HYPERFIELDS, TRUNCATED DVRS, AND VALUED FIELDS

JUNGUK LEE

Abstract. For any two complete discrete valued fields K_{1} and K_{2} of mixed characteristic with perfect residue fields, we show that if the n-th valued hyperfields of K_{1} and K_{2} are isomorphic over p for each n ≥ 1, then K_{1} and K_{2} are isomorphic. More generally, for n_{1}, n_{2} ≥ 1, if n_{2} is large enough, then any homomorphism, which is over p, from the n_{1}-th valued hyperfield of K_{1} to the n_{2}-th valued hyperfield of K_{2} can be lifted to a homomorphism from K_{1} to K_{2}. We compute such n_{2} effectively, which depends only on the ramification indices of K_{1} and K_{2}. Moreover, if K_{1} is tamely ramified, then any homomorphism over p between the first valued hyperfields is induced from a unique homomorphism of valued fields. Using this lifting result, we deduce a relative completeness theorem of AKE-style in terms of valued hyperfields.

We also study some relationships between valued hyperfields, truncated discrete valuation rings, and complete discrete valued fields of mixed characteristic. For a prime number p and a positive integer e and for large enough n, we show that a certain category of valued hyperfields is equivalent to the category of truncated discrete valuation rings of length n and the ramification indices e having perfect residue fields of characteristic p. Furthermore, in the tamely ramified case, we show that a subcategory of this category of valued hyperfields is equivalent to the category of complete discrete valued rings of mixed characteristic (0, p) having perfect residue fields.

1. Introduction

Our main object in this article is a valued hyperfield. A hyperfield is a field-like algebraic structure whose addition is multivalued, and a valued hyperfield is a hyperfield equipped with a valuation. A typical example is the quotient of a valued field by a multiplicative subgroup of the form 1 + 𝔪^{n} for the maximal ideal 𝔪 of the valuation ring, which is called the n-th valued hyperfield of a valued field. In this article, we consider a lifting problem of homomorphisms of the n-th valued hyperfields to homomorphisms of valued fields(See Corollary 3.11 and Theorem 3.18). Also we study relationships between certain categories of valued hyperfields, truncated discrete valuation rings, and discrete valued fields of mixed characteristic(See Theorem 4.8 and Theorem 4.10). At last, we prove a relative completeness theorem of AKE-style in terms of valued hyperfields(See Theorem 5.8).

M. Krasner in [11] introduced a notion of valued hyperfield and used it to do a theory of limits of local fields. In [7], P. Deligne did the theory of limits of local fields in a different way by defining a notion of triple, which consists of truncated discrete valuation rings and some additional data. Typical examples of a valued

The author is supported by the Narodowe Centrum Nauki grant no. 2016/22/E/ST1/00450. The author thanks the Institut Henri Poincaré for their nice hospitality during the author’s participating a trimester ”Model Theory, Combinatorics and Valued fields” held at the IHP. The author thanks Jaiung Jun for bringing J. Tolliver’s result to the author’s attention.

hyperfield and a truncated discrete valuation ring are the n-th valued hyperfield and the n-th residue ring of a valued field respectively, where the n-th residue ring is a quotient of a valuation ring by the n-th power of the maximal ideal. J. Tolliver in [17] showed that discrete valued hyperfields and triples are essentially same, stated by P. Deligne in [7] without a proof. In [14], W. Lee and the author showed that given complete discrete valued fields of mixed characteristic with perfect residue fields, any homomorphism between the n-th residue rings of the valued fields is lifted to a homomorphism between the valued fields for large enough n. From this, we ask the following question.

Question 1.1. Let K_{1} and K_{2} be discrete valued fields of mixed characteristic (0, p) having perfect residue fields. Suppose the n-th valued hyperfields of K_{1} and K_{2} are isomorphic for every n ≥ 1. Then are K_{1} and K_{2} are isomorphic? Moreover, is there N > 0 such that if the N-th valued hyperfiels of K_{1} and K_{2} are isomorphic, are K_{1} and K_{2} isomorphic?

The n-th valued hyperfield H_{n}(K) := K/(1 + 𝔪^{n}) of a valued field K contains automatically information of the multiplicative group K^{×}/(1 + 𝔪^{n}). By the multivalued addition of H_{n}(K), it also contains information of the n-residue ring of K too(See Fact 2.19). We compute the multivalued addition of the n-th valued hyperfields rather explicitly, where the multivalued addition of given two elements in H_{n}(K) is given as a ball and we can compute its center and its radius of the ball explicitly(See Lemma 3.1). In [4] and [12], it was considered some structures, called amc-structures, consisted with information of the n-th residue rings and the multiplicative groups K^{×}/(1+𝔪^{n}). For n ≥ 1, the n-th amc structure of a valued field K is a triple K_{n} := (R_{K}^{2n}, G_{K}^{n}, Θ_{n}), where R_{K}^{n} := R/t^{2(n−1)}𝔪, G_{K}^{n} := K^{×}/(1+t^{n−1}𝔪), and a map Θ_{n} : {x ∈ R_{K}^{2n} : x^{2} ≠ 0} → G_{K}^{n}, a + t^{2n}𝔪 ↦ a(1 + t^{n}𝔪). Here, R is the valuation ring of K, and t = 1 if the residue field is of characteristic 0 and t = p if the residue field is of characteristic p > 0. In [4], S. A. Basrab and F. V. Kuhlmann showed that for a valued field K, and for henselian valued fields L and F which are algebraic extension of K, if L_{n} and F_{n} are isomorphic over K_{n} for each n, then L and F are isomorphic over K(See [4, Corollary 1.4]). Moreover, if L and F are finite extension of K, then it is enough to consider whether L_{n} and F_{n} are isomorphic over K_{n} for large enough n. In [12], F. V. Kuhlmann showed that if L and F are additionally tame extensions of K, then L and F are isomorphic over K if L_{1} and F_{1} are isomorphic over K_{1}(See [12, Lemma 3.1]).

We also study relationships between discrete valued hyperfields, truncated discrete valuation rings, and complete discrete valued fields of mixed characteristic. Fix a prime number p and a positive integer e. Let C_{p,e} be the category of complete discrete valuation rings of mixed characteristic (0, p) having perfect residue fields and ramification index e. Let R_{p,e}^{n} be the category of truncated discrete valuation rings of length n with perfect residue fields such that p is in the e-th power of the maximal ideal but not in the e + 1-th power of the maximal ideal.

Fact 1.2. [14, Theorem 4.7, Example 3.7(2)] For large enough n, there is a lifting functor L : R_{p,e}^{n} → C_{p,e} satisfying several natural conditions. But two categories C_{p,e} and R_{p,e}^{n} are not equivalent.

We ask a question analogous to Fact 1.2 for valued hyperfields.

Problem 1.3. Fix a prime number p and a positive integer e. Let n be a positive integer. Find a suitable category H_{p,e}^{n} of valued hyperfields whose has a lifting functor L^{H} : H_{p,e}^{n} → C_{p,e} satisfying some natural conditions, and which makes H_{p,e}^{n} and C_{p,e} equivalent.

At last, we concern a question of relative completeness of AKE-style in terms of hyper valuedfields. S. A. Basarab in [3] showed that the theory of finitely ramified henselian valued fields of mixed characteristic is determined by the theory of the n-th residue ring for each n and the theory of value group.

Fact 1.4. [3, Corollay 3.1] Let K_{1} and K_{2} be finitely ramified henselian valued fields of mixed characteristic. Let R_{1,n} and R_{2,n} be the n-th residue rings of K_{1} and K_{2} respectively. Let Γ_{1} and Γ_{2} be the value groups of K_{1} and K_{2}. Then the following are equivalent:

(1) K_{1} ≡ K_{2}.

(2) R_{1,n} ≡ R_{2,n} for every n ≥ 1 and Γ_{1} ≡ Γ_{2}.

Question 1.5. Let K_{1} and K_{2} be finitely ramified henselian valued fields mixed characteristic. Let H_{n}(K_{1}) and H_{n}(K_{2}) be the n-th valued hyperfields rings of K_{1} and K_{2} respectively. If H_{n}(K_{1}) ≡ H_{n}(K_{2}) for every n ≥ 1, K_{1} ≡ K_{2}?

In Section 2, we recall basic notations and facts. In Section 3, we answer Question 1.1 positively. We also compute such N effectively. To lift a homomorphism of the n-th valued hyperfields, we consider homomorphisms of valued hyperfields which are over p. In Subsection 3.1, it is enough to check whether there is an isomorphism over p between two first valued hyperfields in the tamely ramified case. More precisely, any homomorphism over p between the first valued hyperfields is induced from a unique homomorphism between given tamely ramified complete discrete valued fields. In Subsection 3.2, we compute such N in Question 1.1 depending only on ramification indices of given complete discrete valued fields. To compute N effectively, we first show that a homomorphism over p of the n-th valued gives a unique homomorphism between Witt rings of given complete discrete valued fields of mixed characteristic with perfect residue fields. Using the structure theorem of totally ramified extension and Krasner’s lemma, we compute N effectively, which is depending only on the ramification index. In general, we can not drop the condition of being over p. In Section 4, we suggest a suitable category Ĥ_{p,e}^{n} of valued hyperfields whose morphisms are isometric homomorphism, and show that there is a lifting functor L^{H} : Ĥ_{p,e}^{n} → C_{p,e} satisfying proper conditions for large enough n. Actually, we show that this category Ĥ_{p,e}^{n} and the category R_{p,e}^{n} are equivalent for large enough n. This essentially comes from the result of J. Tolliver in [17]. We also show that the a subcategory H_{p,e}^{n} of Ĥ_{p,e}^{n} whose morphisms are over p is equivalent to C_{p,e} for every n > e if p does not divide e. In Section 5, using lifting result of homomorphisms of n-th valued hyperfields in Section 3, we give a positive answer of Question 1.5 for the case of perfect residue fields. In this case, it is enough to check whether N-th valued hyperfields are elementary equivalent for large enough N. Specially, if K_{1} and K_{2} are tamely ramified and the first valued hyperfields are elementary equivalent, then K_{1} and K_{2} are elementary equivalent.

