aa r X i v : . [ m a t h . L O ] A p r Dp-finite fields V: topological fields of finite weight
Will JohnsonMay 1, 2020
Abstract
We prove that unstable dp-finite fields admit definable V-topologies. As a conse-quence, the henselianity conjecture for dp-finite fields implies the Shelah conjecture fordp-finite fields. This gives a conceptually simpler proof of the classification of dp-finitefields of positive characteristic.For n ≥
1, we define a local class of W n -topological fields , generalizing V-topologicalfields. A W -topology is the same thing as a V-topology, and a W n -topology is somehigher-rank analogue. If K is an unstable dp-finite field, then the canonical topologyof [5] is a definable W n -topology for n = dp-rk( K ). Every W n -topology has between 1and n coarsenings that are V-topologies. If the given W n -topology is definable in somestructure, then so are the V-topological coarsenings. There are two main conjectures on NIP fields.
Conjecture 1.1 (Henselianity) . Every NIP valued field ( K, v ) is henselian. Conjecture 1.2 (Shelah) . Let K be an NIP field. Then one of the following holds: • K is finite • K is algebraically closed • K is real closed • K admits a non-trivial henselian valuation. The Shelah conjecture is known to imply the henselianity conjecture [2], as well as afull classification of NIP fields [1]. For the Shelah conjecture, we may assume that K issufficiently saturated.These implications continue to hold in the restricted setting of dp-finite fields (fields offinite dp-rank). Our main result is a partial converse to [2]:1 heorem 1.3. The henselianity conjecture for dp-finite fields implies the Shelah conjecturefor dp-finite fields.
For the special case of positive characteristic dp-finite fields, • The henselianity conjecture was proven by a very simple argument in [5], § • The Shelah conjecture was proven by a very complicated argument in [5], § K is a stable dp-finite field, then K is algebraically closed or finite, by Proposition 7.2in [3]. The key to proving Theorem 1.3 is Theorem 1.4.
Let K be an unstable dp-finite field. Then K admits a definable V-topology. If K is sufficiently saturated, then the definable V-topology in Theorem 1.4 yields anexternally definable valuation ring, by Proposition 3.5 in [2]. The henselianity conjectureapplies to externally definable valuation rings, because of general facts about Shelah expan-sions of NIP structures. Therefore, the henselianity conjecture implies the Shelah conjecture(for dp-finite fields).In order to prove Theorem 1.4, we use the canonical topology on K , defined in [5],Remark 6.18. Fact 1.5.
Let K be an unstable dp-finite field. • The canonical topology is a field topology, i.e., the field operations are continuous. • As D ranges over definable subsets of K with dp-rk( D ) = dp-rk( K ) , the sets D − D range over a neighborhood basis of 0 in the canonical topology.See ([6], Corollaries 5.10 and 5.15) for a proof. We define a notion of a W n -topology on a field. These generalize V-topologies; in fact a W -topology is the same thing as a V-topology. Our main results on W n -topologies are thefollowing: Theorem 1.6.
1. If τ is a W n -topology on a field K , then there is at least one V-topology τ ′ coarser than τ .2. If τ is a W n -topology on a field K , then the number of V-topological coarsenings is atmost n .3. If τ is a definable W n -topology on some field ( K, + , · , . . . ) (possibly with extra struc-ture), then every V-topological coarsening is definable.4. Let ( K, + , · , . . . ) be a field of dp-rank n and let τ be the canonical topology. Then τ isa definable W n -topology.
2n particular, the canonical topology is definable. For rank 1, this was proved in [4], andfor rank 2 characteristic 0, this was proved in [9].We also define a class of W n -rings , which generalize valuation rings in the same way that W n -topologies generalize V-topologies. In particular, a W -ring on K is the same thing as avaluation ring on K . Any W n -ring on K induces a W n -topology. Up to Prestel-Ziegler localequivalence, all W n -topologies arise from W n -rings.We avoid using the inflator machinery of [7] for the above results. Nevertheless, thereseems to be a close connection between inflators and W n -rings, which we discuss in §
8. Inparticular, the analysis of 2-inflators in [9] yields a classification of W -topologies on fieldsof characteristic 0: Theorem 1.7.
Let K be a field of characteristic 0, and τ be a field topology. Then τ is a W -topology if and only if τ is one of the following: • a V-topology • a “DV-topology” in the sense of [9], Definition 8.18. • a topology generated by two independent V-topologies. It may be possible to similarly classify W n -topologies using n -inflators. In § W n -rings and W n -topologies. One of these conjectures would implythe Shelah conjecture (for dp-finite fields). We will make heavy use of Prestel and Ziegler’s machinery of local sentences and localequivalence [10]. A “ring topology” on a field K will mean a Hausdorff non-discrete topologyon K such that the ring operations are continuous. All topologies we consider will bering topologies on fields. A ring topology is determined by the set of neighborhoods of 0.Following [10], we identify a (ring) topology τ with its set of neighborhoods of 0 (rather thanits set of open sets).One can consider a topological field ( K, τ ) as a two-sorted structure with sorts K and τ .A local sentence is a first-order sentence in this language, subject to the following constraintson quantification over τ : • If there is universal quantification ∀ U ∈ τ : φ ( U ), then U must occur positively in φ . • If there is existential quantification ∃ U ∈ τ : φ ( U ), then U must occur negatively in φ .These constraints ensure that one can replace the quantifiers ∀ U ∈ τ, ∃ U ∈ τ with quantification over a neighborhood base.3or example, ∀ U ∈ τ ∃ V ∈ τ : (1 + V ) − ⊆ U is a local sentence expressing that the (ring) topology is a field topology.Two topological fields are locally equivalent if they satisfy the same local sentences. Atopological field ( K, τ ) is ω -complete if τ is closed under countable intersections, i.e., anintersection of countably many neighborhoods of 0 is a neighborhood of 0. Every topologicalfield is locally equivalent to an ω -complete topological field ([10], Theorem 1.1).A subset S ⊆ K is bounded if the following equivalent conditions hold: • For every neighborhood U ∋
0, there is nonzero c ∈ K such that cS ⊆ U . • For every neighborhood U ∋
0, there is a neighborhood V ∋ S · V ⊆ U .The equivalence is Lemma 2.1(d) in [10].A ring topology is locally bounded if there is a bounded neighborhood V of 0. In thiscase, the set { cV : c ∈ K × } is a neighborhood basis of 0, by Lemma 2.1(e) in [10].If R is a proper subring of K and K = Frac( R ), then R induces a locally bounded ringtopology τ R on K , for which either of the following families are a neighborhood basis: • The set of rescalings cR , where c ∈ K × . • The set of nonzero ideals of R .Up to local equivalence, every locally bounded ring topology arises in this way ([10], Theo-rem 2.2(a)).We shall need the following variant of Lemma 1.4 and Theorem 2.2 in [10]: Proposition 1.8.
