![PRO-DEFINABILITY of SPACES of DEFINABLE TYPES 11 Where P,Q ∈ L[X] with X = (X1,...,Xm)](http://data.docslib.org/img/7be391f3e0e0f692367302ca6ae3adb2-1.webp)
arXiv:1905.11059v3 [math.LO] 27 May 2020 loe hmt banrslscnenn h oooytype hypot homotopy algebro-geometric strong the under concerning known only results were obtain which to them allowed eeial tbetpson types stable generically oe-hoei vtro h analytification the of avatar model-theoretic a ieapstv nwri aiu otxs omly eobt we Formally, definabil contexts. Theorem uniform various and in pairs answer of closely positive theory is a model give question the this as out of such turned subsets theory It distinguished It other theories. tools. first-order for model-theoretic pro-d holds classical (strict) result different a structural of of structure use Hrushovski-Loeser the of the novelty with back st the equipped the of be Part ACVF) can types. spaces (in bu of have applied– space generally be a can of to techniques seem model-theoretic not usual –where do spaces such that ak1vle field valued 1 Informa a rank provides varieties. which algebraic theory of a analytification Berkovich developed Loeser F. and Hrushovski n ftedffiute osuyBroihsae rmamodel- a from spaces Berkovich study to difficulties the of One n[4,bidn ntemdlter fagbaclyclose algebraically of theory model the on building [14], In 2010 e od n phrases. and words Key 4 h hoyo elcoe audfields valued closed real of theory the (4) 3 h hoyo nt xeso of extension finite a of theory the (3) arithmetic; Presburger theory; (2) o-minimal complete an (1) 6 h hoyo lsdodrddffrnilfields differential ordered closed of theory the (6) 5 opeino h hoyo lerial lsdvalue closed algebraically of theory the of completion a (5) ahmtc ujc Classification. Subject Mathematics osr nfr enblt fdfial ye sfial ach finally argument is pairs. types an embedded definable of using definability pro-definability Uniform implies Loeser. which property a Abstract. rtodrtere,icuigcmlt -iia theorie o-minimal complete including theories, order first aea neetn emti interpretation. geometric interesting dist other an fi of have valued pro-definability prove closed we algebraically Furthermore, and fields. closed real fields, closed R-EIAIIYO PCSO EIAL TYPES DEFINABLE OF SPACES OF PRO-DEFINABILITY u eea taeycnit nsoigta enbetype definable that showing in consists strategy general Our LtrTerm6.0.4) Theorem (Later eso r-enblt fsae fdfial ye nvario in types definable of spaces of pro-definability show We k n nagbacvariety algebraic an and r-enblt,dfial ye,sal meddpairs. embedded stably types, definable Pro-definability, AL UIE OASC N IH YE JINHE AND KOVACSICS CUBIDES PABLO X vralreagbaclycoe audfil extending field valued closed algebraically large a over . Let 1. rmr 21,Scnay0C4 12J25. 03C64, Secondary 12L12, Primary T Introduction eoeo h olwn theories: following the of one be Q p RCVF ( 1 p CF X X an d ; ); over CODF of l,gvnacmlt non-archimedean complete a given lly, ,Pebre arithmetic, Presburger s, lsadcoe ree differential ordered closed and elds nuse usae,sm fwhich of some subspaces, inguished X ee ysuyn lse fstably of classes studying by ieved eae ocascltpc nmodel in topics classical to related enbetpsadee o other for even and types definable k otntby hi association their notably, Most . hysoe o h pc of space the how showed they , ahrcnnclytestructure the canonically rather t stu epigt s fsc a such if ask to tempting thus is . i h olwn result following the ain swr iso h atta their that fact the on lies work ’s u oE rsosiadF. and Hrushovski E. to due fial e,wihgatdthem granted which set, efinable fields d ei on hesis oe-hoei con fthe of account model-theoretic t ftps nti ril we article this In types. of ity r nfrl definable, uniformly are s audfils(CF,E. (ACVF), fields valued d fqaipoetv varieties quasi-projective of utr fadfial set definable a of ructure hoei on fve is view of point theoretic scasclcomplete classical us ACVF X yrslsi [2]. in results by ; p -adically k gives , 2 PABLOCUBIDESKOVASCICSANDJINHEYE
Then, definable types over models of T are uniformly definable. In particular, for every model def M of T and every M-definable set X, the space SX (M) of definable types concentrating on X is pro-definable in Leq (or any reduct in which T admits elimination of imaginaries). The fact that pro-definability follows from uniform definability of types goes back to an argument of E. Hrushovski and F. Loeser in [14] which we present in Proposition 4.2.1. In return, uniform definability of types is obtained from the following criterion which establishes a relation between this property and stably embedded pairs of models of T . Theorem (Later Corollary 6.0.2). Let T be an L-theory such that (1) the theory of stably embedded pairs of models of T is elementary in the language of pairs LP ; def (2) for every small model M |= T and every definable type p ∈ Sx (M), there is a stably embedded (N, M) such that P is realized in N. Then T has uniform definability of types. The previous criterion is derived from a slightly more general version (see Theorem 6.0.1). As a corollary we also obtain pro-definability of some distinguished subspaces of the space of definable types, some of which have an interesting geometric interpretation. In particular, we aim to show that there are spaces of definable types in ACVF that can mimic Huber’s analytification of an algebraic variety in a similar way the space of generically stable types mimics its analytification in the sense of Berkovich. We hope this can serve as a basis towards a model theory of adic spaces. In the same spirit, working in RCVF, there are spaces of definable types which can be seen as the model-theoretic counterpart of the analytification of semi-algebraic sets as recently defined by P. Jell, C. Scheiderer and J. Yu in [16]. This is also very much related to the work of C. Ealy, D. Haskell and J. Maˇr´ıkov´ain [11] on residue field domination in real closed valued fields. This article aims to lay down a foundation for a model-theoretic study of such spaces. In a sequel, we will further explore structural properties of some of these spaces. A natural question to ask is if such spaces are also strict pro-definable (i.e., pro-definable with surjective transition maps). Obtaining strictness is much more subtle and is often related to completions of theories of stably embedded pairs. Results towards a positive answer to this question will also be addressed in subsequent work. The article is laid out as follows. Section 2 sets up the notation and contains the needed model theoretic background. Spaces of definable types are introduced in Section 3. In Section 4.1, we formally define what it means for spaces of definable types to be pro-definable and we show how this property is deduced from uniform definability of types (Proposition 4.2.1. Section 5 focuses on stably embedded pairs and its main objective is to show that the class of such pairs is elementary for all theories above listed (see Theorem 5.2.9. Section 6 contains the proofs of the two theorems above and gathers a couple of open questions.