2. Preliminaries

We introduce basic notations and terminologies which will be used in this paper. We denote a valued field by a tuple (K, R(K), 𝔪_{K}, ν_{K}, k(K), Γ_{K}) consisting of the following data : K is the underlying field, R(K) is the valuation ring, 𝔪_{K} is the maximal ideal of R(K), ν_{K} is the valuation map, k_{K} is the residue field, and Γ_{K} is the value group considering as an additive group. If there is no confusion, we omit K. Hereafter, the full tuple (K, R, 𝔪, ν, k, Γ) will be abbreviated in accordance with the situational need for the components. For γ ∈ Γ, we write Γ_{∗γ} := {x ∈ Γ| x ∗ γ} for ∗ ∈ {≥, >} and 𝔪^{γ} := {x ∈ K| ν(x) ∈ Γ_{>γ}}. Note that for each γ ∈ Γ_{≥0}, 1+𝔪^{γ} is a multiplicative subgroup of K^{×}. For γ ∈ Γ_{≥0}, let R_{γ} := R/𝔪^{γ}. We recall the definitions of ramified valued fields.

Definition 2.1. We say that a valued field (K, ν, Γ) is (absolutely) unramified if char(k) = 0, or char(k) = p and ν(p) is the minimal positive element in Γ for p > 0. We say (K, ν) is (absolutely) ramified if it is not absolutely unramified.

Definition 2.2. Let (K, ν, Γ) be a valued field whose residue field has prime characteristic p.

(1) We say (K, ν, Γ) is (absolutely) finitely ramified if the set {γ ∈ Γ| 0 < γ ≤ ν(p)} is finite. The cardinality of {γ ∈ Γ| 0 < γ ≤ ν(p)} is called the (absolute) ramification index of (K, ν), denoted by e(K, ν) or e(R). If K or ν is clear from context, we write e(K) or e for e(K, ν). For x ∈ R, we write e_{ν}(x) := |{γ ∈ Γ| 0 < γ ≤ ν(x)}|. If there is no confusion, we write e(x) for e_{ν}(x)

(2) Let (K, ν, Γ) be finitely ramified. If p does not divide e_{ν}(p), we say (K, ν) is (absolutely) tamely ramified. Otherwise, we say (K, ν) is (absolutely) wildly ramified.

Note that a discrete valued field having a residue field of characteristic p > 0 is finitely ramified. We say that a discrete valued field (K, ν, Γ) with the residue field having characteristic p > 0 is normalized if Γ is a subgroup of ℝ and ν(p) = 1. From now on, we mean a homomorphism between valued fields is an isometric homomorphism, where a field homomorphism f : K_{1} → K_{2} is called isometric if for a, b ∈ K_{1},

ν_{K_{1}}(a) < ν_{K_{1}}(b) ⇔ ν_{K_{2}}(f(a)) < ν_{K_{2}}(f(b)).

We recall some facts on the Witt ring and Teichmüller representatives of complete discrete valued fields of mixed characteristic(c.f. [16]).

Fact 2.3. (1) Let W (k_{1}) and W (k_{2}) be Witt rings of perfect residue fields k_{1} and k_{2} of characteristic p > 0 respectively. Suppose that there is a homomorphism ϕ : k_{1} ⟶ k_{2}. Then there is a unique lifting homomorphism g : R_{1} ⟶ R_{2} such that g induces ϕ.

(2) Let R be a complete discrete valuation ring of characteristic 0 with perfect residue field k of characteristic p and corresponding valuation ν. Then W(k) can be embedded as a subring of R and R is a free W(k)-module of rank ν(p). Moreover, R = W(k)[π] where π is a uniformizer of R.

(3) Let A be a ring that is Hausdorff and complete for a topology defined by a decreasing sequence 𝔞_{1} ⊃ 𝔞_{2} ⊃ ... of ideals such that 𝔞_{n} · 𝔞_{m} ⊂ 𝔞_{n+m}. Assume that the residue ring A_{1} = A/𝔞_{1} is a perfect field of characteristic p. Then:

(a) There exists one and only one system of representatives h : A_{1} ⟶ A which commutes with p-th powers: h(λ^{p}) = h(λ)^{p}. This system of representatives is called the set of Teichmüller representatives.

(b) In order that a ∈ A belong to S = h(A_{1}), it is necessary and sufficient that a be a p^{n}-th power for all n ≥ 0.

(c) This system of representatives is multiplicative which means

h(λμ) = h(λ)h(μ)

for all λ, μ ∈ A_{1}.

(d) S contains 0 and 1.

(e) S \ {0} is a subgroup of the unit group of A.

Next we introduce a notion of valued hyperfields.

Definition 2.4. [11, Definition 1.2 and 1.4]

(1) A hyeprfield is an algebraic structure (H, +, ·, 0, 1) such that (H^{×}, ·, 1), where H^{×} := H \ {0}, is an abelian group and there is a multivalued operation + : H × H → 2^{H} for the power set 2^{H} of H satisfying the followings:

(a) 0 · α = 0 for all α ∈ H.

(b) (Associative) (α + β) + γ = α + (β + γ) for all α, β, γ ∈ H.

(c) (Commutative) α + β = β + α for all α, β ∈ H.

(d) (Distributive) (α + β) · γ ⊂ α · γ + β · γ for all α, β, γ ∈ H.

(e) (Identity) α + 0 = {α} for all α ∈ H.

(f) (Inverse) For any α ∈ H, there is a unique −α ∈ H such that 0 ∈ α + (−α).

(g) For all α, β, γ ∈ H, α ∈ β + (−γ) if and only if β ∈ α + γ.

(2) A valued hyperfield is a hyperfield (H, +, ·, 0, 1) equipped with a map ν from H to Γ ∪ {∞} for an ordered abelian group Γ such that

(a) For α ∈ H, ν(α) = ∞ if and only if α = 0;

(b) For all α, β ∈ H, ν(α · β) = ν(α) + ν(β);

(c) For all α, β ∈ H, ν(α + β) ≥ min{ν(α), ν(β)};

(d) For all α, β ∈ H, ν(α + β) consists of single element unless 0 ∈ α + β; and

(e) There is ρ_{H} ∈ Γ such that either α + β is a closed ball of radius ρ_{H} + min{ν(α), ν(β)} for all α, β ∈ H, or α + β is a open ball of radius ρ_{H} + min{ν(α), ν(β)} for all α, β ∈ H^{×}.

For B ⊂ H and α ∈ H, define α + B := ⋃_{β∈B} α + β(*). The associativity of + means that given α, β, γ ∈ H, we have (α + β), (β + γ) ⊂ H and α + (β + γ) = (α+β)+γ in the sense of (∗). We say that H is discrete if Γ is a discrete subgroup of ℝ.

For α_{0}, …, α_{k} ∈ H, we write ∑^{H} α_{i} for (α_{0} +⋯+α_{k}) ⊂ H. Since the multivalued operation + is associative, the notion of ∑^{H} is well-defined.

Definition 2.5. Let (H_{i}, +_{i}, ·_{i}, 0_{i}, 1_{i}, ν_{i}) be a valued hyperfield for i = 1, 2. A map f from H_{1} to H_{2} is called a homomorphsim if the followings hold:

(1) f(0_{1}) = 0_{2} and f(1_{2}) = 1_{2}.

(2) f(α ·_{1} β) = f(α) ·_{2} f(β) for all α, β ∈ H_{1}.

(3) f(α +_{1} β) ⊂ f(α) +_{2} f(β) for all α, β ∈ H_{1}.

(4) For all α, β ∈ H_{1},

ν_{1}(α) ≤ ν_{1}(β) ⇔ ν_{2}(f(α_{1})) ≤ ν_{2}(f(β)).

Let Hom(H_{1}, H_{2}) be the set of homomorphism from H_{1} to H_{2}.

Note that the definition of homomorphisms in Definition 2.5 is weaker than one in [17, Definition 1.5]. We call homomorphisms in [17, Definition 1.5] an isometric homomorphism.

Definition 2.6. [17, Definition 1.5] Let (H_{i}, +_{i}, ·_{i}, 0_{i}, 1_{i}, ν_{i}) be a valued hyperfield for i = 1, 2. Suppose ν_{1}(H_{1}^{×}) = ν_{2}(H_{2}^{×}). A map f from H_{1} to H_{2} is called an isometric homomorphism if the followings hold:

(1) f(0_{1}) = 0_{2} and f(1_{2}) = 1_{2}.

(2) f(α ·_{1} β) = f(α) ·_{2} f(β) for all α, β ∈ H_{1}.
(3) f^{−1}(α +_{2} β) = f^{−1}(α) +_{1} f^{−1}(β) for all α, β ∈ f(H_{1}).
(4) For all α ∈ H_{1}, ν_{1}(α) = ν_{2}(f(α)).
(5) f^{−1}(1_{2}) is a ball.

Let Iso(H_{1}, H_{2}) be the set of isometric homomorphisms from H_{1} to H_{2}. Note that (3) implies that for α, β ∈ H_{1} and f ∈ Hom(H_{1}, H_{2}), f(α +_{1} β) ⊂ f(α) +_{2} f(β) so that Iso(H_{1}, H_{2}) ⊂ Hom(H_{1}, H_{2}).

Definition 2.7. Let γ ∈ Γ_{≥0}. The γ-hyperfield of K is a hyperfield (K/(1 + 𝔪^{γ}), +_{H}, ·_{H}), denoted by H_{γ}(K), such that for each a(1+𝔪^{γ}), b(1+𝔪^{n}) ∈ K/1+𝔪^{n},

(1) a(1 + 𝔪^{γ}) ·_{H} b(1 + 𝔪^{γ}) := ab(1 + 𝔪^{γ}),

(2) a(1 + 𝔪^{γ}) +_{H} b(1 + 𝔪^{γ}) := {(x + y)(1 + 𝔪^{γ})| x ∈ a(1 + 𝔪^{γ}), y ∈ b(1 + 𝔪^{γ})},

and Conventionally, we write 0 for 0(1 + 𝔪^{γ}) and 1 for 1(1 + 𝔪^{γ}). The valuation ν of K induces a map ν_{H} on K/(1 + 𝔪^{γ}) sending ν_{H}(a(1 + 𝔪^{γ})) to ν(a). We call (K/(1 + 𝔪^{γ}), +_{H}, ·_{H}, ν_{H}) the valued γ-hyperfield. For γ ≤ λ ∈ Γ_{>0}, we have that (1+𝔪^{λ}) ⊂ (1+𝔪^{γ}) and it induces a projection H_{γ}^{λ} : H_{λ}(K) → H_{γ}(K), a(1+𝔪^{λ}) ↦ a(1 + 𝔪^{γ}). We write H_{γ} : K → H_{γ}(K), x ↦ [x]_{γ}.