Let ( K, τ ) be a field with a Hausdorff non-discrete ω -complete locallybounded ring topology. Let U be a bounded neighborhood of 0. Let R be the subring generatedby U . Then R is a bounded neighborhood of 0, Frac( R ) = K , and τ = τ R .Proof. Recursively define U ⊆ U ⊆ U ⊆ · · · by • U = U ∪ { , }• U i +1 = U i ∪ ( U i − U i ) ∪ ( U i · U i ).Then each U i is a bounded set, by Lemma 2.1 in [10]. By the comment at the start of § U i ’s is bounded. This union is R . Also, R ⊇ U , so R is aneighborhood of 0. Then R is a bounded neighborhood, so the family of sets { cR : c ∈ K × } is a neighborhood basis of 0, by Lemma 2.1(e) in [10]. To see that Frac( R ) = K , note thatfor any a ∈ K × , the neighborhood R ∩ aR must strictly exceed { } (as the topology isnon-discrete). This implies that a ∈ Frac( R ). More accurately, the local sentence is the following: ∀ U ∈ τ ∃ V ∈ τ ∀ x ∈ V ∃ y ∈ U : (1 + x )(1 + y ) = 1 . Rings of finite weight
Let R be a ring and M be an R -module. Let c-rk( M ) or c-rk R ( M ) denote the cube rank of M as an R -module ([7], Definition 6.6). Cube rank is an element of N ∪ {∞} and can becharacterized in two ways: • By Remark 6.7 in [7], c-rk( M ) ≥ n if and only if there are submodules N ′ ≤ N ≤ M such that the subquotient N/N ′ is isomorphic to a direct sum of n non-trivial R -modules. • By Proposition 7.3 in [9], c-rk( M ) ≥ n if and only if there are m , . . . , m n ∈ M suchthat no m i is generated by the others: ∀ i : m i / ∈ R · m + · · · + R · m i − + R · m i +1 + · · · + R · m n . Remark . Cube rank was called “reduced rank” in [5, 7, 9], and is probably a well-knownconcept to people who study modules or lattices. At a minimum, cube rank seems relatedto “uniform dimension” and “hollow dimension” in module theory.By Proposition 6.9 in [7], cube rank has the following properties: • c-rk( M ) > M is non-trivial. • c-rk( M ⊕ N ) = c-rk( M ) + c-rk( N ). • If N is a submodule, quotient, or subquotient of M , then c-rk( N ) ≤ c-rk( M ). • In a short exact sequence 0 → N → M → N ′ → , we have c-rk( M ) ≤ c-rk( N ) + c-rk( N ′ ).Moreover, c-rk( − ) is the smallest N ∪ {∞} -valued function with these properties ([7], Propo-sition E.2). Lemma 2.2. If M is an R ′ -module and R ⊆ R ′ , then c-rk R ( M ) ≥ c-rk R ′ ( M ) .Proof. If c-rk R ′ ( M ) ≥ n , witnessed by m , . . . , m n , then for any i , m i / ∈ X j = i R ′ · m j ⊇ X j = i R · m j . Remark . Occasionally, we will also need cube rank on modular lattices. If Λ is a modularlattice, then c-rk(Λ) can be characterized in one of several equivalent ways. • c-rk(Λ) ≥ n if there is a strict n -cube in M , in the sense of Definition 9.13 in [5].5 c-rk(Λ) ≤ n if for any a , . . . , a n ∈ Λ, there is i such that a ∧ · · · ∧ a n = a ∧ · · · ∧ b a i ∧ · · · ∧ a n . • c-rk(Λ) ≤ n if for any a , . . . , a n ∈ Λ, there is i such that a ∨ · · · ∨ a n = a ∨ · · · ∨ b a i ∨ · · · ∨ a n . These definitions are equivalent by Proposition 6.3 of [7]. If a, b are two elements of Λ with a ≥ b , then c-rk Λ ( a/b ) will denote the cube rank of the interval [ b, a ], a sublattice of Λ. W n -rings Let R be an integral domain with fraction field K . Lemma 2.4. c-rk R ( R ) = c-rk R ( K ) .Proof (cf. Lemma 10.25 in [9]). c-rk R ( R ) ≤ c-rk R ( K ) because R is a submodule of K . Con-versely, suppose c-rk R ( K ) ≥ n . Then there are m , . . . , m n ∈ K such that for any i , m i / ∈ X j = i R · m j . Take non-zero s ∈ R such that sm i ∈ R for all i . Then the set { sm , . . . , sm n } showsc-rk R ( R ) ≥ n . Definition 2.5.
The weight of R , written wt( R ) is the value c-rk R ( R ) = c-rk R ( K ). We saythat R is a W n -ring (on K ) if wt( R ) ≤ n . Proposition 2.6. R is a W -ring if and only if R is a valuation ring.Proof. By definition, c-rk R ( R ) ≤ x, y ∈ R , x ∈ R · y or y ∈ R · x. This is the definition of a valuation ring.
Lemma 2.7.
Let R be a W n -ring on K . Let L/K be a finite extension of degree d . Let R ′ be a subring of L containing R . Then R ′ is a W dn -ring.In particular, if R ′ is a subring of K containing R , then R ′ is a W n -ring.Proof. By Lemmas 2.2 and 2.4,c-rk R ′ ( R ′ ) ≤ c-rk R ′ ( L ) ≤ c-rk R ( L ) = c-rk R ( K d ) = d · c-rk R ( K ) ≤ dn. Corollary 2.8.
Let R be a W n -ring. Let m be a maximal ideal. Then the localization R m isa W n -ring. .3 Maximal ideals and the integral closure Proposition 2.9.
Let R be a W n -ring. Then R has at most n maximal ideals. In particular, W n -rings are semilocal.Proof. If m , . . . , m n +1 are distinct maximal ideals of R , then R/ ( m ∩ · · · ∩ m n +1 ) ∼ = ( R/ m ) × · · · × ( R/ m n +1 )by the Chinese remainder theorem, and so wt( R ) = c-rk R ( R ) ≥ n + 1. Corollary 2.10. If R is a W n -ring on K , and R = K , then the Jacobson radical of R isnon-trivial. Proposition 2.11.
Let O , . . . , O n be pairwise incomparable valuation rings on a field K .Then the intersection T ni =1 O i is a ring of weight n .Proof. Lemma 6.5 in [7].In the language of [6], multivaluation rings have finite weight.
Proposition 2.12.
Let R be a ring of weight n on a field K . Then the integral closure of R (in K ) is a multivaluation ring, an intersection of at most n valuation rings on K .Proof. Let ˜ R denote the integral closure. Let P be the class of valuation rings on K con-taining R . On general grounds, ˜ R = T P . By Proposition 2.11 and Lemma 2.7, P containsno antichains of size n + 1. By Dilworth’s theorem, P is a union of n chains C , . . . , C n . Eachintersection T C i is a valuation ring, and \ P = n \ i =1 (cid:16)\ C i (cid:17) . Remark . If R is non-trivial, i.e., R ⊆ K , then the integral closure is non-trivial. To seethis, take p a maximal ideal of R . Then p = 0. Take nonzero x ∈ p . An easy argumentshows that 1 /x is not integral over R . Let R be a W n -ring on a field K . Suppose R is non-trivial, i.e., R = K . By Example 1.2 in[10], R induces a Hausdorff non-discrete locally bounded ring topology on K , in which thefamily { cR : c ∈ K × } is a neighborhoods basis of 0. Equivalently, the non-zero ideals of R are a neighborhoodbasis of 0. 7 roposition 3.1. The topology induced by R is a field topology, i.e., division is continuous.Proof. This holds because the Jacobson radical of R is non-trivial, as in the proof of The-orem 2.2(b) in [10]. In more detail, to verify that division is continuous it suffices to provethe following: for any non-zero ideal I ≤ R , there is a non-zero ideal I ′ ≤ R such that(1 + I ′ ) − ≤ I. Let J be the Jacobson radical of R . By Corollary 2.10, J = 0. Let I ′ = I ∩ J . Theintersection is non-zero as R is a domain. Then for any x ∈ I ′ , we have x ∈ I ′ = ⇒ x ∈ J = ⇒ x ∈ R × = ⇒ − x x ∈ I ′ = ⇒ − x x ∈ I = ⇒
11 + x ∈ I. Definition 3.2.
Let K be a field. A W n -set is a subset S ⊆ K such that for any x , . . . , x n +1 ∈ K , there is an i ≤ n + 1 such that x i ∈ x · S + · · · + x i − · S + x i +1 · S + · · · + x n +1 · S. Note that an integral domain R is a W n -ring if and only if R is a W n -set in Frac( R ). Definition 3.3. A W n -topology on a field K is a (Hausdorff non-discrete) locally boundedring topology on K such that for every neighborhood U ∋
0, there is c ∈ K × such that c · U is a W n -set. Remark . The class of W n -topologies is a local class , cut out by finitely many local sen-tences in the sense of [10]. Specifically, it is cut out by the axioms of locally boundednon-trivial Hausdorff ring topologies plus the local sentence ∀ U ∈ τ ∃ c = 0 ∀ x , . . . , x n +1 n _ i =1 x i ∈ X j = i c · x j · U ! . Remark . A locally bounded ring topology is a W n -topology if and only if some boundedneighborhood V ∋ W n -set. Indeed, if such a V exists, then for any neighborhood U ∋ c ∈ K × such that cU ⊇ V , because V is bounded. Then cU is a W n -set because V is a W n -set. Conversely, suppose the topology is a W n -topology. Let U be a boundedneighborhood of 0. Then there is c ∈ K × such that cU is a W n -set. But cU is again abounded neighborhood of 0. Proposition 3.6.
