The model theory of residue rings of models of Peano Arithmetic: The prime power case
aa r X i v : . [ m a t h . L O ] J a n The model theory of residue rings of models ofPeano Arithmetic: The prime power case
P. D’Aquino ∗ and A. Macintyre † February 2, 2021
Abstract
In [25] the second author gave a systematic analysis of definability anddecidability for rings M /p M , where M is a model of Peano Arithmetic and p is a prime in M . In the present paper we extend those results to the moredifficult case of M /p k M , where M is a model of Peano Arithmetic, p is aprime in M , and k >
1. In [25] work of Ax on finite fields was used, here weuse in addition work of Ax on ultraproduct of p -adics. Boris Zilber asked the second author the following question:If M is a nonstandard model of full Arithmetic, and n is a nonstandard elementof M congruent to 1 modulo all standard integers m , does the ring M /n M interpretArithmetic?The problem arose in work of Zilber on commutator relations in quantum me-chanics, see [29].Although until recently little attention had been given to rings M /n M , onewould naturally expect that M /n M is much simpler than M , for example being ∗ Dipartimento di Matematica e Fisica, Universit`a della Campania “L. Vanvitelli”, viale Lincoln,5, 81100 Caserta, Italy, e-mail: [email protected] † Partially supported by a Leverhulme Emeritus Fellowship.School of Mathematics, University of Edinburgh, King’s Buidings, Peter Guthrie Tait Road Edin-burgh EH9 3FD, UK, e-mail: [email protected]: models of Peano Arithmetic, ultraproducts, local rings, valuations and generalizations nternally finite in M , and thus M should not be interpretable in M /n M , soZilber’s Problem should have a negative answer.This is in fact how things turn out. From the outset of our work we lookedat much more general problems connected to definability in the rings M /n M , toemphasize the vast gulf between the residue rings M /n M and M . We chose to workwith models M of Peano Arithmetic, where one inevitably encounters G¨odelianphenomena, and contrast M with the ring M /n M where one is in a stronglynon-G¨odelian setting, mainly because of Ax’s work from [5], already exploited byMacintyre in [25] to give a detailed analysis of the case when n is prime.After a lecture of Macintyre in 2015 Tom Scanlon pointed out that mere non-interpretability can easily be obtained via the interpretation of M /n M as a non-standard initial segment [0 , n − M on M /n M . This is certainly the first completely clear solution to Zilber’sProblem. But it seems to have no implications for getting close to exact bounds forcomplexity of definitions in the M /n M . Our work on the M /n M has three clearly defined stages, of increasing difficulty. Stage 1. n prime. Then if n is a standard prime p , M /n M ∼ = F p , while if n is a nonstandard prime then M /n M is a pseudofinite field (in Ax’s sense [5]) ofcharacteristic 0. See [25] for this case, and the cited properties. Pseudofinite fieldsof characteristic 0 are models of the theory of all finite prime fields, and this theory P rimeF in has a nice set of first-order axioms which we now sketch briefly.A field K is a model of P rimeF in if1) K is perfect2) K has exactly one extension of each finite degree n d ) if Γ is an absolutely irreducible plane curve of degree d and | K | > ( d − then Γ has a point in K d ) if | K | ≤ ( d − then | K | = p for some prime p ≤ ( d − for each 0 < d ∈ N .The theory is decidable, and all models are pseudofinite, in the sense weakerthan Ax’s, namely that each model is elementarily equivalent to an ultraproduct,possibly principal, of finite fields. Finally, each model of the theory is elementarilyequivalent to some M /n M , where M ≡ Z .Noninterpretability of M in any model of the theory can be seen via Hrushovski’sresult that pseudofinite fields are simple see [22].2 tage 2. n = p k , p prime, and k >