For A ⊂ K, let H_{γ}(A) := {a(1 + 𝔪^{γ}) : a ∈ A} and for a ∈ K, we write [a]_{n} for a(1 + 𝔪^{γ}) ∈ H_{γ}(K). If γ is obvious, we write [a] for [a]_{γ}. Note that the valuation ν_{H} : H_{γ}^{×}(K) → Γ is a group epimorphism with the kernel H_{γ}(R^{×}).

Remark 2.8. The quotient group H_{γ}^{×}(K)/H_{γ}(R^{×}) is an ordered group isomorphic to Γ.

Definition 2.9. Let K_{1} and K_{2} be valued fields whose residue fields are of characteristic p > 0. Let H_{γ}(K_{1}) and H_{λ}(K_{2}) be valued hyper fields of K_{1} and K_{2} respectively. We say a homomorphism f : H_{γ}(K_{1}) → H_{λ}(K_{2}) is over p if f([p]) = [p].

Note that there is an isometric isomorphism which is not over p(See Example 4.5). In Section 4, we see how two sets of isometric homomorphisms and of isometric homomorphisms over p are different(See Theorem 4.8 and Theorem 4.11).

Notation 2.10. Let (K, ν) be a finitely ramified valued field and let π be a uniformizer of K. Let n ≥ m be positive integers. We write 𝔪^{n} := 𝔪^{ν(π^{n−1})}. We write R_{n}(K) := R(K)/𝔪^{n} and H_{n}(K) := K/(1 + 𝔪^{n}). We call R_{n}(K) the n-th residue ring of K and H_{n}(K) the n-th valued hyperfield of K. We write [x]_{n} := x(1 + 𝔪^{n}) for x ∈ K. We denote H_{m}^{n} and H_{m} for H_{ν(π^{m−1})}^{ν(π^{n−1})} and H_{ν(π^{m−1})} respectively.

We now recall some results on hyperfields and (Deligne’s) triples in [17].

Definition 2.11. [17, Definition 1.8] A truncated discrete valuation ring, in short a truncated DVR, is a principal Artinian local ring. Let R be a truncated DVR of length l. For x ∈ R, we define ν_{R}(x) = sup{i ∈ ℕ| x ∈ 𝔪_{R}^{i}}, where 𝔪_{R} is the maximal ideal of R. Then ν_{R}(R) = {0, 1, …, l − 1} ∪ {∞}, and ν_{R}(x) = ∞ if and only if x = 0.

Definition 2.12. [17, Definition 1.9] A (Deligne’s) triple (R, M, ϵ) consists of a truncated DVR R, a free R-module M of rank 1, and a R-module homomorphism ϵ : M → R whose image is the maximal ideal of R.

Definition 2.13. [17, Definition 1.10] Let T_{1} = (R_{1}, M_{1}, ϵ_{1}) and T_{2} = (R_{2}, M_{2}, ϵ_{2}) be triples. A morphism of triples (r, f, η) : T_{1} → T_{2} consists of a homomorphism f : R_{1} → R_{2}, an integer r, called the ramification index, and an R_{1}-module homomorphism η : M_{1} → M_{2}^{⊗r} such that

• f ∘ ϵ_{1} = ϵ_{2} ∘ η; and

• η induces a R_{2}-module isomorphism from M_{1} ⊗_{R_{1}} R_{2} to M_{2}^{⊗r}.

The composition of morphisms (r_{1}, f_{2}, η_{1}) and (r_{2}, f_{2}, η_{2}) of triples is given by

(r_{1}, f_{2}, η_{1}) ∘ (r_{2}, f_{2}, η_{2}) = (r_{1}r_{2}, f_{1} ∘ f_{2}, η_{1}^{⊗r_{2}} ∘ η_{2}).

Remark 2.14. [17, Remark 4.5] Let T = (R, M, ϵ) be a triple. Let M be a free R-module of rank 1 and Π be a generator. For each k ∈ ℤ, the tensor power M^{⊗k} is a well-defined R-module of rank 1. More precisely,

M^{⊗k} = \begin{cases}R(Π^{⊗k}) & if k > 0 \\ R & if k = 0 \\ Hom_{R}(M^{⊗(−k)}, R) = R(Π^{⊗k}) & if k < 0 \end{cases} ,

where for k < 0, Π^{⊗k} ∈ Hom_{R}(M^{⊗(−k)}, R) is a unique homomorphism sending Π^{⊗(−k)} to 1. Define a map ν_{T} : ⋃_{k∈ℤ} M^{⊗k} → ℤ ∪ {∞}, called a valuation map of

T, as follows: For x = rΠ^{⊗k} ∈ M^{⊗k}, ν_{T} (x) = ν_{R}(r) + k.

Notation 2.15. Let (H, ν) be a discrete valued hyperfield. Denote θ_{H} := min{ν(x)| ν(x) > 0, x ∈ H}.

Remark/Definition 2.16. [17, Remark 4.1] Let H be a discrete valued hyperfield. There is a positive integer l such that ρ_{H} = lθ_{H}. Such l is called the length of H, denoted by l(H).

Example 2.17. For a discrete valued field K, we have that l(H_{n}(K)) = n.

Definition 2.18. Let (H, ν) be a discrete valued hyperfield. For η ∈ ℝ, define an equivalence relation ≡_{η} on H as follows: For α, β ∈ H, α ≡_{η} β if and only if ν(α − β) ≥ η. For α ∈ H, we write [α]_{η} for the ≡_{η}-class of α. If η is obvious, we omit it. Denote O_{H} := {α ∈ H| ν(α) > 0} and 𝔪_{H} := {α ∈ H| ν(α) ≥ θ_{H}}.

Fact 2.19. [17, Sections 4 and 5] Let H be a discrete valued hyperfield. Let R = O_{H}/ ≡_{ρ_{H}}, M = 𝔪_{H}/ ≡_{ρ_{H}+θ_{H}}, and ϵ : M → R, [α]_{ρ_{H}+θ_{H}} ↦ [α]_{ρ_{H}} for α ∈ H. Then,

(1) Tr(H) = (R, M, ϵ) is a triple, and

(2) R is of length l(H) and the maximal ideal 𝔪_{R} is M/ ≡_{ρ_{H}}. We write R_{l}(H) for R. If H or l is obvious, we write R_{l} or R for R_{l}(H). For discrete valued hyperfields H′ and H”, and for f ∈ Iso(H, H′) and g ∈ Iso(H′, H′′), we have that (1) Tr(f) and Tr(g) are homomorphisms between Tr(H), Tr(H′), and Tr(H′′)

respectively, and

(2) Tr(g ∘ f) = Tr(g) ∘ Tr(f).

Remark/Definition 2.20. Let H be a discrete valued hyperfield and Tr(H) = (R, M, ϵ). The set O_{H}/ ≡_{θ_{H}} forms a field and it is isomorphic to R/𝔪_{R}. The field O_{H}/ ≡_{θ_{H}} is called the residue field of H, denoted by k(H). Moreover, for each g ∈ Iso(H_{1}, H_{2}), it induces a homomorphism k(g) : k(H_{1}) → k(H_{2}).

Proof. To show O_{H}/ ≡_{θ_{H}} forms a field, mimic the proof of [17, Lemma 4.2]. And consider a map f : O_{H}/ ≡_{θ_{H}} → R sending [α]_{θ_{H}} to [α]_{ρ_{H}} + 𝔪_{R}. Since θ_{H} ≤ ρ_{H}, f is well-defined and it induces an isomorphism. Let H_{1} and H_{2} be discrete valued hyperfield and let g : H_{1} → H_{2} be an isometric homomorphism. Then it induces Tr(g) : Tr(H_{1}) → Tr(H_{2}), which induces a homomorphism k(g) from k(H_{1}) to k(H_{2}). □

Remark 2.21. [17, Section 6] Let T = (R, M, ϵ) be a triple with the valuation map ν_{T}. Define U(T) := {0} ∪ ⋃_{i∈ℤ}{x ∈ M^{⊗i}| ν_{T} (x) = i}. Then U(T) is a discrete

valued hyperfield. Moreover, if T = Tr(H) for a discrete valued hyperfield H, then U(T) ≅ H by an isometric isomorphism after rescaling ν_{T} (Π) = θ_{H} for a generator Π of M. Moreover the assignments Tr and U are funtorial. For discrete valued hyperfields H_{1} and H_{2}, each f ∈ Iso(H_{1}, H_{2}) induces a morphism Tr(f) ∈ Hom(Tr(H_{1}), Tr(H_{2})). For triples T_{1} and T_{2}, each g ∈ Hom(T_{1}, T_{2}) induces a morphism U(g) ∈ Iso(U(T_{1}), U(T_{2})).

Example 2.22. [17, Example 4.9] Let K be a discrete valued field. Let H_{n}(K) be the n-th valued hyperfield of K for a positive integer n. Then Tr(H_{n}(K)) ≅ (R(K)/𝔪_{K}^{n}, 𝔪_{K}/𝔪_{K}^{n+1}, ϵ), where 𝔪_{K} is the maximal ideal of the valuation ring R(K) and the map ϵ is induced by the inclusion 𝔪_{K} ⊂ R(K).

We introduce some algebraic and model theoretic structural theorems in terms of the n-th valued rings for finitely ramified valued fields.

Remark/Definition 2.23. Let R be a complete discrete valuation ring of mixed characteristic. Let π be a uniformizer of R and ν corresponding valuation of R. Let L and K be the fraction fields of R and W(k) respectively. We denote the maximal number

max {ν(π − σ(π)) : σ ∈ Hom_{K}(L, L^{alg}), σ(π) ≠ π}

by M(R)_{π} or M(L)_{π}. This M(R)_{π} does not depend on the choice of π and we write M(R) = M(L).

Let (F, ν_{F}) be a complete discrete valuation ring of mixed characteristic having the same ramification index with L. Suppose there is an homomorphism from L to F. Then M(F) = M(L)

Remark 2.24. [14, Theorem 3.9] Let (K_{1}, ν_{1}) and (K_{2}, ν_{2}) be complete discrete valued fields of mixed characteristic (0, p). Then we have that

M(K_{1})e_{ν_{1}}(p)e_{ν_{2}}(p) ≤ e_{ν_{2}}(p)(1 + e_{ν_{1}}^{2}(p)).