Let R be a non-trivial W n -ring on a field K . Then the induced fieldtopology is a W n topology. roof. The bounded neighborhood R is a W n set. Lemma 3.7.
Let K be a field with a W n -topology. Suppose that K is ω -complete as in [10].Then the topology is induced by a W n -ring R . Moreover, given any fixed bounded set S , wemay assume S ⊆ R .Proof. Let U be a bounded neighborhood of 0. Scaling U by an element of K × , we mayassume that U is a W n -set. By Proposition 1.8, U ∪ S is contained in a bounded subring R ⊆ K , and R induces the topology. Then R is a W n -ring (= W n -set) on K , because itcontains the W n -set U . Corollary 3.8.
Let K be a field with a ring topology τ .1. τ is a W n -topology if and only if ( K, τ ) is locally equivalent to a field with a topologyinduced by a W n -ring.2. If τ is a W n -topology, then τ is a field topology.3. If τ is a W n -topology, then τ is a W m topology for m > n .4. τ is a W -topology if and only if τ is a V-topology. Definition 3.9.
The weight of a ring topology τ is the minimal n such that τ is a W n -topology, or ∞ if no such n exists. We write the weight as wt( τ ).Then τ is a W n -topology if and only if wt( τ ) ≤ n . Let K be some highly saturated field, possibly with extra structure. Recall that a topologyis definable if it has a (uniformly) definable basis of opens. In the case of a ring topology,it suffices to produce a definable neighborhood basis of 0. In the case of a locally boundedring topology, it suffices to produce a definable bounded neighborhood U , as the definablefamily { aU : a ∈ K × } is then a definable neighborhood basis of 0.As in § X, Y ⊆ K are co-embeddable if there are a, b ∈ K × such that aX ⊆ Y and bY ⊆ X . In a locally bounded ring topology, the boundedneighborhoods of 0 form a single co-embeddability class. Proposition 4.1.
Let τ be a W n -topology on K . Suppose that there is U ⊆ K such that • U is a bounded neighborhood of 0 with respect to τ . • U is ∨ -definable or type-definable. • U is a subgroup of ( K , +) . hen U is co-embeddable with a definable set, and τ is a definable topology.Proof. Rescaling U , we may assume that U is a W n -set. Take m minimal such that U is a W m -set. Then there are b , . . . , b m ∈ K such that for any i , b i / ∈ X j = i b j U. Claim . For any i , b i is not in the closure of P j = i b j U . Proof.
In fact, P j = i b j U is closed. If m = 1, then P j = i b j U = { } , which is closed becausethe topology is Hausdorff. If m >
1, then P j = i b j U is an additive subgroup of K , and aneighborhood of 0. Therefore it is a clopen subgroup. (cid:3) Claim
Let S be the set of x ∈ K such that m _ i =1 b i ∈ xU + X j = i b j U ! If U is type-definable (resp. ∨ -definable), then S is type-definable (resp. ∨ -definable), and K \ S is ∨ -definable (resp. type-definable). Claim . If x / ∈ S , then x ∈ P mj =1 b j U . Proof.
Otherwise, the set { b , . . . , b m , x } witnesses that U is not a W m -set. (cid:3) Claim
Claim . There is a neighborhood V ∋ V ∩ S = ∅ . Proof.
By Claim 4.2, each b i is not in the closure of P j = i b j U . By Lemma 2.1(d) in [10],( b i + V · U ) ∩ X j = i b j U = ∅ for small enough V . We can choose V to work across all b i . Then if x ∈ V ∩ S , there is some i such that b i ∈ xU + X j = i b j U ∅ 6 = ( b i + xU ) ∩ X j = i b j U ⊆ ( b i + V · U ) ∩ X j = i b j U, contradicting the choice of V . (cid:3) Claim
Claims 4.3 and 4.4 imply that V ⊆ K \ S ⊆ m X j =1 b j U.
10y Lemma 2.1 in [5], the right hand side is bounded. Thus K \ S is a bounded neighborhoodof 0. So K \ S is co-embeddable with U . One of { U, K \ S } is type-definable, and othe other is ∨ -definable. As in Remark 6.17 of [9], this implies that U is co-embeddable with a definableset D . Then D is a definable bounded neighborhood of 0, and τ is a definable topology. Remark . It may be possible to drop the peculiar assumption that U is an additivesubgroup in Proposition 4.1, but a more convoluted argument would be needed. ∨ -definable W-rings Let K be some highly saturated field, possibly with extra structure. Proposition 4.6.
Let R be a non-trivial ∨ -definable W n -ring on K . Then R is co-embeddablewith a definable set D . Consequently, the field topology induced by R is definable.Proof. Proposition 4.1 applied to the neighborhood R itself. Proposition 4.7.
Let R be a ∨ -definable W n -ring on K . Let ˜ R be the integral closure. Then ˜ R is ∨ -definable.Proof. ˜ R is the union of the ∨ -definable sets S n := { x ∈ K | ∃ y , . . . , y n ∈ R : x n +1 = y + y x + · · · + y n x n } . Proposition 4.8.
Let R be a ∨ -definable W n -ring on a monster field K . Let p be one of themaximal ideals of R . Then the localization R p is also ∨ -definable.Proof. Let p , . . . , p n enumerate the maximal ideals of R , with p = p . For every subset S ⊆ { , . . . , n } , let a S be an element of p such that a S ∈ p j ⇐⇒ j ∈ S. The a S exist by the Chinese remainder theorem. Claim . For x ∈ R , the following are equivalent: • x / ∈ p • There is S such that 1 / ( x + a S ) ∈ R . Proof. If x ∈ p , then x + a S ∈ p for every S , so there is no S with x + a S ∈ R × .Conversely, if x / ∈ p , then we can find a S such that x ∈ p i ⇐⇒ a S / ∈ p i for i = 1 , . . . , n . Then x + a S / ∈ p i for any i , so x + a S ∈ R × . (cid:3) Claim As R is ∨ -definable, it follows that R \ p is ∨ -definable. Then R p = { x/s : x ∈ R and s ∈ R \ p } is ∨ -definable as well. 11 .3 V-topological coarsenings Theorem 4.10.
Let ( K, τ ) be a field with a W n -topology.1. There is at least one V-topological coarsening of τ .2. There are at most n such coarsenings.3. If ( K, τ ) is a definable topology (with respect to some structure on K ), then everyV-topological coarsening is definable.Proof.
