝔽_{p})| = |Y (𝔽_{p})|, as one may see by counting points of Z, Z \X and subtracting. It follows from Weil’s Riemann hypothesis for curves that X, Y have the same genus, from Faltings that X, Y are isomorphic if the genus is 2 or more, and from Tate that X, Y are isogenous if the genus is one. It was this observation that led Kontsevich and Gromov to ask if X, Y must actually be isomorphic. We show that this is the case below. ^{2} Theorem 13.1. Let X, Y be two smooth d-dimensional subvarieties of a smooth projective n-dimensional variety V, and assume V \ X, V \ Y are biregularly isomorphic. Then X, Y are stably birational, i.e. X × 𝔸^{n−d}, Y × 𝔸^{n−d} are birationally equivalent. If X, Y contain no rational curves, then X, Y are birationally equivalent.

2This already follows from [22], who use different methnods.

While we do not obtain a complete characterization in dimensions > 1, the results and method of proof do show that the answer lies in synthetic geometry and is not cohomological in nature.

Let Var_{Kz} be the category of algebraic varieties over a field K of characteristic 0.

Let [X] denote the class of a variety X in the Grothendieck semigroup K_{+}(Var_{Kz}). We allow varieties to be disconnected. As all varieties will be over the same field K, we will write Var for Var_{Kz}. Let K_{+}Var_{n} be the Grothendieck semigroup of varieties of dimension ≤ n.

For the proof, we view K as a trivially valued subfield of a model of ACV F (0, 0). We work with the theory ACVF_{K}, so that “definable” means K-definable with quantifier-free ACVFformulas.

Note that RES = k^{∗} in ACVF_{K}; the only definable point of Γ is 0, so the only definable coset of k^{∗} is k^{∗} itself.

The residue map is an isomorphism on K onto a subfield K_{RES} of the residue field k. In particular, any smooth variety V over K lifts canonically to a smooth scheme V_{𝒪} = V ⊗_{K}𝒪 over 𝒪, with generic fiber V_{VF} = V_{𝒪} ⊗_{𝒪} VF and special fiber V_{𝒪} ⊗_{𝒪} k = V ⊗_{K}k. We have a reduction homomorphism ρ_{V} : V (𝒪) → V (k). We will write V (𝒪), V (VF) for V_{𝒪} (𝒪), V_{VF}(VF).

Given k ≤ n and a definable subset X of RV^{∗} of dimension ≤ k, let [X]_{k} be the class of X in K_{+}RV[k] ⊆ K_{+}RV[≤ n]. Thus if dim(X) = d we have n − d + 1 classes [X]_{k}, d ≤ k ≤ n, in different direct factors of K_{+}RV[≤ n]. We also use [X]_{k} to denote the image of this class in K_{+}RV[≤ n]/I_{sp}. This abuse of notation is not excessive since for n ≤ N, K_{+}RV[≤ n]/I_{sp} embeds in K_{+}RV[≤ N]/I_{sp} ( Lemma 8.7).

Let SD_{d} be the image of K_{+}RV[≤ d] in K_{+}RV[≤ N]/I_{sp}. Let W D_{d}^{n} be the subsemigroup of RV[n] generated by {[X] : dim(X) ≤ d}, and use the same letter to denote the image in RV[≤ N]/I_{sp}. Let F D^{n} = SD_{n−1} + W D_{n−1}^{n}. We write a ∼ b (F D_{d}^{n}) for: (∃u, v ∈ F D_{d}^{n})(a + u = b + v). More generally, for any subsemigroup S′ of a semigroup S, write a ∼ b (S′) for (∃u, v ∈ S)(a + u = b + v).

We write K(RV[≤ n])/I_{sp} for the groupification of K_{+}(RV[≤ n])/I_{sp}

Lemma 13.2. Let V be a smooth projective k- variety of dimension n, X a definable subset of V (k). Then

D ∫ [ρ_{V} ^{−1}(X)] = [X]_{n}

Proof. Let X = (X, f) where f : X → RV^{n} is a finite-to-one map. We have to show that [𝕃X] = [ρ_{V} ^{−1}(X)] in K_{+}(VF[n]), i.e. that 𝕃X, ρ_{V} ^{−1}(X) are definably isomorphic. By Lemma 2.3 this reduces to the case that X is a point p. Find an open affine neighborhood U of V such that ρ_{V} ^{−1}(p) ⊆ U(𝒪), and U admits an étale map g : V → 𝔸^{n} over k. Now U(𝒪) ≃ 𝒪^{n} ×_{res,g} U(k). This reduces the lemma to the case of affine space, where it follows from the definition of 𝕃. □

Lemma 13.3. Let X be a a K-variety of dimension ≤ d.