To determine whether two complete discrete valued fields of mixed characteristic with perfect residue fields are isomorphic, it is enough to check whether the n-th residue rings are isomorphic for large enough n.

Fact 2.25. [14] Let K_{1} and K_{2} be complete discrete valued fields of mixed characteristic with perfect residue fields. Let R_{1,n} and R_{2,n} be the n-th residue rings of K_{1} and K_{2} respectively. For n > e_{ν_{2}}(p)(1 + e_{ν_{1}}^{2}(p)), if R_{1,n} and R_{2,n} are isomorphic, then K_{1} and K_{2} are isomorphic. More generally, for positive integers n_{1} and n_{2} with n_{2} > e_{ν_{2}}(p)(1 + e_{ν_{1}}^{2}(p)), any homomorphism from R_{1,n_{1}} to R_{2,n_{2}} is lifted to a homomorphism from K_{1} to K_{2}.

For model theory of valued fields, we take the following languages for valued fields and their related structures. Let L_{K} = {+, −, ·; 0, 1; |} be a ring language with a binary relation | for valued fields, where we interpret the binary relation | as a|b if ν(a) ≤ ν(b) for a, b ∈ K. Let L_{k} = {+′, −′, ·′; 0′, 1′} be the ring language for residue fields, and L_{Γ} = {+^{∗}; 0^{∗}; <} be the ordered group language for value groups. For each n ≤ 1, let L_{R_{n}} = {+_{n}, −_{n}, ·_{n}; 0_{n}, 1_{n}} be the ring language for the n-th residue ring. For n = 1, we identify L_{R_{1}} = L_{k}. We use a language L_{vhf} := {0, 1, ·, +, |} for valued hyperfields, where 0, 1 are constant symbols, · is a binary function symbol, + is a ternary predicate, and | is a binary relation. For a valued hyperfield (H, ·, +, ν), 1^{H} is interpreted as the identity of the multiplication, 0^{H} as the identity of the addition, ·^{H} as the multiplication function on H. For α, β, γ ∈ H, (α, β, γ) ∈ +^{H} if and only if γ ∈ α + β, and (α, β) ∈ |^{H} if and only if ν(α) ≤ ν(β). For the convention, (α, 0^{H}) ∈ |^{H} for all α in H.

J. Ax and S. Kochen in [1], and Y. Ershov in [8] independently proved that the first order theories of unramifieid valued fields of characteristic 0 are determined by the first order theories of their residue fields and valued groups.

Fact 2.26. [1, 8](Ax-Kochen-Ershov principle) Let (K_{1}, k_{1}, Γ_{1}) and (K_{2}, k_{2}, Γ_{2}) be unramified henselian valued fields of characteristic zero. Then we have that

K_{1} ≡ K_{2} if and only if k_{1} ≡ k_{2} and Γ_{1} ≡ Γ_{2}.

In [3], S. A. Basarab generalized Fact 2.26 to the case of finitely ramified valued fields and in [14], W. Lee and the author improved the result of Basarab in the case of perfect residue fields.

Fact 2.27. [3, 14] Let (K_{1}, ν_{1}, k_{1}, Γ_{1}) and (K_{2}, ν_{2}, k_{2}, Γ_{1}) be henselian valued fields of mixed characteristic (0, p) with finite ramification indices. Suppose k_{1} and k_{2} are perfect fields. Let n > e_{ν_{2}}(p)(1 + e_{ν_{1}}^{2}(p)). The following are equivalent:

(1) K_{1} ≡ K_{2};

(2) Γ_{1} ≡ Γ_{2} and R_{n}(K_{1}) ≡ R_{n}(K_{2}).

3. Lifting

In this section, we aim to show that for large enough n, any homomorphism over p between the n-th valued hyperfields of discrete complete valued fields can be lifted to a homomorphism of given valued fields.

Lemma 3.1. Let a, b ∈ K and a_{0}, …, a_{k} ∈ K. Fix γ ∈ Γ_{≥0}.

(1) ⋃[a]_{γ} = a(1 + 𝔪^{γ}) = {x| ν(x − a) ≥ γ + ν(a)}.

(2) ⋃[a]_{γ} +_{H} [b]_{γ} = {x ∈ K| ν(x − (a + b)) > γ + min{ν(a), ν(b)}}.
(3) 0 ∈ ⋃[a]_{γ} +_{H} [b]_{γ} if and only if ⋃[a]_{γ} +_{H} [b]_{γ} = 𝔪^{γ+min{ν(a),ν(b)}}.
(4) (a_{0} + … + a_{k}) ∈ ∑^{H}[a_{i}]_{γ}.

(5) Suppose b ∈ ⋃ ∑^{H}[a_{i}]_{γ} and a_{0}, …, a_{k} ∈ R. Then b = (a_{0} + … + a_{k}) + d

for some d ∈ 𝔪^{γ}.

Proof. Let a, b ∈ K and a_{0}, …, a_{k} ∈ K.

(1) For x ∈ K, we have that

x ∈ a(1 + 𝔪^{γ}) ⇔ x = a + ad, d ∈ 𝔪^{γ}
⇔ ν(x − a) = ν(ad) = ν(a) + ν(d), d ∈ 𝔪^{γ}
⇔ ν(x − a) > ν(a) + γ.

(2) (⊆) Let x ∈ ⋃[a]_{γ} +_{H} [b]_{γ} so that thee are c, d ∈ 𝔪^{γ} such that x = a + b + ac + bd. Then we have that

ν(x − (a + b)) = ν(ac + bd) ≥ min{ν(ac), ν(bd)} > γ + min{ν(a), ν(b)}.

(⊇) Take x ∈ K such that ν(x−(a+b)) > γ+min{ν(a), ν(b)}. WLOG we may assume that ν(a) ≥ ν(b). Then there is c ∈ 𝔪^{γ} such that x = (a+b)+bc = a+b(1+c), and [x]_{γ} ∈ [a]_{γ} +_{H} [b]. So, we conclude that x ∈ ⋃[a]_{γ} +_{G} [b]_{γ}.

(3) By (2), ⋃[a]_{γ} +_{H} [b]_{γ} = (a + b) + 𝔪^{γ+min{ν(a),ν(b)}}. Thus we have that

0 ∈ ⋃[a]_{γ} +_{H} [b]_{γ} ⇔ ν(a + b) > γ + min{ν(a), ν(b)}
⇔ (a + b) ∈ 𝔪^{γ+min{ν(a),ν(b)}}
⇔ ⋃[a]_{γ} +_{H} [b]_{γ} = 𝔪^{γ+min{ν(a),ν(b)}}.

(4) and (5) come from (2). □

Proposition 3.2. Let K_{1} and K_{2} be valued fields whose residue fields are of characteristic p > 0. Let H_{γ}(K_{1}) and H_{λ}(K_{2}) be valued hyper fields of K_{1} and K_{2} respectively. Let f ∈ Hom(H_{γ}(K_{1}), H_{λ}(K_{2})) be over p. Then for all n ∈ ℤ, f([n]) = [n]. We denote Hom_{ℤ}(H_{γ}(K_{1}), H_{λ}(K_{2})) for the set of all homomorphisms over p.

Proof. Let f ∈ Hom(H_{γ}(K_{1}), H_{λ}(K_{2})) be over p. First, we have f([1]) = [1] because [1] is the multiplicative identity. Since [−1] is the additive inverse of [1], f([−1]) = [−1]. By Lemma 3.1(2), we have f([k]) = [k] for 1 ≤ k ≤ p − 1 inductively. Now choose an integer n arbitrary. Since f([−1]) = [−1] and f([0]) = [0], we may assume that n > 0. Suppose n and p are relatively prime. We write n = a_{0} + a_{1}p + a_{2} + … + a_{m}p^{m} with 0 ≤ a_{i} ≤ p − 1 and a_{0} ≠ 0 for some m ≥ 0. Suppose f([n]) = [n′] for some n′ ∈ K_{2}. Then we have that

[n′] = f([n]) = f([∑_{i} a_{i}p^{i}])

∈ ∑_{i}^{H} f([a_{i}p^{i}]) = ∑_{i}^{H} f([a_{i}])f([p])^{i} = ∑_{i}^{H}[a_{i}][p]^{i} = ∑_{i}^{H}[a_{i}p^{i}].

By Lemma 3.1(5), we have that n′ = ∑_{i} a_{i}p^{i} + d = n + d for some d ∈ 𝔪_{2}^{λ}. Since n ∉ 𝔪_{2}, we have that n′/n = 1 + d/n ∈ 1 + 𝔪_{2}^{λ} and n′(1 + 𝔪_{2}^{λ}) = n(1 + 𝔪_{2}^{λ}). Now suppose p divides n. Write n = n_{0}p^{l} for some l > 0 and for some n_{0} coprime to p. Then we have that f([n]) = f([n_{0}p^{l}]) = f([n_{0}])f([p^{l}]) = [n_{0}][p]^{l} = [n_{0}p^{l}] = [n]. □

Remark 3.3. Let (K_{1}, ν_{1}, k_{1}) and (K_{2}, ν_{2}, k_{2}) be finitely ramified valued fields having the same ramification index e. Let p = char(k_{1}) = char(k_{2}) > 0. Suppose ν_{1} and ν_{2} are normalized, that is, ν_{1}(p) = ν_{2}(p) = 1. Then for any n, m ≥ 1 and f ∈ Hom_{ℤ}(H_{n}(K_{1}), H_{m}(K_{2})), we have that ν_{1}(α) = ν_{2}(f(α)) for every α ∈ H_{n}(K_{1}). And any homomorphism from K_{1} to K_{2} induces an isometric homomorphism from H_{n}(K_{1}) to H_{n}(K_{2}).

In the remaining part of this section, we assume that a complete discrete complete valued field of mixed characteristic (0, p) has the normalized valuation so that ν(K^{×}) ⊂ ℝ and ν(p) = 1. For a henelian valued field (K, ν) of characteristic 0, there is a unique valuation on K^{alg} extending ν and we use the same notion ν for this valuation on K^{alg}.

Remark 3.4. Let S be the set of Teichmüller representatives of a complete discrete valued field K of mixed characteristic having a perfect residue field. Then, H_{1}(S) is a field which is isomorphic to the residue field of K.