1. Let D ⊆ K be a bounded neighborhood of 0 that is a W n -set. Let ( K ∗ , D ∗ ) be a highlysaturated elementary extension of ( K, D ). Then D ∗ defines a W n -topology on K ∗ thatis ω -complete. Let R be the ∨ -definable subring generated by D ∗ . By Proposition 1.8, R is a bounded neighborhood of 0, and a W n -ring because it contains the W n -set D ∗ .So R defines the same topology as D ∗ . Let ˜ R be the integral closure of R , and let p be a maximal ideal of ˜ R . Then the localization ˜ R p is a ∨ -definable valuation ring byPropositions 4.7, 4.8. Therefore it induces a definable V-topology on ( K ∗ , D ∗ ). Thisdefinable V-topology is coarser than the topology induced by R or D ∗ , because ˜ R p ⊇ R .The statement “there is a definable V-topology coarser than the topology induced by D ∗ ” is expressed by a disjunction of first-order sentences, so it holds in the elementarysubstructure ( K, D ).2. Let σ , . . . , σ m be distinct V-topological coarsenings of τ . We claim m ≤ n . ByRemark 1.5 in [10], we may assume that all the topologies are ω -complete. (Localsentences can assert that τ is or isn’t coarser than τ ′ .) Then τ is induced by a ring R of weight at most n . Additionally, R is bounded with respect to σ i . Therefore,there is a valuation ring O i inducing σ i and containing R , by Lemma 3.7. As i varies,the valuation rings O i induce pairwise distinct topologies, so they must be pairwiseincomparable. As in the proof of Proposition 2.12, this implies m ≤ n .3. Let σ be any V-topological coarsening of τ . Let D and B be bounded neighborhoodsof 0 in τ and σ respectively, with D definable in the given structure. After rescaling,we may assume that D is a W n -set and B is a W -set. Because σ is coarser than τ ,the set B is a neighborhood of 0 in τ , and so there is c such that cD ⊆ B . Therefore D is τ -bounded. Replacing B with B ∪ D , we may assume D ⊆ B .Let ( K ∗ , D ∗ , B ∗ ) be a highly saturated elementary extension. Let R D and R B be thesubrings generated by D ∗ and B ∗ . Then • R D is a W n -subring of K ∗ , ∨ -definable in the reduct ( K ∗ , D ∗ ), and co-embeddablewith D ∗ . • R B is a W -subring (i.e., valuation ring) on K ∗ , ∨ -definable in ( K ∗ , D ∗ , B ∗ ), andco-embeddable with B ∗ . 12 R B ⊇ R D . Therefore R B contains the integral closure f R D . Writing this integralclosure as an intersection of valuation rings O ∩ · · · ∩ O n , there must be some i such that R B ⊇ O i , by Corollary 6.8 in [6].As in Part 1, the valuation ring O i is ∨ -definable in the reduct ( K ∗ , D ∗ ). By Propo-sition 4.6, there is some C ⊆ K ∗ definable in ( K ∗ , D ∗ ), and co-embeddable with O i .Then C, O i , R B , B ∗ are all co-embeddable. (The inclusion R B ⊇ O i forces the twovaluation rings to induce the same topology, hence to be co-embeddable.)The statement “some set definable in ( K ∗ , D ∗ ) is co-embeddable with B ∗ ” is expressedby a disjunction of first-order sentences, so it holds in the elementary substructure( K, D, B ). Therefore there is C ⊆ K definable in ( K, D ) and co-embeddable with B .So the V-topology σ is definable in ( K, D ).While we are here, we note an analogue of Proposition 3.5 in [2].
Proposition 4.11.
Let ( K , + , · , . . . ) be a sufficiently saturated field, possibly with extra struc-ture, and let τ be a definable W n -topology on K . Then τ is induced by a ∨ -definable, externallydefinable W n -ring R on K .Proof. Let D be a definable bounded neighborhood of 0. Rescaling, we may assume D is a W n -set. Passing to a reduct, we may assume the language is countable. Let K be a countableelementary substructure defining D . Let R be the union of all K -definable bounded sets.Then R is a ∨ -definable subring, by Lemma 2.1(b-c) in [10]. Also, R ⊇ D , so R is a W n -ring.By the remark at the start of [10], §
2, a union of countably many bounded sets in K is stillbounded. Therefore R is bounded. Then R is a bounded neighborhood of 0, so R inducesthe topology. Claim . If S , . . . , S n are K -definable bounded subsets, then there is c ∈ K × such that S ∪ · · · ∪ S n ⊆ cD . Proof.
There is c ∈ K × such that S ∪· · ·∪ S n ⊆ cD , by Lemma 2.1(e) in [10]. As S , . . . , S n , D are K -definable, we can choose c ∈ K × . (cid:3) Claim
It follows that R can be written as a directed union S c ∈ K × cD . Therefore R is externallydefinable and ∨ -definable. Definition 5.1.
Let K be a field. A golden lattice on K is a collection Λ of subgroups of( K, +), satisfying the following criteria: (Lattice) Λ is a bounded sublattice of Sub Z ( K ). In other words • , K ∈ Λ • If G, H ∈ Λ, then G ∩ H, G + H ∈ Λ.13
Scaling)
Λ is closed under the action of K × : if G ∈ Λ and c ∈ K × , then cG ∈ Λ. (Rank) Λ has finite cube rank. (Intersection)
Let Λ + = Λ \ { } . Then Λ + is closed under finite intersections. (Non-degeneracy) Λ is strictly bigger than { , K } . Example 5.2. If R is a W n -ring on K , then Sub R ( K ) is a golden lattice on K . Example 5.3.
Proposition 10.1 in [5] says that for certain dp-finite fields K (cid:23) K , the latticeof type-definable K -linear subspaces of K is a golden lattice.For the remainder of §
5, we assume the following:
Assumption 5.4. Λ is a golden lattice of rank r on a field K , and Λ + is the unboundedsublattice Λ \ { } . Lemma 5.5.
There is A ∈ Λ + such that c-rk Λ ( K/A ) = r = c-rk(Λ) .Proof (cf. Proposition 10.1(7) in [5]). If r = 1, then we can take an A ∈ Λ other than 0 and K , by the Non-degeneracy Axiom. Then c-rk( K/A ) ≥
1, because
A < K .Suppose r >
1. Let A be the base of a strict r -cube in Λ. By Proposition 9.15 in [5], thereexists a sequence B , B , . . . , B r in Λ such that each B i > A , and the B i are “independent”over A , meaning that ( B + · · · + B i ) ∩ B i +1 = A for all i < r . Taking i = 1, we see that B ∩ B = A. If A = 0, this contradicts the Intersection Axiom of golden lattices. Therefore A >
0, and A ∈ Λ + . Then the strict r -cube shows c-rk( K/A ) = r . Definition 5.6.
A finite set S ⊆ K is said to guard a group A ∈ Λ if for every B ∈ Λ, B ⊇ S = ⇒ B ≥ A. Lemma 5.7. If A ∈ Λ and c-rk Λ ( K/A ) = r , then A is guarded by some finite set S .Proof (cf. Proposition 10.4(2) in [5]). Increasing A , we may assume that A is the base of astrict r -cube in Λ. By Proposition 9.15 in [5], there are B , B , . . . , B r ∈ Λ such that each B i > A , and the B i are independent over A , in the sense that( B + · · · + B i − ) ∩ B i = A for 1 ≤ i < r . Take g i ∈ B i \ A , and let S = { g , . . . , g r } . We claim S guards A . Suppose C ∈ Λ and C ⊇ S . Let B ′ i = ( C + A ) ∩ B i . Then B ′ i ≤ B i , B ′ i ≤ C + A , and A ≤ B ′ i .Moreover, g i ∈ S ⊆ C ⊆ C + A , and g i ∈ B i , so g i ∈ B ′ i . As g i / ∈ A , it follows that A < B ′ i .For 1 ≤ i < r , we have A ≤ ( B ′ + · · · + B ′ i − ) ∩ B ′ i ≤ ( B + · · · + B i − ) ∩ B i = AA = ( B ′ + · · · + B ′ i − ) ∩ B ′ i .