1. This is covered in the present paper, and isindispensable for Stage 3.
Stage 3. n arbitrary. This will be considered in the last paper of the series. Weuse the factorization theory in M of n as an internal finite product of prime powers,and an internal representation of M /n M as Q p | n M /p v p ( n ) M (see [11], [7]). Thisgets combined with an internalized Feferman-Vaught Theorem (see [21]), and thework of Stage 2, to get analogues of the main results of Stage 1 and 2, and therebygive a thorough analysis in all cases of the definability theory and axiomatizabilityof the M /n M , with spin-offs on pseudofiniteness.It is notable in Stage 2 that the M /p k M are Henselian local rings, and modelsof a natural set of axioms involving TOAGS (see [12]), truncated ordered abeliangroups.We make heavy use of many classical results on Henselian valuation rings, and truncate them to get results for our M /p k M . This is not routine. Though we knowa huge amount about the logic of Henselian fields we know rather little about thelogic of general Henselian local rings (even familiar finite local rings [16]). M /p k M M standard Already for M standard there are nontrivial issues of decidability (and definability).Let n = p k , where p is a prime and k ≥
1. It is well-known that Z /p k Z is an henselianlocal ring [26]. The ideals in the ring of p -adic integers Z p are generated by powersof p . Moreover, Z /p k Z is isomorphic in a natural way to Z p /p k Z p . For any fixedprime p there is an existential formula in the language of rings defining Z p in Q p . Auniform definition of Z p in Q p without reference to the prime needs an existential-universal formula, see [10]. Ax in [5] proved that the theory of the class of valuedfields Q p as p varies among the primes is decidable. So, the theory of the class of all Z /p k Z as p varies over all primes and k over all positive integers, is also decidable. Remark 1.
Whether or not M is standard, if n is divisible by more than oneprime, M /n M is not local. If n is divisible by finitely many primes then M /n M is semilocal. In general for M nonstandard, M /n M is nonstandard semilocal if n is divisible by infinitely many primes (and there are always such n when M isnonstandard). 3 emark 2. We note that for any M , and prime p , standard or nonstandard, M isnot Henselian with respect to the p -adic valuation v p on M . First, suppose p = 2.If v p is Henselian then 1 + p is a square, so 1 + p = y for some y ∈ M . Hence,( y − y + 1) = p , and so eithera) y − y + 1 = p b) y − − y + 1 = − p c) y − p and y + 1 = 1d) y − − p and y + 1 = − p = 3. b) and c) are impossible. d) implies p = 3.So suppose p = 3. Then, if Hensel’s lemma holds, 1 + 2 · P A implies 7 is not a square.Finally, assume p = 2. If Hensel’s lemma holds then 1 + 8 · h is a square for all h , so 17 is a square, impossible in P A . M nonstandard Each of the rings Z /p k Z can be interpreted in M in a uniform way for each prime p and each positive integer k , as can each of the natural maps Z /p k +1 Z → Z /p k Z .But it not possible to interpret Z p in M as the inverse limit of the Z /p k Z ’s with theassociated maps. Los’ theorem implies that if D is a nonprincipal ultrafilter on the set of primesthen the ultraproduct Q D Z p is an Henselian valuation ring, whose maximal ideal is Q D µ p where µ p is the maximal ideal of Z p . Also the value group of the ultraproductof the local domains is the ultraproduct of the value groups of the Z p ’s, and soan ultrapower of Z . Ax’s results in [5] are needed to prove that the residue field Q D Z p / Q D µ p = Q D Z p /µ p is a pseudofinite field.We will first analyze the basic algebraic properties of each of the quotients M /p k M , and we will show that also for M nonstandard M /p k M is a local Henselianring. We will then appeal to classical results of model theory of Henselian fields (see[20], [6]) to understand the theory of the class of all M /p k M for p prime and k positive in M . We will make use also of some constructions and ideas in [15].The valuation v p induces a “valuation” (which we will denote by v ) on the quo-tient ring M /p k M . The residue field of M (cid:14) p k M is either F p if p is standard, or acharacteristic 0 pseudofinite field if p is nonstandard.4 heorem 2.1. For general prime p and k > , M (cid:14) p k M is a local Henselian ring,and the unique maximal ideal is principal.Proof: The units in M (cid:14) p k M are m + ( p k ) where m is prime to p . Clearly, thenon units form an ideal, and this is the unique maximal ideal of M (cid:14) p k M and isgenerated by p + ( p k ).Let f ( x ) be a monic polynomial over M , and α ∈ M such that v p ( f ( α )) > v p ( f ′ ( α )) = 0. We want to show that there exists β ∈ M such that f ( β ) = 0and v p ( α − β ) >