Acknowledgements. We would like to thank: Sergei Starchenko and Kobi Peterzil for shar- ing with us the idea of using pairs as an approach to study spaces of definable types; Martin Hils for interesting discussions around the subject; Ludomir Newelski for sharing with us Example 5.2.11; and Elliot Kaplan for pointing us out a proof of Theorem 6.0.3 for CODF. P. Cubides Kovacsics was partially supported by the ERC project TOSSIBERG (Grant Agreement 637027), ERC project MOTMELSUM (Grant agreement 615722) and individ- ual research grant Archimedische und nicht-archimedische Stratifizierungen h¨oherer Ordnung, funded by the DFG. J. Ye was partially supported by NSF research grant DMS1500671. PRO-DEFINABILITYOFSPACESOFDEFINABLETYPES 3
2. Preliminaries and notation 2.1. Model theoretic background. We will use standard model-theoretic notation with the following specific conventions. Let L be a first order language (possibly multi-sorted) and T be a complete L-theory. The sorts of L are denoted by bold letters D. Given a variable x, we let Dx denote the sort where x ranges. If M is a model of T and D be a sort of L, we let D(M) denote the set of elements of M which are of sort D. Given an ordered tuple of variables x = (xi)i∈I (possibly infinite), we extend this notation and set
Dx(M)= Dxi (M) iY∈I By a subset of M we mean a subset of , where D is any sort in L. Let C be a subset of M (i.e. the union of all D(M), D a L-sort). The language L(C) is the language L together with constant symbols for every element in C. Given an L(C)-definable subset X ⊆ Dx(M), we say that a type p ∈ Sx(C) concentrates on X if p contains a formula defining X. We denote by SX (C) the subset of Sx(C) consisting of those types concentrating on X. For a C-definable function f : X → Y the pushforward of f is the function
f∗ : SX (C) → SY (C) tp(a/C) 7→ tp(f(a)/C). We let U be a monster model of T . As usual, a set is said to be small if it is of cardinality smaller than |U|. A type p(x) is a global type if p ∈ Sx(U). Following Shelah’s terminology, a subset X ⊆ Dx(M) is ∗-C-definable if there is a small collection Θ of L(C)-formulas ϕ(x) (where only finitely many xi occur in each formula) such that X = {a ∈ Dx(M) | U |= ϕ(a), ϕ ∈ Θ}. When the tuple of variables is finite, we also say X is ∞-definable (or type- definable). We let dcl and acl denote the usual definable and algebraic closure model-thoeretic opera- tors. Given a tuple of variables x of length n, a C-definable set X ⊆ Dx(U) and a subset A of U, we let X(A) := X ∩ dcl(A)n. 2.2. Definable types. We will mainly study definable types over models of T . Given a model M of T and a subset A ⊆ M, recall that a type p ∈ Sx(M) is A-definable (or definable over A) if for every L-formula ϕ(x,y) there is an L(A)-formula dp(ϕ)(y) such that for every c ∈ Dy(M)
(E1) ϕ(x, c) ∈ p(x) ⇔ M |= dp(ϕ)(c).
The map ϕ(x,y) 7→ dp(ϕ)(y) is called a scheme of definition for p, and the formula dp(ϕ)(y) is called a ϕ-definition for p. We say p ∈ Sx(M) is definable if it is M-definable. Given any set B containing M, there is a canonical extension of p to Sx(B), denoted by p|B and given by p|B := {ϕ(x, b) | N |= dp(ϕ)(b)}, where N is any model of T containing B. We refer the reader to [19, Section 1] for proofs and details of these facts. The following lemma is also folklore.
2.2.1. Lemma. If tp(a1/M) is definable and a2 ∈ acl(Ma1), then tp(a2/M) is definable.
Let ϕ(x; y) be a partitioned formula. A formula ψ(y, zϕ) is a uniform definition for ϕ (in def T ) if for every model M of T and every p ∈ Sx (M) there is c = c(p, ϕ) ∈ Dzϕ (M) such that ψ(y, c) is a ϕ-definition for p. We say T has uniform definability of types if every partitioned L-formula ϕ(x; y) has a uniform definition, which we write d(ϕ)(y, zϕ). The following is a routine coding exercise. 4 PABLOCUBIDESKOVASCICSANDJINHEYE
2.2.2. Lemma. Let ϕ(x; y) be a partitioned L-formula. Suppose there are finitely many L- def formulas ψ1(y, z1),...,ψn(y, zn) such that for every model M of T and every p ∈ Sx (M), there are i ∈ {1,...,n} and c ∈ Dzi (M) such that ψi(y, c) is a ϕ-definition for p. Then ϕ(x; y) has a uniform definition.
3. Completions by definable types In this section we define various “completions” of definable sets, using definable types. The main one consists precisely of the set of all definable types which concentrate on a given definable set.