14o the B ′ i are an independent sequence in the interval [ A, C + A ] ⊆ Λ. As the B ′ i are strictlygreater than A , it follows that c-rk Λ (( C + A ) /A ) ≥ r. On the other hand, by properties of cube rank (Proposition 9.28(1) in [5]) we have r = c-rk(Λ) ≥ c-rk Λ (( C + A ) / ( C ∩ A )) = c-rk Λ (( C + A ) /A ) + c-rk Λ ( A/ ( A ∩ C )) ≥ r + c-rk Λ ( A/ ( A ∩ C )) . Therefore c-rk Λ ( A/ ( A ∩ C )) = 0, implying A = A ∩ C . Thus C ⊇ A . This shows that A isguarded by S . Lemma 5.8. If S is a finite set and A ∈ Λ + , there is c ∈ K × such that cS ⊆ A .Proof. Let S = { b , . . . , b n } . Each b − i A is an element of Λ + , by the Scaling Axiom. By theIntersection Axiom, T ni =1 b − i A is in Λ + , hence non-zero. Take non-zero c ∈ T ni =1 b − i A . Then cb i ∈ A for all i . Equivalently, cS ⊆ A . Theorem 5.9. If Λ is a golden lattice on K , then Λ + = Λ \ { } is a neighborhood basis ofa W -topology on K . If Λ has rank r , then the topology is a W r -topology.Proof. We check the relevant local sentences (copied straight out of [10]). The variables
U, V, W will range over Λ + .First, Λ + is a filter base, by the Intersection Axiom: ∀ U ∀ V ∃ W : W ⊆ U ∩ V. Second, we verify non-discreteness: ∀ U : { } ( U. (1)This holds by definition of Λ + .Third, we check Hausdorffness: ∀ x ∈ K × ∃ V : x / ∈ V. (2)By the Non-degeneracy Axiom, there is some V ∈ Λ such that 0 < V < K . Then V ∈ Λ + ,and there is some x ∈ K × such that x / ∈ V . For any x ∈ K × , we have x = ( xx − ) x / ∈ ( xx − ) V ∈ Λ + , by the Scaling Axiom.Next, we check continuity of addition and subtraction: ∀ U ∃ V : V − V ⊆ U. (3)15ndeed, we can take V = U , since the elements of Λ are subgroups.Equations (1-3) ensure that we have a non-discrete Hausdorff group topology on ( K, +).Next, we check continuity of multiplication by a constant: ∀ U ∀ x ∃ V : xV ⊆ U. (4)If x = 0, we can take any V . Otherwise, we take V = x − U , using the Scaling Axiom.Next, we check continuity of multiplication near (0 , ∀ U ∃ V : V · V ⊆ U. (5)This will take a little work. By Lemma 5.5, there is V ∈ Λ + such that c-rk Λ ( K/V ) = r .By Lemma 5.7, there is a finite set S = { a , . . . , a n } ⊆ K guarding V . Let V = T ni =1 a − i U ,and V = V ∩ V . By the Scaling and Intersection Axioms, V and V are in Λ + . Suppose x, y ∈ V . Then x ∈ V and y ∈ V . For i = 1 , . . . , n , we have x ∈ a − i Ua i ∈ x − U. Thus S ⊆ x − U . Now x − U ∈ Λ by the Scaling Axiom, so S ⊆ x − U = ⇒ V ⊆ x − U = ⇒ y ∈ x − U = ⇒ xy ∈ U, as S guards V . As x, y were arbitrary elements of V , we have shown (5).Equations (4-5) now show that Λ + defines a ring topology on K .Next we check that the ring topology is locally bounded ∃ U ∀ V ∃ c ∈ K × : cU ⊆ V. (6)To verify this, use Lemma 5.5 to find U ∈ Λ + such that c-rk Λ ( K/U ) = r . By Lemma 5.7,there is a finite set S ⊆ K that guards U . Given any V ∈ Λ + , Lemma 5.8 gives c ∈ K × suchthat cS ⊆ V . Then cS ⊆ V ⇐⇒ S ⊆ c − V = ⇒ U ⊆ c − V ⇐⇒ cU ⊆ V. This proves (6).Lastly, we must verify the W r -condition. Take U as in the proof of (6), with U ∈ Λ + , andc-rk Λ ( K/U ) = r . Then U is a bounded neighborhood of 0. After rescaling, we may assume1 ∈ U . By Remark 3.5, it suffices to show that U is a W r -set. Let a , . . . , a r +1 be elementsof K . Because Λ has rank r , there is some i such that a U + · · · + a r +1 U = a U + · · · + a i − U + a i +1 U + · · · + a r +1 U. This differs from the local sentence given between Lemma 2.1 and Theorem 2.2 in [10]. But it isequivalent, by Lemma 2.1(d). a i = a i ∈ a i U ⊆ r +1 X j =1 a j U = X j = i a j U. As the a i ’s were arbitrary, we have shown that U is a W r -set.From the proof of Theorem 5.9, we extract the following useful fact: Lemma 5.10.
Let Λ be a golden lattice on K , of rank r . If A ∈ Λ and c-rk Λ ( K/A ) = r ,then A is bounded in the topology induced by Λ . Let T be a complete, dp-finite, unstable theory of fields (possibly with extra structure). Proposition 6.1.
Let K be a highly saturated monster model of T . • There is a small field K (cid:22) K such that the group J K of K -infinitesimals is co-embeddable with a definable set D . • The canonical topology on K is a definable W n -topology.Proof. Let k (cid:22) K be a magic subfield (Definition 8.3 in [6]), meaning that for every type-definable k -linear subspace G ⊆ K , we have G = G . Let Λ be the lattice of type-definable k -linear subspaces of K . By Proposition 10.1(1,2,6,7) in [5], this lattice is a golden lattice(Definition 5.1). Let Λ + be the non-zero elements of Λ. Let n be c-rk(Λ) ≤ dp-rk( K ). ByTheorem 5.9, Λ + is a neighborhood basis of 0 for some W r -topology τ on K . Take V ∈ Λ + abounded neighborhood of 0. By Proposition 10.1(4) in [5], there is a small field K such that J K ∈ Λ + and J K ⊆ V . Then J K is a bounded neighborhood of 0. By Proposition 4.1, J K isco-embeddable with a definable set D , and the W r -topology τ is defined by D . It remainsto show that τ is the canonical topology.After rescaling D , we may assume J K ⊆ D ⊆ eJ K for some e ∈ K × . Now J K is a filtered intersection of the K -definable canonical basicneighborhoods. Shrinking D , we may assume that D is a K -definable canonical basic neigh-borhood. Claim . If U is a K -definable canonical basic neighborhood, then there is c ∈ K × suchthat cD ⊆ U . Proof.
The sets U and D are definable over K , and K (cid:22) K , so it suffices to find c ∈ K × .Take c = e − : e − D ⊆ J K ⊆ U. (cid:3) Claim K to K . Therefore,if U is any K -definable canonical basic neighborhood, then there is c ∈ K × such that cD ⊆ U .It follows that D defines the canonical topology on K , which must agree with the definable W n -topology τ . Theorem 6.3. If K is an unstable field with dp-rk( K ) = n , then the canonical topology on K is a definable W n -topology.Proof. The proof of ([9], Theorem 6.27) applies here.Theorem 4.10 then yields
Corollary 6.4.
Every unstable dp-finite field admits a definable V-topology.
By Proposition 3.5 in [2], definable V-topologies on sufficiently saturated fields are in-duced by externally definable valuation rings. Given that the Shelah expansion of a dp-finitestructure is dp-finite, we conclude the following:
Corollary 6.5.
The henselianity conjecture for dp-finite fields implies the Shelah conjecturefor dp-finite fields.
This gives a smoother proof of the Shelah conjecture for positive characteristic dp-finitefields, where the henselianity conjecture is known (Theorem 2.8 in [5]).We can say the following more precise version of Corollary 6.4:
Theorem 6.6.
Let K be an unstable dp-finite field. The definable V-topologies on K areexactly the V-topological coarsenings of the canonical topology on K .Proof. Let τ be the canonical topology. If τ is a V-topological coarsening of τ , then τ isdefinable by Theorem 4.10. Conversely, suppose that τ is a definable V-topology. Let B bea definable bounded neighborhood. After rescaling B , we may assume that for any x, y ∈ K , x ∈ By or y ∈ Bx as this is the definition of a W -topology. Taking y = 1, we see that for any x , x ∈ B or 1 ∈ Bx, or equivalently, B contains one of x or 1 /x . Then B must have full dp-rank, as two copiesof it cover K . Then B − B is a neighborhood in τ , by Corollary 5.10 in [6]. Now the set B − B is a bounded neighborhood in τ , so the family of sets { a · ( B − B ) : a ∈ K × } is a neighborhood basis of 0 for τ . All these sets are neighborhoods in τ , and so τ must becoarser than τ . So τ is one of the V-topological coarsenings of τ .18 .1 Three conjectures If τ , . . . , τ n are field topologies on K , then they generate a minimal common refinement τ . This topology τ is also a field topology—except that it may be discrete. A basis ofneighborhoods is given by { U ∩ · · · ∩ U n : U ∈ τ , U ∈ τ , . . . , U n ∈ τ n } . Conjecture 6.7.
Let ( K, τ ) be a W-topological field of characteristic 0. Then τ is gen-erated by jointly independent topologies τ , . . . , τ n , and each τ i has a unique V-topologicalcoarsening.