(1) ^{D} ∫ (X(VF)) ∈ SD_{d} = K_{+}(RV[≤ d])/I_{sp}

(2) If X is a smooth complete variety of dimension d then ^{D} ∫ X(VF) = [X]_{d}.

(3) If X is a variety of dimension d then ^{D} ∫ X(VF) ∼ [X]_{d} (F D^{d}).

Proof. (1) is obvious, since dim(X(VF)) ≤ d.

(2) By Grothendieck’s valuative criterion for properness, X(VF) = X(𝒪). We thus have a map ρ_{V} : X(VF) = X(𝒪) → X(k). For α ∈ X(k), let X_{α}(VF) = ρ_{V} ^{−1}(α).

Since X is smooth of dimension d it is covered by Zariski open neighborhoods U admitting an étale map f_{U} : U → 𝔸^{d}, defined over K; let 𝒮 be a finite family of such pairs (U, f_{U}), with ∪_{(U,f_{U})∈𝒮}U = X. We may choose a definable finite-to-one f : X → 𝔸^{d}, defined over K, such that for any x ∈ X, for some pair (U, f_{U}) ∈ 𝒮, f(x) = f_{U}(x). We have 𝕃([X]_{d}) = 𝕃(X, f) = VF^{d} ×_{rv,f} X(k). We have to show that 𝕃(X, f) is definably isomorphic to X(VF). By Lemma 2.3 it suffices to show that for each α ∈ X(k), VF^{d} ×_{rv,f} {a} is α-definably isomorphic to X_{α}(VF). Now VF^{d} ×_{rv,f} {a} =_{α} rv^{−1}(f(α)). We have f(α) = f_{U}(α) for some

(U, f) ∈ 𝒮 with α ∈ U. Since f_{U} is étale, it induces a bijective map U_{α}(VF) → rv^{−1}(f(α)). But X_{α}(VF) = U_{α}(VF), so the required isomorphism is proved.

(3) If X, Y are birationally equivalent, then [X]_{d} ∼ [Y]_{d} (W D_{<d}^{d}), while X(VF), Y (VF) differ by VF-definable sets of dimension < d, so ^{D} ∫ (X(VF)) ∼ ^{D} ∫ (Y (VF)) (SD_{d}). Using the resolution of singularities in the form: every variety is birationally equivalent to a smooth nonsingular one, we are done by (2). With a more complicated induction we should be able to dispense with this use of Hironaka’s theorem. □

Lemma 13.4. Let V be a smooth projective K-variety, X, Y closed subvarieties, Let F : V \ X → V \ Y a biregular isomorphism. Let V_{𝒪} , V_{VF}, V_{k}, F_{VF}, etc. be the objects obtained by base change. Then F_{VF} induces a bijection V (VF) \ X(VF) → V (VF) \ Y (VF), and

F_{VF}(ρ_{V} ^{−1}(X) \ X(VF)) = ρ_{V} ^{−1}(Y) \ Y (VF)

Proof. The first statement follows from the Lefschetz principle since VF is algebraically closed.

Since V is projective, V (VF) = V (𝒪), and one can define for v ∈ V the valuative distance d(v, X), namely the greatest α ∈ Γ such that the image of x in V (𝒪/α) lies in X(𝒪/α).

Let F be the Zariski closure in V^{2} of the graph of F. Then F ∩ (V \ X) × (V \ Y) is the graph of F. In fact, in any algebraically closed field L, we have

(29)  if a ∈ V (L) \ X(L) and (a, b) ∈ F(L) then b ∈ V (L) \ Y (L)

and conversely.

Suppose for the sake of contradiction that in some M ⊧ ACV F_{K} there exist a ∈ ρ_{V} ^{−1}(X), b ∉ ρ_{V} ^{−1}(Y), (a, b) ∈ F. So d(a, X) = α > 0, d(b, Y) = 0. Let

C = {γ ∈ Γ : (∀n ∈ ℕ)nγ < α}

We may assume by compactness that C(M) ≠ ∅. Let

I = {y ∈ 𝒪(M) : val(y) ∉ C}

so that I is a prime ideal of 𝒪(M). Let L be the field of fractions of 𝒪(M)/I. Let ā, b̄ be the images of a, b in L. Then (ā, b̄) ∈ F̄, and ā ∈ X, b̄ ∉ Y; contradicting (29) □