0. We use inside M the standard approximation procedure. Theinformal recursion (as it would be done in N ) puts β = α , and β = β + ǫ , where ǫ should be chosen judiciously from M . So, f ( β ) = f ( β + ǫ ) = f ( β ) + f ′ ( β ) ǫ + o ( ǫ ). Choose ǫ = − f ( α ) f ′ ( α ) (an infinitesimal). Hence, v p ( f ( β )) = v p ( f ( β + ǫ )) ≥ ( v p ( f ( β )) >
0, and v p ( f ′ ( β )) = v p ( f ′ ( β ) + ǫ f ′′ ( β ) + o ( ǫ )) = 0. We iterate thisprocedure which can be coded in M , and we get a sequence β j ’s of elements of M ,such that v p ( f ( β j )) > v p ( f ′ ( β j )) = 0, v p ( f ( β j )) < v p ( f ( β j +1 )), and v p ( β − β j ) > v p ( f ( β m )) ≥ k for some m , then β m is a solution of f in M (cid:14) p k M . By thePigeonhole Principle this always happens since otherwise we would have a definableinjective map from M into an initial segment [0 , p k ), clearly a contradiction.Note that the condition that the unique maximal ideal is principal is elementary.In general, in a local ring the unique maximal ideal need not be principal. The principal ideals of M (cid:14) p k M are generated by p j for 0 < j ≤ k , and are linearlyordered by the divisibility condition with minimum (0) and maximum ( p ).On the ring M (cid:14) p k M there is a truncated valuation v with values in [0 , k ], definedby v ( m + ( p k )) = (cid:26) v ( m ) if v ( m ) < kk if v ( m ) ≥ k satisfying1. v ( x + y ) ≥ min( v ( x ) , v ( y ))2. v ( xy ) = min( k, v ( x ) + v ( y ))3. v (1) = 04. v (0) = k
5e can construe v as a map to a “truncated ordered abelian group”, hence-forward called TOAG. In [12] axioms which are true in initial segments of orderedabelian groups are identified. We list them here. Some are obvious, while othersneed some calculations. In [12] it is also shown that models of these axioms canbe realized as initial segments of ordered abelian groups. Notice that in [12] it isnot specified if the order is discrete. We are mostly interested in discrete orders andsome extra axioms will be added later. The language contains two binary operations+ and . − , a binary relation ≤ , and two constants 0 and τ . Axioms:
1. + is commutative2. x + 0 = x and x + τ = τ x ≤ y and x ≤ y imply x + x ≤ y + y
4. + is associative5. x + y = x + z < τ implies y = z ( cancellation rule )6. If x ≤ y < τ then there is a unique z with x + z = y
7. there are z such that z < τ and x + z = τ , and τ . − x = min { z : x + z = τ } τ . − ( τ . − x ) = x
9. If 0 ≤ x, y < τ and x + y = τ then y . − ( τ . − x ) = x . − ( τ . − y )10. If x + ( y + z ) = τ and y + x < τ then x . − ( τ . − ( y + z )) = z . − ( τ . − ( x + y ))11. If y + z < τ , x + ( y + z ) = τ , x + y = τ , and z + ( y . − ( τ . − x )) < τ
12. If y + x = τ and y + z < τ then z + ( y . − ( τ . − x )) < τ
13. If y + z = y + x = τ and z +( y . − ( τ . − x )) < τ then x +( y . − ( τ . − z )) = x . − ( τ . − y )+ z ))14. If y + z = y + x = τ and x + ( y . − ( τ . − z )) = τ then z + ( y . − ( τ . − x )) = τ
15. If y + z = y + x = x + ( y . − ( τ . − z )) = τ then ( y . − ( τ . − x )) . − ( τ . − z ) =( y . − ( τ . − z )) . − ( τ . − x ).A proof of the following fundamental result is in [12].6 heorem 3.1. Let [0 , τ ] be a TOAG with + , . − and ≤ . Then there is an orderedabelian group (Γ , ⊕ , ≤ Γ ) with P the semigroup of non-negative elements, and anelement τ Γ in P such that ([0 , τ ] , + , ≤ ) is isomorphic to ([0 , τ Γ ] , ⊕ , ≤ Γ ) , where thestructure on [0 , τ Γ ] is the one induced by Γ . Our primary interest in this paper is in models of Presburger, and extra condi-tions are needed for characterizing the TOAGs which are initial segments of modelsof Presburger. We expand the language for TOAG with an extra constant symbol1. The extra axioms are also sufficient as shown in the following theorem in [12].