The only real evidence for Conjecture 6.7 is the classification of W -topologies in § Theorem 6.8.
Conjecture 6.7 implies the Shelah conjecture for dp-finite fields.Proof.
By Corollary 6.5, it suffices to prove the henselianity conjecture for dp-finite fields.Suppose the henselianity conjecture fails. By the usual techniques , we get a dp-finite mul-tivalued field ( K, O , O ), where O and O are independent non-trivial valuation rings. ByLemma 2.6 in [5], K has characteristic 0. Let τ be the canonical topology on the structure( K, O , O ). Note that O and O define two distinct V-topological coarsenings of τ , byTheorem 6.6.Applying Conjecture 6.7 to τ , we decompose τ into independent τ , . . . , τ m . We claim m >
1. Otherwise, τ = τ , and then τ has a unique V-topological coarsening, a contradiction.So m ≥
2. Because each τ i is Hausdorff, there are U i ∈ τ i such that (1 + U i ) ∩ ( − U i ).Let U = U ∩ · · · ∩ U m . Then U ∈ τ . Claim . For every V ∈ τ , we have 1 + V (1 + U ) . Proof.
Shrinking V , we may assume V = V ∩ · · · ∩ V n , where each V i ∈ τ i . By continuity ofmultiplication, there are W i ∈ τ i such that (1 + W i ) ⊆ V i . Shrinking the W i , we mayassume W i = − W i and W i ⊆ U i . By joint independence of the τ i , we can find x ∈ ( − W ) ∩ (1 + W ) ∩ · · · ∩ (1 + W m ) . Then ± x ∈ W i for all i , so x ∈ V i for all i . Thus x ∈ V . On the other hand, x / ∈ (1 + U ) . Otherwise, one of x or − x is in 1 + U . • If x ∈ U , then x ∈ U . But x ∈ − W ⊆ − U . So the two sets 1 + U and − U fail to be disjoint. • If − x ∈ U , then − x ∈ U . But − x ∈ − − W ⊆ − U . Then the two sets1 + U and − U fail to be disjoint.Either way, this contradicts the choice of the U i . (cid:3) Claim See the proofs of Lemmas 9.7, 9.8 in [6], or Propositions 6.3, 6.4 in [9]. K be a saturated elementary extension of K , and let J K be the group of K -infinitesimals.By Proposition 5.17(4) in [6], 1 + J K ⊆ (1 + J K ) , where (1 + J K ) denotes the image of1 + J K under the squaring map. The two sets can be written as filtered intersections1 + J K = \ { V : V a K -definable canonical basic neighborhood } (1 + J K ) = \ { (1 + U ) : U a K -definable canonical basic neighborhood } . Therefore the following local sentence holds in K with its canonical topology τ : ∀ U ∈ τ ∃ V ∈ τ : 1 + V ⊆ (1 + U ) . This contradicts Claim 6.9.
Conjecture 6.10. If K is a perfect field of positive characteristic, and τ is a W n -topologyon K , then τ is generated by n independent V-topologies. This would imply the positive-characteristic case of the “valuation-type conjecture” (Con-jecture 10.1 in [6]), which says that the canonical topology on an unstable dp-finite field isa V-topology. This is false in characteristic 0 ( §
10 in [9]), but may still hold in positivecharacteristic.The evidence for Conjecture 6.10 is that it holds for W -topologies in odd characteristic,by combining Proposition 5.32 of [9] with the methods of § Conjecture 6.11. If R is a W n -ring on an algebraically closed field, then R is NTP , andthe burden of R is at most n . For example, this holds for • Multivaluation rings, by [8]. • The diffeovaluation W -rings constructed in [9], specifically the rings Q and R of § W -rings by ([9], Lemma 8.23), and have burden ≤ We prove a few miscellaneous results. In § § n independent V-topologies. We show that • The class of such field topologies is a local class. (This is probably well-known toexperts, and useful in § • Every such topology has weight n .Lastly, in § .1 Coarsenings of W-topologies Lemma 7.1.
Let τ, τ ′ be two ring topologies on K , with τ ′ coarser than τ (i.e., τ ′ ⊆ τ ). If τ is a W n -topology, then τ ′ is a W n -topology. Proof.
Suppose not. There are local sentences expressing that τ ′ is coarser than τ , that τ isa W n -topology, and that τ ′ is not a W n -topology. By Theorem 1.1 (and Remark 1.5) of [10],we may assume that ( K, τ, τ ′ ) is ω -complete. By Lemma 3.7, τ is induced by a W n -ring R on K .We claim that R is τ ′ -bounded. Indeed, if U ∈ τ ′ , then U ∈ τ , so there is c ∈ K × suchthat cR ⊆ U . By Lemma 2.1(d) in [10], the τ ′ -boundedness of R means that ∀ U ∈ τ ′ ∃ V ∈ τ ′ : V · R ⊆ U. (7) Claim . For every U ∈ τ ′ , there exists smaller M ∈ τ ′ such that M is an R -submodule of K . Proof.
Define a descending sequence of τ ′ -neighborhoods U = U ⊇ U ⊇ U ⊇ · · · , choosing U n +1 small enough that U n +1 ∪ ( U n +1 − U n +1 ) ∪ ( U n +1 · R ) ⊆ U n . This is possible using (7) and the fact that τ ′ is a group topology on ( K, +). Let M = T n U n .Then M ⊆ U = U , and M ∈ τ ′ by ω -completeness. Lastly, M is an R -submodule of K byconstruction. (cid:3) Claim
Let Λ + be the set of all R -submodules of K which are τ ′ -neighborhoods of 0. LetΛ = Λ + ∪ { } . Then Λ is a golden lattice (Definition 5.1): • Λ + is clearly closed under intersections and joins, proving the Lattice and IntersectionAxioms. • The Scaling Axiom holds because τ ′ is a ring topology. • The Rank Axiom holds because Sub R ( M ) has finite rank, and Λ is a sublattice. • The Non-degeneracy Axiom holds by applying Claim 7.2 to any neighborhood U ∈ τ ′ strictly smaller than K .By Theorem 5.9, Λ + -defines a W n -topology τ ′′ on K . Then τ ′′ ⊆ τ ′ , by definition of Λ + . Onthe other hand, τ ′ ⊆ τ ′′ by Claim 7.2. Thus τ ′ is a W n -topology, a contradiction. Lemma 7.3.