3.1. The definable completion. Let X ⊆ Dx(U)bea C-definable set and A be a small set. def The definable completion of X over A, denoted SX (A), is the space of A-definable global types which concentrate on X. For a tuple of variables x we write Sdef (A) for Sdef (A). x Dx(U) def 3.1.1. Remark. Note that there is a small notation clash with respect to parenthesis: SX (A) is not a subset of SX (A) (the former is contained in SX (U)). However, when A is a model and X is A-definable, the restriction map SX (U) → SX (A) defines a one-to-one correspondence def between SX (A) and the set of definable types in SX (A). The inverse of such a map is given by extension of definable types over models as above defined. It is an easy exercise to check that the pushforward of a definable type is again definable. We will use the following functorial notation: 3.1.2. Lemma. Let X and Y be C-definable sets and f : X → Y be a C-definable map. Let def def def f denote the restriction to SX (A) of the pushforward map f∗. Then f defines a map def def def f : SX (A) → SY (A) for every small set A containing C. 3.1.3. Remark. def (1) Note that if f is injective, so is f∗ and hence f . However, the transfer of surjectivity from f to f def is more subtle. We will say that T has surjectivity transfer precisely if for every surjective definable function f : X → Y , the function f def is surjective. Note that if T has definable Skolem functions, then it has surjectivity transfer. Indeed, let g : Y → X def def def be a definable section of f. Then, given any type p ∈ SY , we have that f (g (p)) = p, so f def is surjective. We do not know whether CODF has surjectivity transfer. (2) By [14, Lemma 4.2.6], a stronger results holds for o-minimal theories and ACVF. Indeed, the function f def is surjective even when X and Y are definable subsets of products of imaginary sorts. Adapting the argument given in [14, Lemma 4.2.6(a)], the same result holds for RCVF. 3.2. Other completions by definable types. 3.2.1. Bounded types. Let T be an o-minimal theory and M be a model of T . Given an elementary extension M N, we say that N is bounded by M if for every b ∈ N, there are c1, c2 ∈ M such that c1 6 b 6 c2. Let A be a small subset of U and X be a definable set. def A type p ∈ SX (A) is bounded if for any small model M containing A and every realization a |= p|M, there is an elementary extension M N with a ∈ X(N) and such that N is bounded by M. Let T be either RCVF or a completion of ACVF. Let A be a small subset of U. A type def p ∈ SX (A) is bounded if for any small model M containing A and every realization a |= p|M, Γ(acl(Ma)) is bounded by Γ(M), where Γ denotes the value group sort. PRO-DEFINABILITYOFSPACESOFDEFINABLETYPES 5
3.2.2. The bounded completion. Let T be either an o-minimal theory, a completion of ACVF or RCVF. Let A be a small subset of U and X be a definable set. The bounded completion of X over A, denoted X‹(A), is the set of bounded global A-definable types. 3.2.3. Types orthogonal to Γ. Let T be a complete theory extending the theory of valued fields def (e.g., RCVF or a completion of ACVF). Let A be a small subset of U. A type p ∈ Sx (A) is said to be orthogonal to Γ if for every model M containing A and every realization a |= p|M, Γ(M) = Γ(acl(Ma)). 3.2.4. The orthogonal completion. Let T be either a completion of ACVF or RCVF. Given “ a definable set X, the orthogonal completion of X, denoted by X(A), is the set of global A-definable types concentrating on X which are orthogonal to Γ. By [14, Proposition 2.9.1], when T is a completion of ACVF, the set X“(A) also corresponds to the set of definable types which are generically stable and is also called the stable completion of X. Note that this equivalence does not hold in RCVF as in this theory every generically stable type is a realized type. 3.2.5. Remark. (1) If T is an o-minimal expansion of the theory of real closed field, then every bounded defin- able type is a realized type. However, this is not the case for general o-minimal theories. For example, the type of an element arbitrarily close to zero in DOAG is bounded. (2) Let T be one of the theories listed in 3.2.2 and M be a model of T . If tp(a/M) is bounded, then tp(b/M) is bounded for every b ∈ acl(M, a). (3) Let T be either a completion of ACVF or RCVF, and let M a model of T . If tp(a/M) is orthogonal to Γ, then tp(b/M) is orthogonal to Γ for every b ∈ acl(M, a). ‹ (4) Given a definable function f : X → Y , then the pushforward f∗ of f restricted to X (resp. “ ‹ “ to X) has image in Y (resp. Y ). Similarly as for the definable completion, one uses the ‹ “ more functorial notation f (resp. f) to denote the restriction of f∗ to X (resp. X).
3.2.6. Geometric interpretation.e For Tb a completion of ACVF, let V be a variety over a complete rank 1 valued field F . In [14], V“ is introduced as a model-theoretic analogue of ‹ the Berkovich analytification V an of V . Similarly, our aim is to view V as a model-theoretic analogue of the Huber analytification of V . When T is RCVF, V“ is a good candidate to be the model-theoretic counterpart of the analytification of semi-algebraic sets defined by Jell, Scheiderer and Yu in [16]. The set V“ also corresponds to the set of residue field dominated ‹ types as defined by Ealy, Haskell and Maˇr´ıkov´ain [11]. The space V (in RCVF) seems to def suggest there is an analogue of Huber’s analytification of semi-algebraic sets. Finally, SV can be viewed as a model-theoretic analogue of the “space of valuations on V ”. As mentioned in the introduction, we will present more structural results concerning these spaces in a sequel of this article.
4. Spaces of definable types as pro-definable sets 4.1. Pro-definability. 4.1.1. Pro-definable sets. Let (I, ≤) be a small upwards directed partially ordered set and C be a small subset of U. A C-definable projective system is a collection (Xi,fij) such that (1) for every i ∈ I, Xi is a C-definable set; (2) for every i, j ∈ I such that i > j; fij : Xi → Xj is C-definable; (3) fii is the identity on Xi and fik = fjk ◦ fij for all i > j > k. 6 PABLOCUBIDESKOVASCICSANDJINHEYE
A pro-C-definable set X is the projective limit of a C-definable projective system (Xi,fij) X := lim X . ←− i i∈I We say that X is pro-definable if it is pro-C-definable for some small set of parameters C. Pro-definable sets can also be seen as ∗-definable sets. By a result of Kamensky [17], we may identify X and X(U). 4.1.2. Pro-definable morphisms. Let X = lim X and Y = lim Y be two pro-C-definable ←−i∈I i ←−j∈J j sets with associated C-definable projective systems (Xi,fii′ ) and (Yj, gjj′ ). A pro-C-definable morphism is the data of a monotone function d: J → I and a family of C-definable functions ′ ′ {ϕij : Xi → Yj | i > d(j)} such that, for all j > j in J and all i > i in I with i > d(j) and i′ > d(j′), the following diagram commutes
fii′ Xi Xi′
ϕij ϕi′j′
gjj′ Yj Yj′ . By [17], taking U-points, the above data defines a map ϕ: X(U) → Y (U) satisfying the following commutative diagram
πi fii′ X Xi Xi′
ϕ ϕij ϕi′j′
πj gjj′ Y Yj Yj′ , where πi and πj denote the canonical projections πi : X → Xi and πj : Y → Yj. Two pro- ′ ′ definable maps (ϕij , d) and (ϕij, d ) are identified if they induced the same map from X(U) to Y (U). 4.2. Completions by definable types as pro-definable sets. As stated in the introduc- tion, we would like to equip spaces of definable types with a pro-definable structure. Let us explain the precise meaning of this in the case of the definable completion. The other completions are handled analogously. We say that definable types are pro-definable in T if for every model M of T , for every M- definable set X, for M-definable function f : X → Y and every elementary extension N, there N def are a pro-M-definable set PX (M), a bijection hX : SX (N) → PX (N) and a pro-M-definable ′ morphim fN : PX (N) → PY (N) making the following diagram commute hN def Y SY (N) PY (N) def ′ f fN
hN def X SX (N) PX (N) i
i hM def Y SY (M) PY (M) f def f ′ hM M def X SX (M) PX (M) PRO-DEFINABILITYOFSPACESOFDEFINABLETYPES 7
′ Note that the map f• is completely determined by the commutativity of the diagram. The condition we impose states that such a map has to be a pro-M-morphism. Similarly, we say def def a subfunctor C(•) of S• is pro-definable if S• is the above diagram is replaced by C(•). It is in this sense that we say that the bounded and orthogonal completions are pro-definable (in both cases we have a subfunctor by Remark 3.2.5).