Theorem 3.2.
A TOAG [0 , τ ] with least positive element is a Presburger TOAGif and only if it satisfies the following conditions1. [0 , τ ] is discretely ordered and every positive element is a successor;2. for every positive integer n and each x ∈ [0 , τ ] there is a y ∈ [0 , τ ] and aninteger m < n such that x = ( y + . . . + y ) | {z } n times + (1 + . . . + 1) | {z } m times . Let
P resT OAG be the set of axioms in the language containing +, . − , ≤ , 0 and τ for Presburger TOAGs together with all the remainders axioms as above. Theorem 3.3. P resT OAG is model complete.2.
P resT OAG is not complete. The complete extensions are in natural corre-spondence with ˆ Z , where to ( f ( p )) f ( p ) ∈ Z p corresponds P resT OAG + τ ≡ f ( p )( mod p k ) for every k ≥ .3. The theory of models of P resT OAG has quantifier-elimination in the languageaugmented by the relations ≡ n (divisibility by n ) for each n ≥ .Proof: 1. Let S , S be models of P resT OAG and assume there is a monomorphism S ֒ → S . Since τ is in the language, S and S have the same top element. Let S be an initial segment of a model S of Presburger. Let λ , . . . , λ m ∈ S . We claimthe type of h λ , . . . , λ m i in S is the same as the type of h λ , . . . , λ m i in S . By theelimination of quantifiers for Presburger Arithmetic the type of h λ , . . . , λ m i overany S is determined by order and congruence conditions in terms of τ and 1, andso it is the same in both S and S . Clearly the conditions got from f ∈ ˆ Z are consistent. The completeness comeseasily from Theorem 3.2 and the elimination of quantifiers for Presburger Arithmetic. This follows from elimination of quantifiers for Presburger Arithmetic.7
Henselian local rings
In Section 3 we considered a valuation v on the Henselian local ring M /p k M withvalues onto the Presburger TOAG [0 , k ]. In addition, the residue field of the localring is either F p if p is standard, and otherwise a characteristic 0 pseudofinite field.The union of these alternatives says exactly that the residue field is a model ofthe theory P rimeF in . Moreover, v ( p ) = 1 if p is standard. A natural problem iswhether a Henselian local ring with these properties is a quotient of the valuationring of a Henselian field with the same residue field, and value group a model ofPresburger Arithmetic, and in addition v ( p ) = 1 if p is standard.The first order theory of a Henselian valued field ( K, v ), in the language of valuedfields, is well understood in the particular cases when the value group is a model ofPresburger and the residue field is either a characteristic 0 pseudofinite field or theresidue field is F p and v ( p ) = 1 (i.e. K is unramified).In these cases, T h ( K, v ) is completely determined by the theory of the residuefield and the theory of the value group. As a consequence, one gets that
T h ( K, v )is decidable if and only if the theory of the residue field and the theory of the valuegroup are decidable.
Theorem 4.1.