Proof. Let H = (H_{1}(K), +_{1}, ×_{1}) be the first valued hyperfield of K. For each a, b ∈ S, there is a unique c ∈ S such that [c] ∈ [a] +_{1} [b]. So, (H_{1}(S), +, ×_{1}) forms a field where for a, b ∈ S, [a] + [b] = [c] if [c] ∈ [a] +_{1} [b]. Consider a map sending [a] to a + 𝔪, where 𝔪 is the maximal ideal of the valuation ring of K and this induces an isomorphism from H_{1}(S) to the residue field of K. □

Lemma 3.5. Let K_{1} and K_{2} be complete discrete valued fields of mixed characteristic having perfect residue fields. Let S_{1} and S_{2} be the set of Teichmüller representatives of K_{1} and K_{2} respectively, and let f ∈ Hom(H_{n}(K_{1}), H_{m}(K_{2})).

(1) We have f(H_{n}(S_{1})) ⊂ H_{m}(S_{2}) and f ↾_{H_{n}(S_{1})} is injective.

(2) If f is over p, then f ↾_{H_{n}(W(k_{1}))} is induced by a unique homomorphism from W(k_{1}) to W(k_{2}), where W(k_{1}) and W(k_{2}) are Witt subrings of K_{1} and K_{2} respectively.

Proof. (1) The proof is similar to the proof of [14, Lemma 3.3]. Let R_{1} and R_{2} be the valuation rings of K_{1} and K_{2} respectively. By Fact 2.3(3), S_{i} is contained in W(k_{i})^{×} where k_{i} is the residue field of K_{i} for i = 1, 2. For each λ ∈ S_{1}, let η_{s} be any representative of f(λ^{1/p^{s}}(1 + 𝔪_{1}^{n})) so that (η_{s})^{p^{s}}(1 + 𝔪_{2}^{m}) = f(λ(1 + 𝔪_{1}^{n})). Since η_{s} is in W(k_{2})^{×}, we have that η_{s}(1 + 𝔪_{2}^{m}) = η_{s} + 𝔪_{2}^{m} and (η_{s})^{p^{s}}(1 + 𝔪_{2}^{m}) = (η_{s})^{p^{s}} + 𝔪_{2}^{m} as a set. For any other representative θ_{s} of f(λ^{1/p^{s}}(1 + 𝔪_{1}^{n})), we have that η_{s} + 𝔪_{2}^{m} = θ_{s} + 𝔪_{2}^{m}. If we write η_{s} = θ_{s} + π_{2}^{n_{2}}a for some a in R_{2}, the following binomial expansion

η_{s}^{p^{s}} = (θ_{s} + π_{2}^{n_{2}}a)^{p^{s}}
= θ_{s}^{p^{s}} + p^{s}θ_{s}^{p^{s}−1}π_{2}^{n}a + ... + (π_{2}^{n}a)^{p^{s}}

shows η_{s}^{p^{s}} − θ_{s}^{p^{s}} ∈ 𝔪_{2}^{s}. Since η_{s+1}^{p} is a representative of f(λ^{1/p^{s}}(1 + 𝔪_{1}^{n})), the calculation above shows that (η_{s}^{p^{s}}) is a Cauchy sequence and lim_{s→∞} η_{s}^{p^{s}} is welldefined in R_{2}. Since η_{s}^{p^{s}}(1+𝔪_{2}^{m}) = f(λ(1+𝔪_{1}^{n})) and 1+𝔪_{2}^{m} is topologically closed in R_{2},

f (λ(1 + 𝔪_{1}^{n})) = ( lim_{s→∞} η_{s}^{p^{s}}) (1 + 𝔪_{2}^{m}).

Similarly, we have

f ((λ)^{1/p}(1 + 𝔪_{1}^{n})) = ( lim_{s→∞} η_{s}^{p^{s−1}}) (1 + 𝔪_{2}^{m}).

Since

lim_{s→∞} η_{s}^{p^{s}} = ( lim_{s→∞} η_{s}^{p^{s−1}})^{p} ,

we obtain

lim_{s→∞} η_{s}^{p^{s}} ∈ S_{2}

by 2.3(3). Therefore, we have that f(H_{n}(S_{1})) ⊂ H_{m}(S_{2}).

By Remark 3.4, the restriction map f ↾ H_{1}(S_{1}) induces an homomorphism between residue fields of K_{1} and K_{2}. So, we have that f ↾ H_{1}(S_{1}) is trivial or injective. Since 1 ∈ S_{1}, we have that f([1]_{n}) = [1]_{m} and f ↾ H_{1}(S_{1}) is not trivial. Therefore, f ↾ H_{1}(S_{1}) must be injective.

(2) Suppose f is over p, that is, f([p]_{n}) = [p]_{m}. Note that each a ∈ W(k_{i}) is uniquely written as ∑_{k≥0} a_{k}p^{k} for a_{k} ∈ S for i = 1, 2. By Fact 2.3(1), Remark 3.4,

and (1), we have a homomorphism f̄: W(k_{1}) → W(k_{2}) such that f([a]_{n}) = [f̄(a)]_{m} for a ∈ S_{1}. Take a ∈ W(k_{1}) and write a = p^{l} ∑_{k≥0} a_{k}p^{k} for l ≥ 0 and a_{k} ∈ S_{1} with

a_{0} ≠ 0. We have that

f([a]_{n}) = [p]_{m}^{l}f([∑_{k≥0} a_{k}p^{k}])_{m}
= [p]_{m}^{l}f([a_{0} + a_{1}p + a_{2}p^{2} + ⋯ + a_{s}p^{s}])_{m},

where s = max{n, m}. It is enough to show that

f([ ∑_{0≤k≤s} a_{k}p^{k}]_{n}) = [ ∑_{0≤k≤s} f̄(a_{k})p^{k}]_{m}

for a_{0} ≠ 0 and s ≥ max{n, m}. Take a = ∑_{0≤k≤s} a_{k}p^{k} ∈ W(k_{1}) with a_{0} ≠ 0 and

s ≥ max{n, m}. Then,

f([a]) ∈ ∑_{0≤k≤s}^{H} f([a_{k}]_{n})[p]_{m}^{k}

= ∑_{0≤k≤s}^{H}[f̄(a_{k})]_{m}[p]_{m}^{k}

= ∑_{0≤k≤s}^{H}[f̄(a_{k})p^{k}]_{m}.

So, we have that

f([a]_{n}) = [ ∑_{0≤k≤s} f̄(a_{k})p^{k} + d]_{m}
= [f̄(a) + d]

for some d ∈ 𝔪_{2}^{m} by Lemma 3.1(5). Since (f̄(a) + d)/f̄(a) = 1 + d/f̄(a) ∈ 1 + 𝔪_{2}^{m}, we have that [f̄(a) + d]_{m} = [f̄(a)]_{m}. Therefore, f([a]_{n}) = [f̄(a)]_{m} for each a ∈ W(k_{1}). □

3.1. Tamely ramified case. We first recall some embedding lemma in [12] for tame algebraic extensions of valued fields. We say that an algebraic extension (L, ν_{L}) of a henselian valued field (K, ν_{K}) is tame if the following conditions hold: For every finite subextension (F, ν_{F}) of (L, ν_{L}) over (K, ν),

• the residue field extension k_{L} over k_{K} is separable; • if the characteristic of k_{K} is p > 0, then the ramification index (Γ_{F} : Γ_{K})

is prime to p; and

• [F : K] = [k_{F} : k_{K}](Γ_{F} : Γ_{K}). If K is a complete discrete tamely ramified valued field with a perfect residue field k, then K is the tame extension of the fraction field of W(k).

Fact 3.6. [12, Lemma 3.1] Let K be an arbitrary valued field. Let L be an algebraic tame extension of some henselization of K and F be an arbitrary henselian extension of K. Any K_{1}-embedding from L_{1} to F_{1} is induced from a K-embedding from L to F. Furthermore if k_{K} = k_{L}, then any (group) G_{K}^{1}-embedding from G_{L}^{1} to G_{F}^{1} is induced from a K-embedding from L to F.

By adapting the ideas of the proofs of [12, Lemma 3.1], we prove the following theorems.

Theorem 3.7. Let K_{1} and K_{2} be complete discrete valued fields of mixed characteristic (0, p) with perfect residue fields. Suppose K_{1} is tamely ramified. Any homomorphism over p from H_{n}(K_{1}) to H_{m}(K_{2}) is induced from a unique homomorphism from K_{1} to K_{2}. From this, we conclude that there is one-to-one correspondence between Hom_{ℤ}(H_{n}(K_{1}), H_{m}(K_{2})) and Hom(K_{1}, K_{2}).

Proof. Take f ∈ Hom_{ℤ}(H_{n}(K_{1}), H_{m}(K_{2})). Let F_{i} be the fraction field of W(k_{i}) where k_{i} is the residue field of K_{i} for i = 1, 2. By Lemma 3.5(2), f ↾_{H_{1}(F_{1})} is induced from a homomorphism f̄: F_{1} → F_{2}. Since K_{1} is a totally tamely ramified extension of F_{1}, K_{1} = F_{1}(\sqrt[e]{pa}) for some a in W(k_{1})^{×} where e is the ramification index of K_{1} (c.f. Chapter 2 of [13]). Let π_{1} = \sqrt[e]{pa} and f([π_{1}]_{n}) = [π′_{2}]_{m} for some π′_{2} ∈ K_{2} so that [π′_{2}]_{m}^{e} = [pf̄(a)]_{m}. Consider a polynomial P (X) = X^{e} − (pf̄(u))/π_{2}^{′e} ∈ K_{2}[X]. Since P (1) ∈ 𝔪_{2}^{m} and P′(1) ∉ 𝔪_{2}, by Hensel’s lemma, there is unique b ∈ R_{2}^{×} such that b^{e} = (pf̄(a))/π_{2}^{′e} and (b − 1) ∈ 𝔪_{2}^{m}. Let π_{2} := bπ′_{2} so that π_{2}^{e} = pf̄(a). Note that [π_{2}]_{m} = [π′_{2}]_{m} and π_{2} is such a unique zero of the polynomial X^{e} − pf̄(a). So we have a homomorphism f̃ : K_{1} → K_{2} extending f̄ ∪ {(π_{1}, π_{2})} and it induces f. □

We generalize Theorem 3.7 to the case of infinitely tamely ramified valued fields. To do this, we first recall the Ax-Sen-Tate theorem.