The following result, essentially due to E. Hrushovski and F. Loeser [14, Lemma 2.5.1], shows the link between uniform definability of types and pro-definability. We include a proof for the reader’s convenience. 4.2.1. Proposition. Suppose T has uniform definability of types. Then definable types are pro-definable in Leq. In particular, if T has elimination of imaginaries, then definable types are pro-definable in T .
Proof. Fix some model M of T . Given a partitioned L-formula ϕ(x; y), let d(ϕ)(y, zϕ) be eq a uniform definition for ϕ. By possibly using an L -formula, we may suppose that zϕ is a single variable and eq ′ ′ ′ T |= (∀zϕ)(∀zϕ)(∀y)[(d(ϕ)(y, zϕ) ↔ d(ϕ)(y, zϕ)) → zϕ = zϕ]. Let Φ denote the set of partitioned L-formulas of the form ϕ(x; y) where y ranges over all finite tuples of variables. We associate to this data a map τ defined by def (E2) τ : Sx (M) → Dzϕ (M) p 7→ (c(p, ϕ))ϕ∈Φ, ϕY∈Φ where c(p, ϕ) is such that d(ϕ)(y, c(p, ϕ)) is a ϕ-definition for p. The codomain of τ is a pro- definable set (being a small product of definable sets). Thus, since τ is injective, to obtain def pro-definability it suffices to show that τ(Sx (M)) is ∗-definable. Without loss of generality we may suppose that the map d factors through Boolean com- binations, that is, d(ϕ ∧ ψ)= d(ϕ) ∧ d(ψ) and d(¬ϕ)= ¬d(ϕ). Consider the following set of formulas Θ containing, for each L-formula ϕ(x; y) ∈ Φ, the formula θϕ(zϕ) given by
θϕ(zϕ) := (∀y)(∃x)(ϕ(x,y) ↔ d(ϕ)(y, zϕ)).
Given ϕ1(x; y1),...,ϕm(x; ym) in Φ, let y be the tuple (y1,...,ym) and ϕ(x; y) denote the m conjunction i=1 ϕi(x,yi). Since d factors through conjunctions m V (E3) |= (∀zϕ)(∀zϕ1 ) · · · (∀zϕm )(∀y)(d(ϕ)(y, zϕ) ↔ d(ϕi)(y, zϕi )). i^=1 We claim that def τ(SX (M)) = {(cϕ)ϕ ∈ Dϕ | (cϕ)ϕ |=Θ}. ϕY∈Φ def From left-to-right, let p ∈ Sx and θϕ(zϕ) be a formula in Θ. We have that
τ(p) |= θϕ(zϕ) ⇔ (c(p, ϕ))ϕ |= θϕ(zϕ) ⇔ U |= (∀y)(∃x)(ϕ(x,y) ↔ d(ϕ)(y, c(p, ϕ))), and the last formula holds since for every y any realization of p satisfies such a formula. To show the right-to-left inclusion, let (cϕ)ϕ be such that (cϕ)ϕ |= Θ. Consider the set of formulas p(x) := {ϕ(x, b) | U |= d(ϕ)(b, cϕ)}. Let us show that p(x) is an element of Sx(U). Once we show p(x) is consistent, that p ∈ def Sx (M) follows by definition. Let ϕ1(x, b1),...,ϕm(x, bm) be formulas in p(x). Letting 8 PABLOCUBIDESKOVASCICSANDJINHEYE b := (b1, . . . , bm), by the definition of p(x) and (E3) we have that the formula ϕ(x, b) is also in p(x). Moreover, since (cϕ)ϕ |= θϕ we have in particular that
|= (∃x)(ϕ(x, b) ↔ d(ϕ)(b, cϕ)).
Finally, since ϕ(x, b) ∈ p(x), we must have that |= d(ϕ)(b, cϕ), which shows there is an element satisfying ϕ(x, b). By compactness, p(x) is consistent. def Given an M-definable subset X ⊆ Dx(M), we endow SX (M) with the pro-definable def structure inherited from Sx (M). More precisely, if X is defined by a formula ψ(x, a) for some tuple a ∈ M, we have that def τ(SX (M)) = {(cϕ)ϕ ∈ Dϕ | (cϕ)ϕ |=Θ ∪ {d(ψ)(a, zψ )}}. ϕY∈Φ We leave as an exercise to show that the present construction guarantees all the above functoriality properties. 4.2.2. Corollary. Suppose T has uniform definability of types. Let M be a model and X be a def definable subset of M. Then every ∗-definable subset of SX (M) is pro-definable. 4.2.3. Remark. If T has uniform definability of types and has surjectivity transfer for de- finable functions in Leq (see (2) of Remark 3.1.3), then definable types concentrating on a product of imaginary sorts are also uniformly definable. In particular, by Proposition 4.2.1, def if X is a definable subset of some product of imaginary sorts, then SX (M) is pro-definable (in Leq).