For any M (cid:14) p k M there is a ring R such that1) R is a Henselian valuation domain of characteristic and unramified, and thevalue group Γ of R is a Z -group (i.e. a model of Presburger), with initial segment [0 , k ] .2) M (cid:14) p k M is isomorphic to the quotient ring R (cid:14) I for some principal ideal I of R . 3) the residue field of R is naturally isomorphic to the residue field of M (cid:14) p k M .Proof: Using the MacDowell-Specker Theorem (see [24]), choose N a proper ele-mentary end extension of M . Let d = p δ for some δ ∈ N and δ > M . By Theorem2.1 the ring N (cid:14) p δ N is a Henselian local ring with respect to the valuation inducedby the p -adic valuation on N . The set ∆ = { x ∈ N : v p ( x ) > v p ( a ) for all a ∈ M} is a non-principal prime ideal of N and contains p δ . Hence, N (cid:14) ∆ is a domain,a local ring with value group M , and residue field either F p or a characteristic 0pseudofinite field. Moreover, N (cid:14) ∆ is also Henselian since it is a homomorphic imageof N (cid:14) p δ N which is a Henselian ring. Let R = N (cid:14) ∆ then R (cid:14) p k R ∼ = M (cid:14) p k M (herewe use that N is an end extension of M ). (cid:3) R is principal due to the fact that divisibilityis a linear order on valuation domains, and the value group is discrete. As alreadynoticed the maximal ideal of M (cid:14) p k M is principal generated by p + ( p k ).So far we have identified the M (cid:14) p k M as Henselian local rings, with two distinctcases (see [2], [3], [4]): Case 1. p standard, and M (cid:14) p k M isomorphic to some S/αS , where S ≡ Z p , and α ∈ S . In particular, v p ( p ) = 1. Case 2. p nonstandard, and M (cid:14) p k M isomorphic to some S/αS , with S elementar-ily equivalent to a ring of power series with exponents in a Z -group and coefficientsfrom a pseudofinite field of characteristic 0, with α ∈ S .In both cases we have the valuation onto a P resT OAG , and we can link upto the results of [16] where we began to analyze the set of axioms for such rings.Moreover, a trivial compactness argument shows that any local ring modelling thoseaxioms is elementarily equivalent to some M (cid:14) p k M , where M | = P A , see below.We note one important point. The rings M (cid:14) p k M have the special propertyof recursive saturation, see [25], so we cannot replace elementary equivalence byisomorphism in the preceding paragraph. However, there are standard resplendencyarguments which give the converse when the ring S/αS is countable recursivelysaturated.
Note that we still have not proved the converse that any
S/αS as in Case 1 and 2above is elementarily equivalent to some M (cid:14) p k M .Now, we do this and in addition we obtain decidability results. Theorem 4.2.
Suppose S is as in Cases 1 and Case 2 above, and α is a non-unitand α = 0 . Then S/αS is elementarily equivalent to an ultraproduct of Z /p k Z , for p prime and k > .Proof: By Ax-Kochen-Ershov, S is elementarily equivalent to an ultraproduct of Z p ’s. So, S/αS is elementarily equivalent to an ultraproduct of Z p /α p Z p , where α p is a non-zero, nonunit of Z p , and each Z p (cid:14) α p Z p is isomorphic to some Z /p h Z .So, S/αS is elementarily equivalent to an ultraproduct of various Z /p h Z , so it isisomorphic to some M (cid:14) p k M where M is an ultrapower of Z and p is a prime in M and k ∈ M . (cid:3) Corollary 4.3.
The elementary theories of the M (cid:14) p k M are exactly the elementarytheories of the S/αS where S is as in Case 1 and 2. roof: By Theorem 4.1 and Theorem 4.2. (cid:3)
Ax’s decidability results for the class of all Q p gives (via interpretability) decid-ability of the class of all S/αS , and so of the class of all M (cid:14) p k M .We summarize what we have proved so far. Theorem 4.4. M (cid:14) p k M are pseudofinite (or finite).2. The theory of the M (cid:14) p k M is the theory of the Z (cid:14) p h Z .3. The theory of the M (cid:14) p k M is decidable.Proof:
1) and 2) follow from Theorem 4.2. The theory of Z (cid:14) p h Z for a fixed prime p and h > Let R = M /p k M .If k = 1 then R is a model of the theory of finite prime fields (see [25], [16]).The theory P rimeF in of finite prime fields (and thus the theory of the R ’s) hasuniform quantifier elimination in the definitional extension of ring theory got byadding primitives Sol n ( y , . . . , y n ) expressing ∃ t ( y + . . . + y n t n = 0), see [1], and [16]for rings arising from models of P A . Models R and R are elementarily equivalentif and only if they have the same characteristic and agree on all Sol n ( ℓ , . . . , ℓ n ) for ℓ , . . . , ℓ n ∈ Z , see [5].Some T h ( R ) are model-complete, some not. The model-complete ones arethose where R is either F p or characteristic 0, and for each m the unique ex-tension of dimension m is got by adjoining an algebraic number, see [18]. Onegets an abundance of model-complete examples from Jarden’s result in [23] that { σ ∈ Gal ( Q alg ) : F ix ( σ ) is pseudofinite } has measure 1. An example of a nonmodel-complete R is one where Q alg ⊆ R (see [5]).In the case of k > M /p k M is isomorphic to some S/αS where α ∈ S , S is an unramifiedHenselian domain, with a value group a Z -group, residue field a model of the theoryof finite prime fields. Conversely, any such S/αS is elementarily equivalent to some M /p k M for some M model of P A .So we have to investigate axioms, elementary invariants and model-completenessfor all R of the form S/αS , where S and α are as before. Each is valued in aPresburger TOAG. Moreover, the TOAG is interpretable (in L rings ) in R , by taking10he underlying set Γ of the TOAG, to the set of principal ideals of R , linearlyordered by reverse inclusion, so (1) is the least element and (0) is the top element(obviously, we get a linear order since S is a valuation domain). We get a ⊕ onΓ via ( α ) ⊕ ( β ) = ( αβ ), making Γ into a TOAG with (1) as 0-element, and (0)as ∞ -element. The valuation maps α to ( α ). In this way we get a PresburgerTOAG. This is a crucial axiom about R . The 1 of the Presburger TOAG is ( p ) (themaximal ideal). In [16] we show that the elementary type of the TOAG is given bythe Presburger type of the penultimate element of Γ (in our case ( k − µ of R is of course definable as the set of nonunits and theresidue field of R is interpretable. Naturally, we seek “truncations” of the Ax-Kochen-Ershov theorem. On the basis of what we have done above we get thefollowing two basic theorems. Theorem 5.1.
The theory of the class of M /p k M ( p prime and k ≥ ) is axiom-atized by the following conditions1. Henselian local ring, valued in a Presburger TOAG, with residue field a modelof the theory of finite prime fields;2. If the characteristic of the residue field is a prime p then the valuation of p isthe least positive element of the Presburger TOAG. Theorem 5.2.
The elementary theory of an individual M /p k M is uniquely deter-mined by the Presburger type of the penultimate element of the TOAG, and by theelementary theory of the residue field. Given a Presburger type and a residue field,there is an M /p k M with the corresponding invariants. Remark 5.3.
1. In the above we work with the ring language, since the TOAGcondition comes from a condition on principal ideals.2. Obviously there are decidability results parallel to the above (of the form, forexample, if the residue field is decidable, and the Presburger type of the penul-timate element of the TOAG is computable then M /p k M is decidable). The analysis of formulas in the M /p k M , rather than the mere sentences of thepreceding subsection, is quite tricky. It involves a Denef-Pas analysis as in [16] forsentences, but with some complications, and we give only a sketch of the argument.11y the preceding we are dealing (as far as definability is concerned) with rings S/αS with S a Henselian valuation domain with Presburger value group, withresidue field k a model of the theory of finite prime field, and unramified. Wenote the substantial result in [10] that S is uniformly ∃∀ -definable in the languageof rings in its field of fractions K .It is convenient to fix in S an element t with v ( t ) = 1, i.e. the least positiveelement of the value group. We take t = p if k has characteristic p , and t arbitraryotherwise (note then that the type of t has many possibilities). We can assume K is ℵ -saturated and so has an angular component map ac : K → k . This will not bedefinable in general unless we have some saturation. See [27], and [16] for the P A context for the details. Note that our M /p k M are recursively saturated, and so if M is countable we have an ac .We now work in the 3-sorted formalism L Denef − P as , with sorts of K , k , andΓ (field, residue field, and value group with added ∞ ). We have the usual fieldformalism on each of K , k , and on Γ the usual + , − , <, ∞ . In addition we have twotrans-sort primitives, ac and v , the former from K to k , and the latter from K toΓ ∪ {∞} . K has characteristic 0, K has constant t as above with t = p if k hascharacteristic p , and v ( t ) = 1.Our purpose is to analyze the structure of the sets { ( y , . . . , y n ) ∈ ( S/αS ) n : S/αS | = ψ ( y , . . . , y n ) } for a ring formula ψ .For convenience in applying Denef-Pas, we work in K rather than in S , but weexploit the uniform definability of S in K . So, we consider variables x , . . . , x n , α and the formula expressing (in K ) x , . . . , x n , α are in S , and S/αS | = ψ ( x + αS, . . . , x n + αS ) . (1)This is a first order condition. Thus by Denef-Pas it is uniformly, except forfinitely many p (characteristic of k ), equivalent to a formula Θ( x , . . . , x n , α ) in L DP with no bounded K variables (but do not forget the constant t !). The basic formulasout of which Θ is built are:algebraic equations in x + αS, . . . , x n + αS, t over K ;residue field formulas in ac ( τ ℓ ) for various polynomials τ ℓ over Z in x + αS, . . . , x n + αS, t .Presburger formulas in v ( µ m ) for various polynomials µ m in x + αS, . . . , x n + αS, t .In fact, Θ can be taken as a Boolean combination of these basic formulas, as canbe seen by inspection of the proof of Denef-Pas.12or the finitely many exceptional standard primes p , . . . , p n (which can be foundeffectively from ψ , by [5]), S is elementarily equivalent to one of Z p , . . . , Z p n (theanalogous easier argument for sentences is sketched in [16]). By using Macintyre’squantifier elimination for each of Z p , . . . , Z p n , and changing the ac if need be, oneeasily gets the power conditions P ℓ ( x ) to have the required Denef-Pas form. Byusing that p = 0 in k captures the characteristic p condition, one can combinethese finitely many analyses with the one that works except for p , . . . , p n to get theDenef-Pas result for all ψ , giving a new Θ that works always, independent of p .Now we can do further elimination semplification in the other sorts. For thePresburger sort we simply have the classical elimination down to order and con-gruence conditions, provided we have a constant for the least positive element (andabove we have stipulated that v ( t ) = 1 = least positive element). So Presburgerquantifiers get eliminated. This leaves the issue of quantification over k . Recall that k ranges over models of the theory of finite prime fields. By [5] and [1] one easilysees that (uniformly) k has quantifier elimination down to the solvability predicates Sol n .Now recall that we seek elimination results in the S/αS , where the above takesplace in K . However, k depends only on S and not on α , and the Presburgerconditions have the same value in S/αS as in S , for x , . . . , x n proper dividing α , soin fact we have proved (recall our starting point (1)): Theorem 5.4.
Assume the previous conditions on S , and adjoin constant t with v ( t ) = 1 , and t = p if k has characteristic p . Then uniformly in S for any ψ ( x , . . . , x n ) in L ring,t , the language of rings with t , there is another such formula ψ + ( x , . . . , x n , y ) such that if S is ℵ -saturated S has an angular component ac suchthat for all α ∈ S , α = 0 and β , . . . , β n ∈ S properly dividing αS/αS | = ψ ( β + αS, . . . , β n + αS ) ⇔ S | = ψ + ( β , . . . , β n , α ) where ψ + ( x , . . . , x n , y ) is a Boolean combination of three kinds of sorted formulas(where now the sorts are local ring, residue ring, TOAG):1. polynomial equations over Z [ t ] in β + αS, . . . , β n + αS
2. solvability conditions over Z in β + αS, . . . , β n + αS and ac ( β ) , . . . , ac ( β n ) , ac ( t ) = 1 (i.e. using the predicates Sol n )3. Presburger conditions over monomials in v ( β ) , . . . , v ( β n ) in the PresburgerTOAG [0 , v ( t/α )] .Proof. This is sketched above, and is the obvious “truncated” analogue of Denef-Pas.Note that [12] contains the background for 3.13e do not attempt to go any deeper to minimize the role of the ac . This maybe worthwhile, but it is not needed for our last topic below. We work with rings R ≡ S/αS , where S is as above. The TOAG valuation is alge-braically interpretable in R . From Section 4, the elementary type of R is determinedby the elementary type of the residue field, and the Presburger type of the penulti-mate element of the TOAG. For R ≡ R as rings, and an embedding f : R → R ,we want to find out when f is elementary. There is no loss of generality in analyzingthe case when f is a ring inclusion. It has no chance of being