Let R ⊆ R ′ be two rings on K = Frac( R ) . If R is a W n -ring, then This looks easy, given that the local sentence appearing in Remark 3.4 is preserved in coarsenings. Butthe difficulty is showing that τ ′ is locally bounded in the first place. A coarsening of a locally bounded ringtopology need not be locally bounded. For example, the diagonal embedding of Q into Q p Q p induces afield topology on Q that is not locally bounded. This topology is a coarsening of the locally bounded ringtopology τ Z induced by Z . R ′ is a W n − -ring, or • R and R ′ are co-embeddable.Proof. Decreasing n , we may assume n = wt( R ) = c-rk R ( K ). By Lemma 2.2, c-rk R ′ ( K ) ≤ n ,so we may assume wt( R ′ ) = c-rk R ′ ( K ) = n . Then we must show that R and R ′ are co-embeddable. Let Λ and Λ ′ be the lattices of R -submodules and R ′ -submodules of K . ThenΛ and Λ ′ are golden lattices (Definition 5.1). Also, Λ ′ is a sublattice of Λ. By Lemma 5.5,there is A ∈ Λ ′ such that A is the base of a strict n -cube in Λ ′ . Then A is the base of a strict n -cube in Λ as well. Thenc-rk R ( K/A ) = c-rk R ′ ( K/A ) = n = c-rk R ( K ) = c-rk R ′ ( K ) . By Lemma 5.10, A is a bounded neighborhood in both τ R and τ R ′ . Then R is co-embeddablewith A , and A is co-embeddable with R ′ . Proposition 7.4. If τ is a W n -topology on a field K , and τ ′ is a strict coarsening, then τ ′ is a W n − -topology on K .Proof. As in Lemma 7.1, we may assume (
K, τ, τ ′ ) is ω -complete. Then τ is induced bya W n -ring R . The ring R is τ ′ -bounded. By Lemma 3.7, τ ′ is induced by some superring R ′ ⊇ R . Then Lemma 7.3 implies one of the following: • R ′ is a W n − -ring, implying that τ ′ is a W n − -ring. • R and R ′ are co-embeddable, implying that τ = τ ′ . V n -topologies Definition 7.5. A V n -topology on K is a locally bounded ring topology τ such that thefollowing local sentence holds: there are distinct q , . . . , q n ∈ K such that for any U ∈ τ ,there is c ∈ K × such that for all x ∈ K ,( { x } ∪ { / ( x − q i ) : 1 ≤ i ≤ n } ) ∩ cU = ∅ . Remark . An equivalent condition is that there is a bounded neighborhood U ∋ q , . . . , q n such that for every x ∈ K ,( { x } ∪ { / ( x − q i ) : 1 ≤ i ≤ n } ) ∩ U = ∅ . Note that V n -topologies form a local class. Non - V n -topologies form a local class as well. Lemma 7.7. If R is an intersection of n valuation rings on K , then R induces a V n -topologyon K . roof. First of all, R defines a locally bounded field topology because Frac( R ) = K by([6], Proposition 6.2(3)). Suppose τ R fails to be a V n -topology. Passing to an elementaryextension, we may assume that τ R is ω -complete. Let K be a subfield of K of size ℵ .Then K is bounded, by ω -completeness. Let R ′ be the ring generated by R and K .This ring continues to define τ R , by Proposition 1.8. Also R ′ is a multivaluation ring, byProposition 6.10 in [6]. If q , . . . , q n are arbitrary distinct elements of K , and if x ∈ K , then( { x } ∪ { / ( x − q i ) : 1 ≤ i ≤ n } ) ∩ R ′ = ∅ , by Lemma 5.24 in [7]. Lemma 7.8. If ( K, τ ) is an ω -complete V n -topology, then ( K, τ ) is induced by a ring R thatis an intersection of n valuation rings on K .Proof. Let U be a bounded neighborhood, and q , . . . , q n be as in Remark 7.6, so that forany x ∈ K , ( { x } ∪ { / ( x − q i ) : 1 ≤ i ≤ n } ) ∩ U = ∅ . Let K be a countable subfield containing the q i . Then K is bounded, by ω -completeness.By Proposition 1.8, τ is induced by some subring R containing U and K . Then R is a K -algebra, and for every x ∈ K , at least one of the numbers x, / ( x − q ) , . . . , / ( x − q n )lies in R . Then R is an intersection of n valuation rings, by Lemma 5.24 in [7]. Corollary 7.9.
Let K be a field with a ring topology τ .1. τ is a V n -topology if and only if ( K, τ ) is locally equivalent to a field with a topologyinduced by an intersection of n valuation rings.2. If τ is a V n -topology, then τ is a W n -topology, and hence a field topology.3. If τ is a V n -topology, then τ is a V m -topology for m > n .4. τ is a V -topology if and only if τ is a V-topology. Proposition 7.10. If τ , . . . , τ n are distinct V-topologies on a field K , and τ is the topologygenerated by τ , . . . , τ n (as in Corollary 4.3 of [10]), then τ is a V n -topology. On the otherhand, τ is not a W n − -topology, and therefore not a V n − -topology.Proof. As in the proof of Theorem 4.4 of [10], we may assume that (
K, τ , . . . , τ n , τ ) is ω -complete. Then each τ i is induced by a valuation ring O i . One easily sees that the topology τ is induced by R = O ∩ · · · ∩ O n . By Corollary 7.9, τ is a V n -topology.Now suppose that τ is a W n − -topology. Then some W n − -set U is a bounded neighbor-hood of 0 with respect to τ . The set U is bounded with respect to the coarser topologies τ i . By Proposition 1.8, we may coarsen the O i to ensure that U ⊆ O i for each i . Then R = O ∩ · · · ∩ O n contains U , hence is a W n − -ring. This contradicts Proposition 2.11. (The O i are pairwise incomparable, because they are independent .)23 roposition 7.11. Let τ be a V n -topology on a field K . Then τ is generated by n or fewerV-topologies on K .Proof. Let q , . . . , q n and U be as in Remark 7.6, so U is a bounded neighborhood of 0 and ∀ x ∈ K : U ∩ { x, / ( x − q ) , . . . , / ( x − q n ) } 6 = ∅ . Let ( K ∗ , U ∗ ) be a saturated elementary extension of ( K, U ), and let τ ∗ be the topologyinduced by U ∗ . Let K be the countable subfield of K generated by the q i . Let R ⊆ K ∗ be the ring generated by K and U ∗ . Note that R is ∨ -definable over K . As in the proofof Lemma 7.8, the ring R is a multivaluation ring inducing τ ∗ . Therefore R and U ∗ areco-embeddable. Let p , . . . , p m be the maximal ideals of R ; m is the number of valuationrings needed to define R , so m ≤ n . As in the proof of Theorem 4.10.1, each localization R p i is a valuation ring, and the induced V-topology is definable in the structure ( K ∗ , U ∗ ). Theresulting V-topologies generate τ ∗ , because R = T i R p i . Then the statement“ τ is the topology generated by m distinct definable V-topologies in ( K ∗ , U ∗ )”can be expressed by a disjunction of first-order sentences. Therefore it transfers to theelementary substructure ( K, U ).We summarize the situation below:
Theorem 7.12.
For every n , there is a local sentence σ n holding in ( K, τ ) if and only if τ is generated by n independent V-topologies. If ( K, τ ) | = σ n , then ( K, τ ) is a W n -topology,but not a W n − -topology.Remark . A coarsening of a V n -topology is again a V n -topology. This is Lemma 4.4 in[10]. Definition 7.14.
Let τ, τ ′ be two ring topologies on K . Then τ and τ ′ are independent ifevery τ -open set U intersects every τ ′ -open set V .This can be expressed via a local sentence: ∀ x ∀ y ∀ U ∈ τ ∀ V ∈ τ ′ ∃ z : ( z − x ∈ U and z − y ∈ V ) . Note that we can equivalently just say ∀ U ∈ τ ∀ V ∈ τ ′ : U + V = K. Lemma 7.15.
Let R be a W n -ring on K . For i = 1 , , let R i be a subring of K containing R . Then one of the following holds: • There is a V-topology coarser than both τ R and τ R . τ R and τ R are independent.Proof. Let Λ + be the set of R -submodules of M ≤ K satisfying the following equivalentconditions: • M is a neighborhood of 0 in both τ R and τ R . • There are non-zero ideals I ≤ R and I ≤ R such that I , I ⊆ M . • There are non-zero c , c ∈ K such that c R ⊆ M and c R ⊆ M .Then Λ + is an unbounded sublattice of Sub R ( K ), closed under scaling by K × . Let Λ = { } ∪ Λ + . Then Λ is a bounded sublattice of Sub R ( K ). It has rank at most n . Thus Λsatisfies all the axioms of golden lattices (Definition 5.1), except possibly non-degeneracy.If τ R and τ R are independent, we are done. Otherwise, there are ideals I ≤ R and I ≤ R such that I + I < K . Then I + I ∈ Λ, showing that Λ is a golden lattice.By Theorem 5.9, the sets Λ + define a W n -topology τ ′ on K . By definition of Λ, every τ ′ -neighborhood of 0 is a neighborhood of 0 in the topologies τ R and τ R . Thus τ ′ is a commoncoarsening of τ R and τ R . By theorem 4.10, there is a V-topology coarser than τ ′ , hencecoarser than τ R and τ R . Theorem 7.16.