5. Stably embedded pairs and elementarity
Suppose L is a one-sorted language. Let LP be a language extending L by a new unary predicate P . We denote an LP -structure as a pair (N, A) where N is an L-structure and A ⊆ N corresponds to the interpretation of P . Given a complete L-theory T , the LP -theory of elementary pairs of models of T , denoted TP . Given a tuple x = (x1,...,xm), we abuse n of notation and write P (x) as an abbreviation for i=1 P (xi). When L is multi-sorted we let L denote the language which extends L by a new unary predicate PD for every L-sort D. P V Analogously, an LP -structure N is a model of TP if the collection of subsets PD(N) forms an elementary L-substructure of N. We will also denote any such a structure as a pair (N, M) where M ≺ N |= T and for every L-sort D, PD(N)= D(M).
5.1. Stable embedded pairs. Let M ≺ N be an elementary extension of models of T . The extension is called stably embedded if for every L(N)-definable subset X ⊆ Dx(N), the set X ∩ Dx(M) is L(M)-definable. The class of stably embedded models of T , denoted SE(T ), is the class of LP -structures (N, M) such that M ≺ N |= T and the extension M ≺ N is stably embedded. The following lemma is a standard exercise. 5.1.1. Lemma. An elementary extension M ≺ N is stably embedded if and only if for every tuple a in N, the type tp(a/M) is definable. The main objective of this section is to show that for various NIP theories, the class of stably embedded pairs is an LP -elementary class. Stable theories constitute a trivial example of this phenomenon, since the class of stably embedded pairs coincides with the class of elementary pairs. O-minimal theories constitute a less trivial example. Let us first recall Marker-Steinhorn characterizarion of definable types in o-minimal structures. PRO-DEFINABILITYOFSPACESOFDEFINABLETYPES 9
5.1.2. Theorem ([18, Theorem 2.1]). Let T be an o-minimal theory and M be a model of T . Then, p ∈ Sx(M) is definable if and only if for every realization a of p, M is Dedekind complete in M(a) (where M(a) is the prime model of M over a).
Since being Dedekind complete is expressible in LP , we readily obtain: 5.1.3. Corollary. Let T be a complete o-minimal theory. Then the class of stably embedded pairs is LP -elementary. The following result of Q. Brouette shows the analogue result for CODF.
5.1.4. Theorem ([3, Proposition 3.6]). Let M be a model of CODF. Then, p ∈ Sx(M) is definable if and only if for every realization a of p, M is Dedekind complete in the real closure of the ordered differential field generated by M ∪ {a}.
5.1.5. Corollary. Then class of stably embedded pairs of models of CODF is LP -elementary. A similar situation holds in Presburger arithmetic. The following characterization of defin- able types over models in T is due to G. Conant and S. Vojdani (see [5]).
5.1.6. Theorem. Let M be a model of Presburger arithmetic. A type p ∈ Sn(M) is definable if and only if for every realization a |= p, M ≺ M(a) is an end-extension. 5.1.7. Corollary. An elementary pair (N, M) of models of Presburger arithmetic is stably embedded if and only if it is an end-extension. In particular, the class SE(T ) is an elementary class in LP . 5.2. Stable embedded pairs of valued fields. In what follows we gather the corresponding characterization of stably embedded pairs of models for ACVF, RCVF and pCFd. We need first the following notation and terminology for induced structures. Let M be an L-structure eq and D be an imaginary sort in M . We let LD be the language having a predicate PR for every L-definable (without parameters) subset R ⊆ Dn(M). The structure (D(M), LD) in which every PR is interpreted as the set R is called the induced structure on D(M). The sort D is called stably embedded if every Leq-definable subset of D(M) is LD-definable. The following lemma is left to the reader. 5.2.1. Lemma. Let T be an complete L-theory and (N, M) be a stably embedded pair of models of T . Let D be an Leq-sort which is stably embedded (as a sort) in every model of T . Then the pair D(M) ≺ D(N) is stably embedded in LD.
For a valued field (K, v) we let ΓK denote the value group, OK its valuation ring, kK the residue field and res: OK → kK the residue map. Given a valued field extension (K ⊆ L, v) and a subset A := {a1, . . . , an} ⊆ L, we say that A is K-valuation independent if for every n K-linear combination i=1 ciai with ci ∈ K, n P v ciai = min(v(ciai)). i i ! X=1 The extension L|K is called vs-defectless1 if every finitely generated K-vector subspace V of L admits a K-valuation basis, that is, a K-valuation independent set which spans V over K. 5.2.2. Remark. By standard arguments, both the value group and the residue field are stably embedded sorts in ACVF, RCVF and pCFd. As a corollary of Lemma 5.2.1 we obtain that if (K ≺ L, v) is a stably embedded pair of models of any of these theories, then, the pairs ΓK ≺ ΓL and kK ≺ kL are stably embedded in their respective induced structure languages.
1This is the same as separated as in [1] and [9] 10 PABLO CUBIDES KOVASCICS AND JINHE YE
Let us now explain how to show that the class SE(T ) is LP -elementary for T either ACVF, RCVF or pCFd. In all three cases, the result will follow from the following theorem:
5.2.3. Theorem. Let T be either ACVF, RCVF or pCFd. Let K ≺ L be a pair of models of T . Then the following are equivalent (1) the pair K ≺ L is stably embedded (2) the valued field extension L|K is vs-defectless, the pairs ΓK ≺ ΓL and kK ≺ kL are stably embedded. For ACVF the above Theorem is precisely the content of [6, Theorem 1.9]. For RCVF and pCFd the result is new and the corresponding proof is presented in Sections 5.2.5 and 5.2.6. Note that both in ACVF and pCFd the condition on the residue field extension can be removed. Indeed, in ACVF, the induced structure on the residue field is that of a pure algebraically closed field, and therefore (since such a structure is stable) the corresponding extension kK ≺ kL is always stably embedded. In pCFd, the extension kK ≺ kL is finite and hence trivially stably embedded.
5.2.4. Corollary. Let T be either T either ACVF, RCVF or pCFd. Then the class SE(T ) is elementary in the language of pairs. Proof. By Theorem 5.2.3, it suffices to show that the condition stated in part (2) is elementary in the language of pairs. Note that being a vs-defectless extension is an elementary property in the language of pairs, so we only need to show that having stably embedded value group and residue field extensions is an elementary property in the language of pairs. Concerning the value group, both in ACVF and RCVF, the value groups extension is an elementary extension of models of an o-minimal theory (namely, divisible ordered abelian groups). Therefore, being stably embedded is elementary by Corollary 5.1.3 and Remark 5.2.2. For pCFd, the value group extension is an extension of models of Presburger arithmetic, and hence the result follows by Corollary 5.1.7. Regarding the residue field extension, as explained above, it only plays a role for RCVF. In this case, the residue field extension corresponds to an elementary pair of real closed fields. As before, the result follows by Corollary 5.2.9 and Remark 5.2.2. It is worthy to acknowledge that recently, building on the present results, P. Touchard extend some of these results to other classes of henselian fields [22].
5.2.5. Stably embedded pairs of real closed valued fields. The study real closed valued fields in 6 the language Ldiv which corresponds to the language of ordered rings extended by a binary 6 predicate div(x,y) which holds if and only if v(x) 6 v(y). The Ldiv-theory of real closed valued fields RCVF has quantifier elimination (see [4]). Proof of Theorem 5.2.3 for RCVF: Let (K ≺ L, v) be a pair of real closed valued fields. (1) ⇒ (2): By Remark 5.2.2, the pairs ΓK ≺ ΓL and kK ≺ kL are stably embedded. That the extension is vs-defectless follows word for word as for algebraically closed valued fields in [6, Theorem 1.9]. m 6 m (2) ⇒ (1): Let X ⊆ L be an Ldiv-definable set over L. We need to show that X ∩ K is 6 Ldiv-definable over K. By quantifier elimination, we may suppose that X is defined by one of the following formulas (i) v(P (x))v(Q(x)) with either 6 or <, (ii) 0 < P (x), PRO-DEFINABILITY OF SPACES OF DEFINABLE TYPES 11 where P,Q ∈ L[X] with X = (X1,...,Xm). When X is defined by a formula as in (i), one can proceed as in [6, Theorem 1.9], so it remains to show the result for (ii). Since the extension L|K is vs-defectless, there is a K-valuation independent set A := {a1, . . . , an} ⊆ L and polynomials Pi ∈ K[X] with i ∈ I := {1,...,n} such that P (x) = i∈I aiPi(x). By [7, Lemma 2.24], we can further suppose that for every i, j ∈ I, if v(ai) and v(a ) lie in the same coset modulo Γ , then v(a )= v(a ). Moreover, at the expense of P j K i j multiplying Pi by −1, we can suppose that ai > 0 for all i ∈ I. For each ∅= 6 J ⊆ I, let AJ be the set m AJ := {x ∈ K | v( aiPi(x)) = v(aj Pj(x)) if and only if j ∈ J}. Xi∈I By case (i), we may suppose AJ is definable over K. Further, since A is K-valuation inde- m pendent, the sets AJ cover K when J varies over all non-empty subsets of I. Therefore, it suffices to show that X ∩ AJ is definable over K for every J ⊆ I. Let us first show how to reduce to the case where J = I. If J 6= I, then for all x ∈ AJ we have
0 < P (x) ⇔ 0 < aiPi(x)+ aiPi(x) ⇔ 0 < aiPi(x), Xi∈J i∈XI\J Xi∈J and thus we obtain an equivalent formula where for all i ∈ J, v(aiPi(x)) is the same. Therefore without loss of generality it suffices to show the case J = I. Now, since for all x ∈ AI , v(aiPi(x)) = v(aj Pj(x)) for all i, j ∈ I, v(ai) and v(aj ) are in the same coset modulo ΓK, and hence v(ai) = v(aj ) for all i, j ∈ I. Also v(Pi(x)) = v(Pj (x)) for all i, j ∈ I. Multiplying by a suitable constant c ∈ L, we may suppose that v(ai) = 0 for all i ∈ I. Similarly, multiplying ′ by a suitable constant c ∈ K, we may suppose that v(Pi(x)) = 0 for all x ∈ AI . We conclude by noting that in this situation, for all x ∈ AI n n 0 < P (x) ⇔ 0 < res( aiPi(x)) ⇔ 0 < res(ai)res(Pi(x)). i i X=1 X=1 m n Since kK is stably embedded in kL, the set {y ∈ kK | 0 < i=1 res(ai)yi} is definable over k . Lifting the parameters, we obtain that X ∩ A is definable over K. K I P
5.2.6. Stably embedded pairs of models of pCFd. Let K be a finite extension of Qp. Let d be the p-rank of the extension (see [12, Section 2]). The theory pCFd is the theory of K in the language LMac with finitely many new constant symbols as introduced in [12, Theorem 5.6]. Such a theory admits elimination of quantifiers. We need some preliminary lemmas. 5.2.7. Lemma. Let (K ⊆ L, v) be a valued field extension. Suppose that every y ∈ L is of the form y = x + a with a ∈ K and x ∈ L such that |v(x)| > ΓK. Then the extension is vs-defectless. Proof. Let V ⊆ L be a K-vector space of dimension n. Let us show that V contains elements {x1,...,xn} such that each v(xi) lies in a different ΓK-coset. By [12, Lemma 3.2.2], this implies that {x1,...,xn} is a K-valuation basis for V . We proceed by induction on n. Let {y1,...,yn} be a basis for V . For n = 1 the result is trivial (take x1 = y1). Suppose the result holds for all K-vector spaces of dimension smaller than n. Then, by induction, the K-vector space generated by {y1,...,yn−1} contains elements {x1,...,xn−1} such that each v(xi) lies in a different ΓK-coset. Without loss of generality we may assume
v(xn) > v(xn−1)+ΓK > · · · > v(x1)+ΓK. 12 PABLO CUBIDES KOVASCICS AND JINHE YE
If either v(yn) > v(xn−1)+ΓK , v(x1) > v(yn)+ΓK , or
v(xm+1) > v(yn)+ΓK > v(xm)+ΓK, for some 1 6 m
We are ready to show Theorem 5.2.3 for pCFd:
Proof of Theorem 5.2.3 for pCFd: (1) ⇒ (2): That ΓK ≺ ΓL is stably embedded follows again by Remark 5.2.2. It remains to show that the extension is vs-defectless. By Lemma 5.2.7, it suffices to show that every element y ∈ L is of the form x + a for a ∈ K and x ∈ L such that |v(x)| > ΓK. Every element in y ∈ K is of such form taking x = 0, so we may suppose y ∈ L \ K. If |v(y)| > ΓK take a = 0. Otherwise, since by Corollary 5.1.7 ΓL is an end extension of ΓK, we must have v(y) ∈ v(K). Suppose there is no a ∈ K such that |v(y − a)| > ΓK. Thus, for every a ∈ K, v(y − a) ∈ ΓK. But this implies that y is a limit over K, which contradicts that K is stably embedded in L. This shows the extension is vs-defectless.