Definition 3.8. [2] Let (K, ν) be a henselian valued field and (K^{alg}, ν) be the algebraic closure of K. For a ∈ K^{alg}, define

Δ_{K}(a) := min{ν(σ(a) − a)| σ ∈ G_{K}},

where G_{K} is the Galois group of K^{alg} over K.

Fact 3.9. [2, Proposition 1, Proposition 2’] Let (K, ν) be a complete valued field and (K^{alg}, ν) be the algebraic closure of K. Suppose Γ_{K^{alg}} is archimedean. Let K ⊂ F ⊂ K^{alg}.

(1) Suppose K is of mixed characteristic (0, p). Then for all a ∈ K^{alg}, there exists b ∈ F such that

ν(a − b) ≥ Δ_{F} (a) − (p/(p − 1)^{2})ν(p).

(2) Suppose K is of equal characteristic p ≥ 0. Then for all a ∈ K^{alg} and for all γ ∈ (Γ_{K^{alg}})_{>0}, there exists b in the perfect closure of F such that

ν(a − b) ≥ Δ_{F} (a) − γ.

Fact 3.10 (Ax-Sen-Tate Theorem). [9, Proposition 3.8] Let (K, ν) be a complete valued field and (K^{alg}, ν) be the algebraic closure of K with Γ_{K^{alg}} archimedean. Let C be the completion of K^{alg} which is algebraically closed and let L be a perfect complete subfield of C containing K. Then L is the completion of L ∩ K^{alg}.

Proof. See [9, Proposition 3.8] with Fact 3.9. □

Corollary 3.11. Let p be a prime number. Let K be a tamely ramified valued field of mixed characteristic (0, p) with a perfect residue field k, and F be the fraction field of W(k). Suppose K is a subfield of the completion of F^{alg}, and either

• K is an algebraic extension of F, or

• complete. For a complete valued field L of mixed characteristic (0, p) with a perfect residue field, any homomorphism over p from H_{γ}(K) to H_{λ}(L) is induced from a unique homomorphism K to L for any γ ∈ (Γ_{K})_{≥0} and λ ∈ (Γ_{L})_{≥0}.

Proof. Let L be a complete valued field of mixed characteristic (0, p) with a perfect residue field. Let F be the fraction field of W(k). Fix γ ∈ (Γ_{K})_{≥0} and λ ∈ (Γ_{L})_{≥0}.

Suppose K is an algebraic extension of F. Since K is tamely ramified, K = ⋃ K_{i} where K_{i} is a tamely totally ramified finite extension of F. Take f ∈ Hom_{ℤ}(H_{γ}(K), H_{λ}(L)). Set f_{i} := f ↾_{H_{γ}(K_{i})} and we have f := lim_{⟵K_{i}} f_{i} = ⋃ f_{i}.

By Theorem 3.7, f_{i} is induced from a unique homomorphism σ_{i} : K_{i} → L. Take σ := lim_{⟵K_{i}} σ_{i} = ⋃ σ_{i} : K → L and it is a unique homomorphism inducing f.

Suppose K is complete. By Fact 3.10, K is the completion of K′ := K ∩ F^{alg}. Let f′ = f ↾_{H_{γ}(K′)}. By above result, f′ is induced from a unique homomorphism f̃′ : K′ → L. Since K is a completion of K′, f̃′ induces a unique homomorphism f̃ → L, which induces f. □

We can not drop the condition of being over p in Theorem 3.7.

Example 3.12. Consider K_{1} = ℚ_{3}(\sqrt{3}) and K_{2} = ℚ_{3}(\sqrt{−3}), which are not isomorphic by Kummer Theory. Note that their residue fields are isomorphic to 𝔽_{3} and so their Teichmüller representatives are \{−1, 0, 1\}. Let π_{1} = \sqrt{3} and π_{2} = \sqrt{−3}. Every elements of K_{1} is of the form ∑_{i≥n} a_{i}π_{1}^{n} for some integer n and a_{i} ∈ {−1, 0, 1} with a_{n} ≠ 0. So each elements in H_{1}(K_{1}) is of the form π_{1}^{n}a_{n}(1+𝔪_{1}) for some integer n and a_{n} ∈ {−1, 1}(†). The same formulas hold for K_{2}. We will show that Hom(H_{1}(K_{1}), H_{1}(K_{2})) = {f_{0}, f_{1}}, where f_{i} sends [a] to [a] for a ∈ {−1, 1}, [π_{1}] to [(−1)^{i}π_{2}], and [3] to [−3] for i = 1, 2, so that Hom_{ℤ}(H_{1}(K_{1}), H_{1}(K_{2})) = ∅. Note that [1] ≠ [−1]. Let f ∈ Hom(H_{1}(K_{1}), H_{1}(K_{2})). Then f([−1]) = [−1] because f([1]) = [1] and 0 ∈ f([1] + [−1]). Since π_{1} is an uniformizer, f([π_{1}]) = [π_{2}] or = [−π_{2}]. In both cases, f([3]) = f([π_{1}]^{2}) = [π_{2}]^{2} = [π_{2}^{2}] = [−3]. By (†), such f induces an isomorphism between H_{1}(K_{1}) and H_{1}(K_{2}).

Without base fields, even the residue fields of K_{1} and K_{2} are primes fields so that the residue fields are equal, we can not lift a group homomorphism from H_{1}^{×}(K_{1}) to H_{1}^{×}(K_{2}) to a homomorphism from K_{1} to K_{2}(c.f. Fact 3.6).

Example 3.13. Let K_{1} = K_{2} = ℚ_{5}. The set of Teichmüller representatives of ℚ_{5} is \{0, i, i^{2}, i^{3}, i^{4}\} where i = \sqrt{−1}. Consider a group isomorphism f : H_{1}^{×}(ℚ_{5}) → H_{1}^{×}(ℚ_{5}) by mapping [i] ↦ [−i] and [5] ↦ [5]. Then f is never induced from an automorphism of ℚ_{5} since any automorphism of ℚ_{5} sends i to i.

3.2. Generally ramified case. We first introduce of a notion of lifting map of homomorphisms of the n-th valued hyperfields, which is an analogy to a lifting map of homomorphisms of the n-th residue rings in [14, Definition 3.1].

Definition 3.14. Let K_{1} and K_{2} be complete discrete valuation rings of characteristic 0 with perfect residue fields k_{1} and k_{2} of characteristic p respectively. Let π_{i} be a uniformizer of K_{i} and ν_{i} be a corresponding valuation of K_{i} for i = 1, 2. For any homomorphism ϕ : H_{n}(K_{1}) → H_{m}(K_{2}), we say that a homomorphism g : K_{1} → K_{2} is a (n, m)-lifting of ϕ at π_{1} if g satisfies the following:

• There exists a representaive b of ϕ([π_{1}]) which satisfies

ν_{2}(g(π_{1}) − b) > M(K_{1}).

• ϕ_{red,1} ∘ H_{1} = H_{1} ∘ g where ϕ_{red,1} : H_{1}(K_{1}) → H_{1}(K_{2}) denotes the natural reduction map of ϕ.

When such g is unique, we denote g by L_{π_{1},n,m}^{H}(ϕ). When L_{π_{1},n,m}^{H}(ϕ) exists for all ϕ : H_{n}(K_{1}) → H_{m}(K_{2}), we write L_{π_{1},n,m}^{H} : Hom_{ℤ}(H_{n}(K_{1}), H_{m}(K_{2})) → Hom(K_{1}, K_{2}). When n = m, we briefly write L_{π_{1},n,m}^{H} = L_{π_{1},n}^{H} and say that L_{π_{1},n}^{H} is an n-lifting at π_{1}.

The following result is analogous to Proposition 2.9(2) for n-th residue rings in [14].

Proposition 3.15. Let K_{1} and K_{2} be complete discrete valued fields of characteristic 0 with perfect residue fields k_{1} and k_{2} of characteristic p respectively. Let π_{i} be a uniformizer of K_{i} and ν_{i} be a corresponding valuation of K_{i} for i = 1, 2. Let R_{i} be the valuation ring of K_{i} for i = 1, 2. The definition of liftings is independent of the choice of uniformizer of K_{1}. More precisely, saying that g : K_{1} → K_{2} is a (n, m)-lifting of ϕ : H_{n}(K_{1}) → H_{m}(K_{2}) at π_{1} is equivalent to the following:

(1) For any x in R_{1}, there exists a representative b_{x} of ϕ(x(1 + 𝔪_{1}^{n})) which satisfies

ν_{2}(g(x) − b_{x}) > M(K_{1}).
(2) ϕ_{red,1} ∘ H_{1} = H_{1} ∘ g

We write L_{π_{1},n,m}^{H} = L_{n,m}^{H} and say that L_{n,m}^{H} is a (n, m)-lifting. Moreover, there is at most one (n, m)-lifting for m > M(K_{1})e_{2}, where e_{2} is the ramification index of K_{2}.

Proof. The proof is similar to the proof of [14, Proposition 3.5 (2)]. Let S_{i} be the Teichmüller representatives of K_{i} for i = 1, 2. Fix a uniformizer π of K_{1}. Let g : K_{1} → K_{2} be a (n, m)-lifting of ϕ : H_{n}(K_{1}) → H_{m}(K_{2}) at a uniformizer π. Let b ∈ K_{2} be a representative of ϕ([π])) such that ν_{2}(g(x) − b) > M(K_{1}). Note that ν_{2}(b) > 0. Take x ∈ R_{1}. Then x = ∑_{i≥0} λ_{i}π^{i}. Take l > 0 such that ν_{1}(∑_{i>l} λ_{i}π^{i}) > M(K_{1}). Denote x^{≤l} := ∑_{i≤l} λ_{i}π^{i} and x^{>l} ∑_{i>l} λ_{i}π^{i} so that x = x^{≤l} + x^{>l}. Then we have that

ϕ([x]) = [x^{≤l} + x^{>l}])

∈ (ϕ([x^{≤l}] +_{H} ϕ([x^{>l}]))

⊂ ( ∑_{i≤l}^{H} ϕ([λ_{i}])ϕ([π]^{i}) + ϕ([x^{>l}]))

⊂ ( ∑_{i≤l}^{H}[g(λ_{i})][b]^{i} + ϕ([x^{>l}])).