Let τ , τ , τ be three W-topologies on K , with τ finer than τ and τ . Thenat least one of the following holds: • τ and τ are independent. • τ and τ share a common V-topological coarsening.Proof. Suppose τ and τ are dependent. As usual, we can find sets B , B , B such that • B i is a bounded neighborhood of 0 in τ i • B ⊆ B and B ⊆ B • ∈ B , and B is a W n -set for some n .Let ( K ∗ , B ∗ , B ∗ , B ∗ ) be a saturated elementary extension of ( K, B , B , B ). As usual, B ∗ i defines a topology τ ∗ i on K ∗ , and ( K ∗ , τ ∗ , τ ∗ , τ ∗ ) is locally equivalent to ( K, τ , τ , τ ). Inparticular, τ ∗ and τ ∗ are still dependent.Let R i be the ∨ -definable ring generated by B i . As usual, R i generates τ i . Then R con-tains the W n -set B , so R is a W n -ring. Because τ ∗ and τ ∗ are not independent, Lemma 7.15yields a V-topology coarser than both R and R . By Theorem 4.10, this V-topology is de-finable in the structure ( K ∗ , B ∗ , B ∗ , B ∗ ). The statement “there is a definable V-topologycoarser than the topologies induced by B ∗ and B ∗ ” is expressed by a disjunction of first-order sentences, so it holds in the elementary substructure ( K, B , B , B ).25 W n -rings and inflators Lemma 8.1.
Let R be a ring and M be a module. If c-rk R ( M ) ≥ n , then M has a subquotientthat is semisimple of length n .Proof. M has a subquotient isomorphic to L ni =1 N i for some non-trivial R -modules N i . Each N i has a simple subquotient N ′ i . Then L ni =1 N ′ i is a subquotient of M . Proposition 8.2.
Let R be a W n -ring on a field K . Then there is m ≤ n and ideals A ⊆ B ⊆ R such that1. B/A is a semisimple R -module of length m .2. The induced map ς : Dir K ( K ) → Dir R ( B/A )Sub K ( K i ) → Sub R ( B i /A i ) V ( V ∩ B i + A i ) /A i is a malleable m -inflator.3. If ς ′ is any mutation of ς , such as ς itself, and if R ′ is the fundamental ring of R , thenthere is c ∈ K × such that R ⊆ R ′ ⊆ cR. Therefore R ′ is a W n -ring co-embeddable with R .Proof. Let m = wt( R ) = c-rk R ( R ). Take submodules (i.e., ideals) A ⊆ B ⊆ R such that B/A is semisimple of length m . The first point holds. Note m = c-rk R ( K ), by Lemma 2.4.Then A is a “pedestal” in the lattice of R -submodules of K , and the second point follows byTheorems 8.5, 8.9, 8.12 in [7]. If ς ′ is a mutation of ς , then ς ′ is induced by another pedestal A ′ , of the form A ′ = b A ∩ · · · ∩ b k A, for some non-zero b i , by Proposition 10.15 in [7]. The the fundamental ring R ′ is exactly the“stabilizer” R ′ = { x ∈ K : xA ′ ⊆ A ′ } , by Proposition 8.10 in [7]. Certainly R ′ ⊇ R , as A ′ is an R -module. Also, A ′ is a boundedneighborhood of 0, because A is. Therefore, there is non-zero c ∈ A ′ , and there is non-zero c ∈ K × such that c A ′ ⊆ R . Then c c R ′ ⊆ c A ′ R ′ ⊆ c A ′ ⊆ R, showing that R ′ is embeddable into R . 26 .1 W -rings Applying the results of [9], we obtain the following fact about W -rings on fields of charac-teristic 0: Theorem 8.3.
Let K be a field of characteristic 0, and let R be a W -ring on K . Then oneof two things happens:1. R is co-embeddable with a ring of the form Q = { x ∈ K : val( x ) ≥ and val( ∂x ) ≥ } induced by some dense “diffeovaluation data” as in § S and c ∈ K × such that cS ⊆ R .Proof. Let ς be the malleable m -inflator as in Proposition 8.2. Then m = 1 or m = 2. Webreak into two cases: • Suppose no mutation ς ′ of ς is weakly multi-valuation type (Definition 5.27 in [7]). ByProposition 5.19 in [7], m >
1, so m = 2. By Corollary 8.27 in [9], some mutation ς ′ of ς is a “diffeovaluation inflator” (Definition 8.25 in [9]). Let R ′ be the fundamental ringof ς ′ . By Proposition 8.2, R ′ is co-embeddable with R . By the proof of Corollary 8.27in [9], R ′ is the desired set { x ∈ K : val( x ) ≥ ∂x ) ≥ } obtained from thediffeovaluation data. • Suppose some mutation ς ′ of ς is weakly multi-valuation type. By definition, thismeans that the fundamental ring R ′ of ς ′ contains eS for some e ∈ K × and somemultivaluation ring S . Then S is embeddable into R ′ , and by Proposition 8.2, R ′ isembeddable into R . W -topologies in characteristic 0 Lemma 8.4.
Let K be a field of characteristic 0. Let τ be a DV-topology in the sense of[9], Definition 8.18. Then τ is a W -topology, but not a V n -topology for any n .Proof. By definition of “DV-topology,” τ is locally equivalent to a diffeovaluation topologyin the sense of Definition 8.16, [9]. The W -topologies and V n -topologies are local classes,so we may assume τ is a diffeovaluation topology. Let Q and R be Q = { x ∈ K : val( x ) ≥ ∂x ) > } R = { x ∈ K : val( x ) ≥ ∂x ) ≥ } as in § Q and R are rings inducing τ . By Lemma 8.23 in [9], Q is a W -ring,and so τ is a W -topology.Suppose that τ is a V n -topology for some n . Then τ is induced by independent V-topologies τ , . . . , τ m . After passing to an elementary extension of the original diffeovalued27eld, we may assume that each τ i is induced by a valuation ring O i (not necessarily definablefrom the diffeovalued field structure). The fact that the τ i generate τ implies that R isco-embeddable with O ∩ · · · ∩ O m . This contradicts Lemma 8.29 in [9]. Theorem 8.5. If τ is a W -topology on a field of characteristic 0, then exactly one of thefollowing holds:1. τ is a V-topology.2. τ is generated by two independent V-topologies.3. τ is a DV-topology in the sense of [9], Definition 8.18.Moreover, all such topologies are W -topologies.Proof. Theorem 7.12 and Lemma 8.4 show that the three cases are all W -topologies, and thethree cases are mutually exclusive. It remains to show that the three cases are exhaustive.Each of the three cases is closed under local equivalence. For cases (1-2) this is byTheorem 7.12; for case (3) this is by fiat (in Definition 8.18 of [9]). So we may pass to alocally equivalent field. Therefore we may assume that τ is induced by a W -ring R . ByTheorem 8.3, one of two things happens: • R is co-embeddable with some ring of the form R ′ = { x ∈ K : val( x ) ≥ ∂x ) ≥ } induced by some dense diffeovaluation data as in § τ is a DV-topology. • There is a multi-valuation ring S such that aS ⊆ R . Then τ = τ R is a coarsening of τ S .The topology τ S is a V n -topology (Lemma 7.7). By Remark 7.13, τ is a V n -topology.By Theorem 7.12 and the fact that τ is a W -topology, it follows that τ is generatedby one or two independent V-topologies.Because non - V -topologies are a local class, we get an interesting corollary: Corollary 8.6.
DV-topologies (on fields of characteristic 0) are a local class.
Lemma 8.7. If τ is a DV-topology, then τ has exactly one V-topological coarsening.Proof. Otherwise, it would have two coarsenings τ , τ , by Theorem 4.10. Let τ + τ denotethe V -topology generated by τ and τ , as in Corollary 4.3 of [10]. Then τ + τ is coarser orequal to τ . By Theorem 7.12, τ + τ is a W -topology but not a W -topology. Proposition 7.4then forces τ = τ + τ . This contradicts Lemma 8.4.Lemma 8.7 can be used to prove that unstable fields of dp-rank 2 admit unique definableV-topologies (Proposition 6.2 in [9]). 28 cknowledgments. The author would like to thank • Meng Chen, for hosting the author at Fudan University, where this research was carriedout. • Yatir Halevi, whose questions prompted the current paper.
This material is based upon work supported by the National Science Foundation under Award No. DMS-1803120. Any opinions, findings, andconclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National ScienceFoundation.
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