(2) ⇒ (1): Since the pair is vs-defectless, there are no limit points in L over K. Moreover, since ΓK ≺ ΓL is stably embedded, by Corollary 5.1.7, it is an end extension of ΓK. The same argument as in the previous implication shows that every element y ∈ L is of the form x + a for a ∈ K and x ∈ L such that |v(x)| > ΓK. In particular, the convex hull of ΓK in ΓL w is ΓK. Let us show that kL is isomorphic to K. For all y ∈ L \ K such that w(y) = 0, there w is a unique a ∈ K such that v(y − a) > ΓK. Therefore resw(y) = a, which shows that kL is in bijection with K. The result now follows from 5.2.8. We summarize the above results in the following theorem. 5.2.9. Theorem. Let T be one of the following theories PRO-DEFINABILITY OF SPACES OF DEFINABLE TYPES 13
(i) a complete o-minimal theory; (ii) CODF, Presburger arithmetic, the theory of a finite extension of Qp (pCFd), a com- pletion of ACVF, RCVF.
Then, the class SE(T ) is an elementary class in LP . 5.2.10. Question. Is there a natural characterization of the class of complete NIP theories T for which SE(T ) is LP -elementary?
Note that there are NIP L-theories T for which the class SE(T ) is not LP -elementary. The following example is due to L. Newelski.
5.2.11. Example. Let (ai)i<ω be a sequence of irrational numbers ai such that the limit limi→∞ ai = c is again irrational an bigger than every ai. Consider the structure M = (Q,<
, (Pai )i<ω) where Pai is a unary predicate interpreted as the cut {x ∈ Q | x < ai}. The theory T = Th(M) is NIP as any NIP theory extended by externally definable sets is NIP (see [21, Section 3.3]). Now let N = M ∪ {bi | i<ω} where bi realizes the definable type over M determined by the set of formulas
{c 6. Uniform definability via classes of pairs The following theorem provides an abstract criterion for a theory T to have uniform defin- ability of types. 6.0.1. Theorem. Suppose there is an LP -elementary class C such that (i) if (N, M) ∈C, then M |= T ; (ii) if (N, M) ∈C, then N is an L-substructure of an L-elementary extension N ′ of M; (iii) if (N, M) ∈C, and a is a finite tuple in N, then tp(a/M) is definable (where tp(a/M) is defined with respect to N ′); def (iv) for every small model M |= T and every definable type p ∈ Sx (M), there is a pair (N, M) ∈C and a ∈ Dx(N) such that p = tp(a/M). Then T has uniform definability of types. In particular, definable types are pro-definable in T in any reduct of Leq in which T has elimination of imaginaries. Proof. Fix a partitioned L-formula ϕ(x; y). Let (ψi(y, zi))i∈I be an enumeration of all L- formulas having y among their free variables. Suppose for a contradiction that no formula ψi(y, zi) provides a uniform definition for ϕ. This implies, by Lemma 2.2.2, that for every def finite subset J ⊆ I there are a model MJ of T and a type qJ (x) ∈ Sx (MJ ) such that no formula ψi is a ϕ-definition for qJ . Consider for every i ∈ I the LP -formula θi(x) (∀zi ∈ P )(∃y ∈ P )(¬(ϕ(x,y) ↔ ψi(y, zi))). Let Σ(x) := {θi(x) | i ∈ I}∪ TC, where TC is an LP -axiomatization of C. Let us show that Σ(x) is consistent. Let Σ0 be a finite subset of Σ and let J := {i ∈ I | θi(x) ∈ Σ0}. By def assumption, there is some q = qJ (x) ∈ Sx (MJ ) such that no ψi is a ϕ-definition for q. By condition (iv), let (N, MJ ) be an element of C and a ∈ Dx(N) be such that q = tp(a/M). By 14 PABLO CUBIDES KOVASCICS AND JINHE YE the choice of a, we have that θi(a) holds for all i ∈ J. This shows that Σ0 is consistent. Thus, by compactness, Σ is consistent. Let (N ′, M ′) be an element in C and a ∈ N ′ be a realization of Σ. By the definition of Σ, the type tp(a/M ′) is not definable, which contradicts condition (iii). 6.0.2. Corollary. Let T be such that (i) SE(T ) is LP -elementary; def (ii) for every small model M |= T and every definable type p ∈ Sx (M), there is a stably embedded (N, M) such that P is realized in N. Then T has uniform definability of types. In particular, definable types are pro-definable in T in any reduct of Leq in which T has elimination of imaginaries. Proof. This follows directly by Theorem 6.0.1 by taking C = SE(T ). Indeed, note that condition (i) − (iii) of 6.0.1 are trivially satisfied, and condition (iv) corresponds to the present assumption (ii). 6.0.3. Theorem. The following theories have uniform definability of types (in one of their natural languages): (1) Any complete stable theory; (2) Any complete o-minimal theory; (3) Presburger arithmetic; (4) The theory of a finite extension of Qp; (5) RCVF; (6) any completion of ACVF; (7) CODF. Proof. (1) This is a well-known result (see B. Poizat’s paper [20]). Let us give two short different proofs. By [15, Proposition 3.2], generically stable types are uniformly definable in any NIP theory. This implies the result since in a stable theory all types are generically stable. Alternatively, use Corollary 6.0.2: condition (i) is trivially satisfied and for condition (ii) take N = U. Let T be one of the theories form (2)-(6). We apply Corollary 6.0.2. Condition (i) is granted def by Theorem 5.2.9. For condition (ii), let M be a model of T , p ∈ Sx (M) and a ∈ Dx(U) be a realization of p. If T is o-minimal, by Marker-Stenhorn’s theorem (Theorem 5.1.2), we can take N := M(a) since (M(a), M) is already stably embedded. For T a theory from (3)-(6) we take N := acl(Ma). Note that in these cases N is indeed a model of T and the extension (N, M) is stably embedded by Lemma 2.2.1. This completes the result for (2) − (6). For CODF, the structure N := acl(Ma) is a real closed field extension of M but it is not necessarily a model of CODF. To ensure condition (ii) we can apply the following idea of E. Kaplan. Let N ′ be a sufficiently large real closed field containing a and such that the extension (N ′, M) is a stably embedded extension of real closed fields. By [13, Proposition 4.11], there is a derivation δ on N ′ extending the derivation on M such that (N ′, δ) is a model of CODF. By [3, Proposition 3.6], the extension of models of CODF (N ′, M) is stably embedded (in the language of CODF). Alternatively, if L denotes the language of ordered rings, Lδ its extension by the derivation, and Lδ,P the extension of Lδ by a new unary predicate P , one can apply Theorem 6.0.1 to the following Lδ,P -elementary class of pairs (N, M): (1) M |= CODF; (2) (N, M) is an L-stably embedded pair of real closed fields; PRO-DEFINABILITY OF SPACES OF DEFINABLE TYPES 15 (3) N is an Lδ-substructure of an Lδ-elementary extension of M; We leave the details of this second approach to the reader. 6.0.4. Theorem. Let T be an L-theory listed in Theorem 6.0.2 and L′ be any reduct of Leq in which T has elimination of imaginaries. Then, definable types in T are pro-definable in L′. Proof. This follows directly by Theorem 6.0.2 and Proposition 4.2.1. We expect similar results hold for theories of (tame) valued fields with generic derivations as defined in [8] and of o-minimal fields with a generic derivation as defined in [13]. 6.0.5. Corollary. Let T be either an o-minimal theory, a completion of ACVF or RCVF. The bounded completion of a definable set is pro-definable in Leq. In the two latter cases, the orthogonal completion of a definable set is pro-definable in Leq Proof. Both cases are similar, we will just work with the bounded case and leave the others to ‹ def the reader. By Corollary 4.2.2 it suffices to show that X(M) is ∗-definable inside SX (M), for a given M-definable set X. Suppose X ⊆ Dx(M). For every formula ϕ(x,y,z) where y is a Γ- variable, for every a ∈ Dz(M) such that ϕ(x, y, a) defines a function fa : X → Γ(M), and for ‹ every p(x) ∈ X(M), there is γ ∈ Γ(M) such that p(x) contains the formula −γ References [1] W. Baur, Die Theorie der Paare reell abgeschlossener K¨orper, in Logic and algorithmic (Zurich, 1980), vol. 30 of Monograph. Enseign. Math., Univ. Gen`eve, Geneva, 1982, pp. 25–34. [2] V. G. Berkovich, Smooth p-adic analytic spaces are locally contractible, Invent. Math., 137 (1999), pp. 1–84. [3] Q. Brouette, Definable types in the theory of closed ordered differential fields, Arch. Math. Logic, 56 (2017), pp. 119–129. [4] G. Cherlin and M. A. Dickmann, Real closed rings ii. model theory, Annals of Pure and Applied Logic, 25 (1983), pp. 213 – 231. [5] G. Conant and S. Vojdani, Definable types and f-generics in presburger arithmetic. unpublished note available at https://www3.nd.edu/∼gconant/Math/presburger note.pdf. [6] P. Cubides Kovacsics and F. Delon, Definable types in algebraically closed valued fields, MLQ Math. Log. Q., 62 (2016), pp. 35–45. [7] P. Cubides Kovacsics, F.-V. Kuhlmann, and A. Rzepka, On valuation independence and defectless extensions of valued fields, Journal of Algebra, 555 (2020), pp. 69 – 95. [8] P. Cubides Kovacsics and F. Point, Topological fields with a generic derivation. arXiv:1912.07912 [math.LO], 2020. [9] F. Delon, Extensions s´epar´ees et imm´ediates de corps valu´es, J. Symbolic Logic, 53 (1988), pp. 421–428. 16 PABLO CUBIDES KOVASCICS AND JINHE YE [10] , D´efinissabilit´eavec param`etres ext´erieurs dans Qp et R, Proc. Amer. Math. Soc., 106 (1989), pp. 193–198. [11] C. Ealy, D. Haskell, and J. Makov, Residue field domination in real closed valued fields, Notre Dame J. Formal Logic, 60 (2019), pp. 333–351. [12] A. J. Engler and A. Prestel, Valued fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. [13] A. Fornasiero and E. Kaplan, Generic derivations on o-minimal structures. arXiv:1905.07298 [math.LO], 2019. [14] E. Hrushovski and F. Loeser, Non-archimedean tame topology and stably dominated types, vol. 192 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2016. [15] E. Hrushovski and A. Pillay, On nip and invariant measures, Journal of the European Mathematical Society, 013 (2011), pp. 1005–1061. [16] P. Jell, C. Scheiderer, and J. Yu, Real tropicalization and analytification of semialgebraic sets. arXiv:1810.05132 [math.AG], 2017. [17] M. Kamensky, Ind- and pro- definable sets, Ann. Pure Appl. Logic, 147 (2007), pp. 180–186. [18] D. Marker and C. I. Steinhorn, Definable types in o-minimal theories, J. Symbolic Logic, 59 (1994), pp. 185–198. [19] A. Pillay, An introduction to stability theory, vol. 8 of Oxford Logic Guides, The Clarendon Press, Oxford University Press, New York, 1983. [20] B. Poizat, Paires de structures stables, J. Symbolic Logic, 48 (1983), pp. 239–249. [21] P. Simon, A guide to NIP theories, vol. 44 of Lecture Notes in Logic, Association for Symbolic Logic, Chicago, IL; Cambridge Scientific Publishers, Cambridge, 2015. [22] P. Touchard, Stably embedded submodels of henselian valued fields, arXiv preprint arXiv:2005.02363, (2020). Pablo Cubides Kovacsics, Mathematisches Institut der Heinrich-Heine-Universitat¨ Dussel-¨ dorf, Universitatsstr.¨ 1, 40225 Dusseldorf,¨ Germany. Jinhe Ye, University of Notre Dame, Department of Mathematics, 255 Hurley, Notre Dame, IN 46556, USA. E-mail address: [email protected]