There is a representative b_{x} of ϕ([x]) of the form:

b_{x} = ∑_{i≤l} g(λ_{i})b^{i} + d

for some ν_{2}(d) > M(K_{1}). Compute

ν_{2}(g(x) − b_{x}) = ν_{2}( ∑_{i}(g(λ_{i})g(π)^{i}) − b_{x})

= ν_{2}( ∑_{0<i≤l} g(λ_{i})(g(π)^{i} − b^{i}) + (∑_{i>l} g(λ_{i})g(π)^{i} − d))

≥ min{ν_{2}( ∑_{0<i≤l} g(λ)(g(π)^{i} − b^{i})), ν_{2}(∑_{i>l} g(λ_{i})g(π)^{i} − d)}

> M(K_{1})

because ν_{2}(g(π)^{i} − β^{i}) = ν_{2}(g(π) − β) + ν_{2}(g(π)^{i−1} + ⋯ + β^{i−1}) > M(K_{1}), and ν_{2}(∑_{i>l} g(λ_{i})g(π)^{i}), ν_{2}(d) > M(K_{1}).

Now we show moreover part. Assume m > M(K_{1})e_{2} and there are two (n, m)- liftings L, L′ : Hom_{ℤ}(H_{n}(K_{1}), H_{m}(K_{2})) → Hom(K_{1}, K_{2}). Fix f ∈ Hom_{ℤ}(H_{n}(K_{1}), H_{m}(K_{2})). By Lemma 3.5 and (2), we have that L(f) ↾ W(k_{1}) = L′(f) ↾ W(k_{1}) (†). It remains to show that L(f)(π_{1}) = L′(f)(π_{1}). Set π := L(f)(π_{1}) and π′ := L′(f)(π_{1}). By (†), π and π′ are conjugates over the fraction field of W(k_{2}) (‡). By (1), there are two representatives b and b′ of f([π_{1}]) such that

ν_{2}(π − β), ν_{2}(π′ − β′) > M(K_{1}).

Since ν_{2}(b − b′) > m/e_{2} > M(K_{1}), we have that

ν_{2}(π − π′) > max{ν_{2}(π − b), ν_{2}(b − b′), ν_{2}(π′ − b′)}
> M(K_{1}).

By (‡), we conclude that ν_{2}(π − π′) = ∞ and π = π′. □

Remark 3.16. Let K_{1} and K_{2} be complete discrete valued field of mixed characteristic (0, p) with perfect residue fields. Suppose K_{1} is tamely ramified. By Theorem 3.7, for every n ≥ 1 there is a unique bijective n-lifting map

L_{n}^{H} : Hom_{ℤ}(H_{n}(K_{1}), H_{n}(K_{2})) → Hom(K_{1}, K_{2})

such that f([a]_{n}) = [L_{n}^{H}(f)(a)] for f ∈ Hom_{ℤ}(H_{n}(K_{1}), H_{n}(K_{2})) and a ∈ K_{1}.

Fact 3.17 (Krasner’s lemma). Let (K, ν) be henseilan valued field whose value group is contained in ℝ and let a, b ∈ K^{alg}. Suppose a is separable over K(b). Suppose that for all embeddings σ(≠ id) of K(a) over K, we have

ν̃(b − a) > ν̃(σ(a) − a).

Then K(a) ⊂ K(b).

We show that a lifting map of homomorphisms of n-th valued hyperfields for large enough n

Theorem 3.18. Let K_{1} and K_{2} be complete discrete valued field of mixed characteristic (0, p) with perfect residue fields. Let e_{1} and e_{2} be ramification indices of K_{1} and K_{2} respectively. Let H_{n}(K_{1}) and H_{m}(K_{2}) be valued hyper fields of K_{1} and K_{2} respectively. Suppose m > M(K_{1})e_{1}e_{2}. There is a unique lifting map L_{n,m}^{H} : Hom_{ℤ}(H_{n}(K_{1}), H_{m}(K_{2})) → Hom(K_{1}, K_{2}).

Proof. Fix a homomorphism f in Hom_{ℤ}(H_{n}(K_{1}), H_{m}(K_{2})). By Lemma 3.5, f induces a map f ↾_{H_{n}(S_{1})} from H_{n}(S_{1}) to H_{m}(S_{2}). This map f ↾_{H_{n}(S_{1})} induces a homomorphism from k_{1} to k_{2} and by the functoriality of Witt ring, we have a homomorphism f̄: W(k_{1}) → W(k_{2}). Note that f̄ induces a homomorphism from W(k_{1})[X] to W(k_{2})[X] by acting on coefficients and we denote this homomorphism by f̄ also. By Fact 2.3(2), there is a uniformizer π_{1} of K_{1} such that R_{1} = W(k_{1})[π_{1}]. Let q(X) be the irreducible polynomial of π_{1} over the fraction field of W(k_{1}), which is in W(k_{1})[X]. Write q(X) = X^{e} + a_{e−1}X^{e−1} + … + a_{0} where e = ν_{1}(p) is the ramification index of K_{1}. Let π′_{2} ∈ R_{2} such that [π′_{2}]_{m} = f([π_{1}]_{n}). For a ∈ W(k_{1}), we can uniquely write a = a_{0} + a_{1}p + a_{2}p^{2} + … with a_{i} ∈ S_{1}. For l ≥ 0, define a^{≤l} := a_{0} + a_{1}p + … + a_{l}p^{l} and a^{>l} := a − a^{≤l}. And define q^{≤l}(X) := X^{e} + a_{e−1}^{≤l}X^{e−1} + … + a_{0}^{≤l} and q^{>l}(X) = q(X) − q^{≤l}(X), which are in W(k_{1})[X]. Then we have that

0 = f([q(π_{1})]_{n})

= f([q^{≤m}(π_{1}) + q^{>m}(π_{1})]_{n})

∈ f([q^{≤m}(π_{1})]_{n}) +_{H} f([q^{>m}(π_{1})]_{n})

⊂ [π_{2}^{′e}]_{m} +_{H} f[a_{e−1}^{≤m}π_{2}^{′e−1}]_{m} +_{H} … +_{H} f[a_{0}^{≤m}]_{m} +_{H} f([q^{>m}(π_{1})]_{n})

⊂ [π_{2}^{′e}]_{m} +_{H} ∑_{0≤i<e}^{H} ∑_{0≤j≤m}^{H}[π_{2}^{′i}f̄(a_{i}^{j})p^{j}]_{m} +_{H} f([q^{>m}(π_{1})]_{n}).

Note that ν_{2}(q^{>m}(π_{1})) ≥ m/e_{2}. By Lemma 3.1(2), we have that

0 = f̄(q^{≤m})(π′_{2}) + d(∗)

for some d ∈ 𝔪_{2}^{m}. Since q = q^{≤m} + q^{>m}, we have that f̄(q)(π′_{2}) = f̄(q^{≤m})(π′_{2}) + f̄(q^{>m})(π′_{2}) and ν_{2}(f̄(q^{>m})(π′_{2})) ≥ m/e_{2}. From (∗) we have that

0 = f̄(q)(π′_{2}) + d′

for some d′ ∈ 𝔪_{2}^{m}, and ν_{2}(f̄(q)(π′_{2})) ≥ m/e_{2} > M(K_{1})e_{1}. By Krasner’s lemma, there is π_{2} ∈ K_{2} such that f̄(q)(π_{2}) = 0 with ν_{2}(π_{2} − π′_{2}) > M(K_{1}). Define f̃: K_{1} → K_{2}, π_{1} ↦ π_{2} extending f̄, and set L_{n,m}^{H}(f) = f̃. □

Corollary 3.19. Let (K_{1}, ν_{1}, k_{1}, Γ_{1}) and (K_{2}, ν_{2}, k_{2}, Γ_{1}) be finitely ramified complete valued fields of mixed characteristic. Let n > e_{ν_{2}}(p)(1+e_{ν_{1}}^{2}(p)). The following are equivalent:

(1) K_{1} ≅ K_{2};

(2) R_{n}(K_{1}) ≅ R_{n}(K_{2}); and

(3) H_{n}(K_{1}) ≅_{H_{n}({p})} H_{n}(K).

The following is an analogy of [14, Proposition 4.4] for a lifting map of homomorphisms of n-th valued hyperfields, which gives a funtoriality of lifting map in Section 4.

Proposition 3.20. Let (K_{1}, ν_{1}), (K_{2}, ν_{2}), and (K_{3}, ν_{3}) be complete discrete valued fields of mixed characteristic (0, p) with perfect residue fields. Suppose K_{1} and K_{2} have the same ramification index. Suppose m, k > max{e_{ν_{2}}(p)(1+e_{ν_{1}}^{2}(p), e_{ν_{3}}(p)(1+ e_{ν_{2}}^{2}(p))}. Let f ∈ Hom_{ℤ}(H_{n}(K_{1}), H_{m}(K_{2})) and g ∈ Hom_{ℤ}(H_{m}(K_{2}), H_{k}(K_{3})). Then L_{n,k}^{H}(g ∘ f) = L_{m,k}^{H}(g) ∘ L_{n,m}^{H}(f).

Proof. Take f ∈ Hom_{ℤ}(H_{n}(K_{1}), H_{m}(K_{2})) and g ∈ Hom_{ℤ}(H_{m}(K_{2}), H_{k}(K_{3})). First, we have that

(g ∘ f) ∘ H_{1} = g ∘ (f ∘ H_{1})
= g ∘ (H_{1} ∘ L_{n,m}^{H}(f))
= (g ∘ H_{1}) ∘ L_{n,m}^{H}(f)
= (H_{1} ∘ L_{m,k}^{H}(g)) ∘ L_{n,m}^{H}(f)
= H_{1} ∘ (L_{m,k}^{H}(g) ∘ L_{n,m}^{H}(f)).

It remains to show that for a uniformizer π_{1} of K_{1}, there is a representative b of g ∘ f([π_{1}]) such that

ν_{3}(L_{m,k}^{H}(g) ∘ L_{n,m}^{H}(f)(π_{1}) − b) > M(K_{1}).

Take a representative b_{1} of f([π_{1}]) such that ν_{2}(L_{n,m}^{H}(f)(π_{1})−b_{1}) > M(K_{1}), and we have that ν_{3}(L_{m,k}^{H}(g)∘L_{n,m}^{H}(f)(π_{1})−L_{m,k}^{H}(g)(b_{1})) > M(K_{1}). Next consider a representative b_{2} of g([b_{1}])(= g (f([π_{1}]))) such that ν_{3}(L_{m,k}^{H}(g)(b_{1})−b_{2}) > M(K_{2}). Since K_{1} and K_{2} have the same ramification indices, we have that M(K_{1}) = M(K_{2}). Thus we conclude that ν_{3}(L_{m,k}^{H}(g)∘L_{n,m}^{H}(f)(π_{1})−b_{2}) > M(K_{1}), and set b := b_{2}. □

4. Valued hyperfields, truncated DVRs, and valued fields

In this section, we figure out relationships between valued hyperfields, truncated DVRs, and complete discrete valued fields of mixed characteristic. We first define a notion of finitely ramified valued hyperfields.