Proof of Theorem 13.1. By Lemma 13.4, there exists a definable bijection ρ_{V} ^{−1}(X) \ X → ρ_{V} ^{−1}(Y) \ Y. Applying ^{D} ∫ : K(VF[n]) → K(RV[≤ n])/I_{sp}, and using Lemmas 13.2 and 13.3, we have [X]_{n} − [X]_{d} = [Y]_{n} − [Y]_{d}. Applying the first retraction K(RV[≤ n])/I_{sp} → K(RES[n]) of Theorem 10.5, we obtain

[X_{n}] − [X × 𝔸^{n−d}]_{n} = [Y]_{n} − [Y × 𝔸^{n−d}]_{n}

in !K(RES[n]) = K(Var_{n}). So

[X × 𝔸^{n−d} ∪^{.} Y]_{n} + [Z] = [Y × 𝔸^{n−d} ∪^{.} X]_{n} + [Z]

for some Z with dim(Z) ≤ n, where now the equality is of classes in K_{+}Var_{n}. Counting birational equivalence classes of varieties of dimension n, we see that X × 𝔸^{n−d}, Y × 𝔸^{n−d} must be birationally equivalent. The last sentence follows from the lemma below. □

Lemma 13.5. Let X, Y be varieties containing no rational curve. Let U be a variety such that there exists a surjective morphism 𝔸^{m} → U. If X × U, Y × U are birationally equivalent, then so are X, Y.

Proof. For any variety W, let ℱ(W) be the set of all rational maps g : 𝔸^{1} → W. Write dom(g) for the maximal subset of 𝔸^{1} where g is regular; so dom(g) is cofinite in 𝔸^{1}. Let R_{W} = {(g(t), g(t′)) ∈ W^{2} : g ∈ ℱ(W), t, t′ ∈ dom(g)}. Let E_{W} be the equivalence relation generated by R_{W}, on points in the algebraic closure. R_{W} , E_{W} may not be constructible in general, but in the case we are concerned with they are: Claim Let W ⊆ X × U be a Zariski dense open set. Let π : W → X be the projection. Then π(w) = π(w′) iff (w, w′) ∈ E_{W} iff (w, w′) ∈ R_{W}.

Proof. If g ∈ ℱ(U), then π ∘ g : dom(g) → X is a regular map; hence by assumption on X it is constant. It follows that if (w, w′) ∈ R_{U} then π(w) = π(w′), and hence if (w, w′) ∈ E_{U} then π(w) = π(w′). Conversely, assume w′, w′′ ∈ W and π(w′) = π(w′′); then w′ = (x, u′), w′ = (x, u′′) for some x ∈ X, u′, u′′ ∈ U. Let U_{x} = {u ∈ U : (x, u) ∈ W }. Since W is open, U_{x} is open in U. Let h : 𝔸^{m} → U be a surjective morphism; let h(v′) = u′, h(v′′) = u′′. The line through v′, v′′ intersects h^{−1}(U_{x}) in a nonempty open set. This gives a regular map f from the affine lines, minus finitely many points, into U, passing through u′, u′′. So t ↦ (x, f(t)) gives a rational map from 𝔸^{1} to W, passing through (w′, w′′); and so (w′, w′′) ∈ R_{U} and certainly in E_{U}. □

Using the Claim, we prove the lemma. Let W_{X} ⊆ X × U , W_{Y} ⊆ Y × U be Zariski dense open, and F : W_{X} → W_{Y} a biregular isomorphism. Then F takes E_{W_{X}} to E_{W_{Y}}. Moving now to the category of constructible sets and maps, quotients by constructible equivalence relations exist, and W_{X}/E_{W_{X}} is isomorphic as a constructible set to W_{Y} /E_{W_{Y}}. Let π_{X} : W_{X} → X, π_{Y} : W_{Y} → Y be the projections. By the Claim, W_{X}/E_{W_{X}} = π_{X}(W_{X}) =: X′. Similarly W_{Y} /E_{W_{Y}} = π_{Y} (W_{Y}) =: Y′. Now since W_{X}, W_{Y} are Zariski dense, so are X′, Y′. Thus X, Y contain isomorphic Zariski dense constructible sets, so they are birationally equivalent. □

Remark The condition on X, Y may be weakened to the statement that they contain no rational curve through a generic point; i.e. that there exist proper subvarieties (X_{i} : i ∈ I) defined over K, such that for any field L ⊃ K, any rational curve on X ×_{K} L is contained in some X_{i} ×_{K} L.

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INTEGRATION IN VALUED FIELDS 117

Institute of Mathematics, the Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel.

   E-mail address: ehud@math.huji.ac.il,kazhdan@math.huji.ac.il