elementary if the leastpositive element of the TOAG of R is not the least positive element of the TOAGof R . So we work in L ring,t , the language of rings with t , with corresponding as-sumptions on R and R . Thus there is a natural inclusion of residue fields k → k ,provided each R i satisfies that t generates the maximal ideal. This we now assume(recall that the corresponding maximal ideals µ and µ are both generated by t ).Note one cost of adjoining t is that our work has to take some account of part of“the type of t ”. We indicate, as we go along, what is involved.Our purpose is to show that if the embedding k → k is elementary then sois R → R . Our convention about t ensures that the induced map on TOAGs iselementary.Now, neither R nor R need have an ac , as required for the Denef-Pas anal-ysis, and we have to resort to “tricks of the saturation trade” to reduce to thecase when R has an ac which restricts to R . If there is any counterexample toour claim that if k → k is elementary then so is R → R , we select such acounterexample (witnessed by a particular residue-field formula), and by a standardcompactness/saturation argument produce a counterexample with stronger proper-ties, namely those given in the next two paragraphs.We are assuming R ⊆ R (and in fact a local extension because of the t -convention), R ∼ = S/αS , R ≡ R in the rings language, S is ℵ -saturated, and k (cid:22) k (note that since ac ( t ) can be chosen as 1, we need only consider k (cid:22) k in the ring language), and some formula W ( x , . . . , x n ) witnessing that R R (in L ring,t ), i.e. there are some c , . . . , c n in R so that R | = W ( c , . . . , c n ) and R | = ¬ W ( c , . . . , c n ).Then we have to work (quite hard) to get an ac on R restricting to one on R .This involves modifying R in general. First get ac on R using the ℵ -saturation of R , recall that S is ℵ -saturated and by [8] has a normalized cross-section and thusan ac (appropriately normalized) which truncates to R .14ow go from R to an | R | + -saturated elementary extension R . This by anobvious adaptation of Cherlin’s argument in [8] give an ac on R extending the ac on R . Now use R → R , which still satisfies the original condition that ( R , R )did. By Theorem 5.4 the truncated Denef-Pas version gives us R (cid:22) R since k (cid:22) k , where k is the residue field of R , and all polynomial equations f ( η , . . . , η n , ac ( η ) , . . . , ac ( η n )) = 0 , with η , . . . , η n ∈ R maintain their truth value between R and R (trivially). Butsince R (cid:22) R and R (cid:22) R we must have R (cid:22) R .So we have now the analogue of the result due to Ziegler [28] in generality. Theorem 5.5.
Assume R ⊆ R with the t condition to guarantee k ⊆ k . Then R (cid:22) R if and only if k (cid:22) k . Finally this gives us a model-completeness result.
Theorem 5.6.
The theory of R in the t -formalism is model-complete if and only ifthe theory of the residue field k is. If p is standard, the preceding shows that each M (cid:14) p k M is interpretable in anultrapower of Z p . By [17] the ultrapower is N IP , and so does not interpret even I ∆ (which has IP ), a much weaker system than Peano Arithmetic see [11].When p is nonstandard, M (cid:14) p k M interprets M (cid:14) p M which has IP (see [19]), so M (cid:14) p k M has IP . However, M (cid:14) p k M lives in the N T P enviroment of neostability[9], since M (cid:14) p k M is interpretable (by the preceding) in the ring of power series in M (cid:14) p M with value group a model of Presburger. By [9] this ring of power serieshas N T P .However, any model M of I ∆ has T P , as we see by the following construction.Let a nm be p mn for n, m positive standard integers, and p n a prime in M . Considerthe formula (of ring theory) ϕ ( x, y ) saying that y is a power of a prime p , and v p ( x ) = v p ( y ). From [11] this is given in M by a ∆ -formula. Now,1. the set { ϕ ( x, a nm ) : m ∈ N } is inconsistent, for fixed n ;2. for any f : ω → ω the set { ϕ ( x, a nf ( n ) ) : n ∈ N } is consistent.1. This is clear, since the type forces v p n ( x ) = m , for all m .2. One shows that for each n , the set { ϕ ( x, a nf ( n ) ) : n ≤ n } is realized in M , infact by the element b = p f (0)0 · . . . · p f ( n ) n .Since T P is preserved by interpretation, we have15 heorem 6.1. For each k ≥ no model of I ∆ is interpretable in any M (cid:14) p k M . Concluding remarks.
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