Definition 4.1. Let (H, ν) be a discrete valued hyperfield of length l. Let R_{l}(H) = R. Suppose the residue field k(H) has the characteristic p ≥ 0.

(1) We say H is of equal characteristic (p, p) if p = 0 in R. Otherwise, we say H is of mixed characteristic (0, p).

(2) Suppose H is of mixed characteristic (0, p). We say H is finitely ramified if ν_{R}(p) = e < ∞, and e is called the ramification index of H.

(3) Suppose H is finitely ramified. We say H is normalized if θ_{H} = 1/e where e is the ramification index of H.

Note that any discrete valued hyperfield of mixed characteristic need not be finitely ramified.

Proposition 4.2. Let K be a discrete valued field. Let K_{1}, K_{2}, and K_{3} be discrete valued fields of mixed characteristic (0, p) having the same ramification index e.

(1) T_{n}(K) = (R_{n}(K), M_{n}(K), ϵ_{n}) is a triple, where M_{n}(K) := 𝔪_{K}/𝔪_{K}^{n+1} and the map ϵ_{n} is induced by the inclusion 𝔪_{K} ⊂ R(K).

(2) For every triple T = (R_{n}(K), M, ϵ), we have T ≅ T_{n}(K).

(3) Let n > e. Each morphism f ∈ Hom(R_{n}(K_{1}), R_{n}(K_{2})) induces a morphism T_{n}(f) ∈ Hom(T_{n}(K_{1}), T_{n}(K_{2})). For f ∈ Hom(R_{n}(K_{1}), R_{n}(K_{2})) and g ∈ Hom(R_{n}(K_{2}), R_{n}(K_{3})), we have T_{n}(g ∘ f) = T_{n}(g) ∘ T_{n}(f).

Proof. (1) It is clear.

(2) Let R := R_{n}(K). Given a triple T = (R, M, ϵ), let Π be a generator of M. Since ϵ(M) = 𝔪_{R}, ϵ(Π) = π + 𝔪_{K} for a uniformizer π of K. Define a map η : M → 𝔪_{K}/𝔪_{K}^{n+1}, Π ↦ π/𝔪_{K}^{n+1}. Then the triple (1, Id_{R}, η) gives an isomorphism from T to T_{n}(K).

(3) Let f ∈ Hom(R_{n}(K_{1}), R_{n}(K_{2})). Let π_{1} and π_{2} be uniformizers of K_{1} and K_{2} respectively. By the choice of n, f(π_{1}/𝔪_{K_{1}}^{n}) generates the maximal ideal of R_{n}(K_{2}) so that f(π_{1}/𝔪_{K_{1}}^{n}) = (a_{f}π_{2})/𝔪_{K_{2}}^{n+1} for a unit a_{f} ∈ R(K_{2})^{×}. Define a map η_{f} : M_{n}(K_{1}) → M_{n}(K_{2}) sending π_{1}/𝔪_{K_{1}}^{n+1} to (a_{f}π_{2})/𝔪_{K_{2}}^{n+1}. Then the triple T_{n}(f) := (1, f, η_{f}) gives a morphism from T_{n}(K_{1}) to T_{n}(K_{2}). Note that T_{n}(f) does not depend on the choices of π_{1} and π_{2} because M_{n}(K_{1}) and M_{n}(K_{2}) are free R_{n}(K_{1}) and R_{n}(K_{2})-modules of rank 1 respectively.

Now we show T_{n} is commute with the composition. Let π_{3} be a uniformizer of K_{3}. Fix f ∈ Hom(R_{n}(K_{1}), R_{n}(K_{2})) and g ∈ Hom(R_{n}(K_{2}), R_{n}(K_{3})). There are a_{f} ∈ R(K_{2})^{×}, b_{g}, b_{g∘f} ∈ R(K_{3})^{×} such that f(π_{1}/𝔪_{K_{1}}^{n}) = (a_{f}π_{2})/𝔪_{K_{2}}^{n}, g(π_{2}/𝔪_{K_{1}}^{n}) = (b_{g}π_{3})/𝔪_{K_{3}}^{n}, and g ∘ f(π_{1}/𝔪_{K_{1}}^{n}) = (b_{g∘f}π_{3})/𝔪_{K_{3}}^{n}. Let b_{f} ∈ R(K_{3})^{×} such that g(a_{f}/𝔪_{K_{2}}^{n}) = b_{f}/𝔪_{K_{3}}^{n}. Then we have that b_{g∘f}/𝔪_{K_{3}}^{n} = (b_{g}b_{f})/𝔪_{K_{3}}^{n}. By choosing b_{g} and b_{f} properly, we may assume that b_{g}b_{f} − b_{g∘f} ∈ 𝔪_{K_{3}}^{n+1} (†). Define η_{g∘f} : M_{n}(K_{1}) → M_{n}(K_{3}), π_{1}/𝔪_{K_{1}}^{n+1} ↦ (b_{g∘f}π_{3})/𝔪_{K_{3}}^{n+1}. By (†), we have that η_{g∘f} = η_{g} ∘ η_{f}. Therefore,

T_{n}(g ∘ f) = (1, g ∘ f, η_{g∘f})
= (1, g ∘ f, η_{g} ∘ η_{f})
= (1, g, η_{g}) ∘ (1, f, η_{f})
= T_{n}(g) ∘ T_{n}(f).
□

Corollary 4.3. Let H be a finitely ramified discrete valued hyperfield of mixed characteristic with the perfect residue field and let e be the ramification index. Suppose l(:= l(H)) > e(1 + ν_{R}(e)) so that e is not zero in R(:= R_{l}(H)). Then there is a unique complete discrete valued field K (up to isomorphic) such that H_{l}(K) ≅ H.

Remark/Definition 4.4. Let H_{1} and H_{2} be finitely ramified discrete valued hyperfield of mixed characteristic (0, p) with perfect residue fields. Let Tr(H_{1}) = (R_{1}, M_{1}, ϵ_{1}) and Tr(H_{2}) = (R_{2}, M_{2}, ϵ_{2}), and we identify H_{1} = U(Tr(H_{1})) and H_{2} = U(Tr(H_{2})). Suppose they have the same length l and the same ramification index e so that n = ν_{1}(e) = ν_{2}(e) ∈ {0, 1, …, l − 1} ∪ {∞}, where ν_{1} = ν_{R_{1}} and ν_{2} = ν_{R_{2}}. Suppose l > e. Since l > e, p is not zero in R_{l} and R_{2} and ν_{1}(p) = ν_{2}(p) = e. So p = a_{1}π_{1}^{e} in R_{1} and p = a_{2}π_{2}^{e} in R_{2} for some units a_{1}, a_{2} and some uniformizers π_{1}, π_{2} in R_{1} and R_{2} respectively. Take Π_{i} ∈ M_{i} such that ϵ_{i}(Π_{i}) = π_{i} for i = 1, 2. Then ν_{Tr(H_{i})}(a_{i}Π_{i}^{⊗e}) = e for i = 1, 2. We say a homomorphism f : H_{1} → H_{2} is over p if f(a_{1}Π_{1}^{⊗e}) = a_{2}Π_{2}^{⊗e}. Denote Hom_{ℤ}(H_{1}, H_{2}) for the set of all homomorphisms from H_{1} to H_{2}, which are over p.

Example 4.5. Let K_{1} = K_{2} = ℚ_{3}(\sqrt{3}) and 𝔪 be the maximal ideal of the valuation ring ℤ_{3}[\sqrt{3}]. Note that R := R_{4}(K_{1}) = R_{4}(K_{2}) ≅ (ℤ_{3}/9ℤ_{3})[x]/(x^{2}−3). Then there is an isomorphism f : a + bx ↦ a + 4bx in Hom(R_{4}(K_{1}), R_{4}(K_{2})). Then f induces an isomorphism T_{4}(f) : (R, 𝔪/𝔪^{4}, ϵ_{4}) → (R, 𝔪/𝔪^{4}, ϵ_{4}) and it induces an isometric isomorphism U(T_{4}(f)) : H_{4}(K_{1}) → H_{4}(K_{2}), \sqrt{3}(1 + 𝔪^{4}) ↦ 4\sqrt{3}(1 + 𝔪^{4}), which is not in Hom_{ℤ}(H_{4}(K_{1}), H_{4}(K_{2})). Suppose U(T_{4}(f)) is over p. Then 3(1 + 𝔪^{4}) = U(T_{4}(f))(3(1 + 𝔪^{4})) = U(T_{4}(f))((\sqrt{3})^{2}(1 + 𝔪^{4})) = (4\sqrt{3})^{2}(1 + 𝔪^{4}) = 4^{2}3(1 + 𝔪^{4}). So, we have (1 + 𝔪^{4}) = 4^{2}(1 + 𝔪^{4}), which is impossible, because 15 ∉ 𝔪^{4} = 9ℤ_{3}.

Now we introduce some categories of valued hyperfields, truncated DVRs, and valuation rings and we study relationships between them. We recall two categories of truncated DVRs and valuation rings, which were used to generalize the functoriality of unramified valuation rings in [14, Section 4]. For a prime number p and a positive integer e, let C_{p,e} be a category consisting of the following data :

• Ob(C_{p,e}) is the family of complete discrete valuation rings of mixed characteristic having perfect residue fields of characteristic p and the ramification index e; and

• Mor_{C_{p,e}}(R_{1}, R_{2}) := Hom(R_{1}, R_{2}) for R_{1} and R_{2} in Ob(C_{p,e}). Let R_{p,e}^{n} be a category consisting of the following data :

• For n ≤ e, Ob(R_{p,e}^{n}) is the family of truncated DVRs \overline{R} of length n with perfect residue fields of characteristic p, and for n > e, Ob(R_{p,e}^{n}) is the family of truncated DVRs \overline{R} of length n with perfect residue fields of characteristic p such that p ∈ \overline{𝔪}^{e} \ \overline{𝔪}^{e+1} where \overline{𝔪} is the maximal ideal of \overline{R}; and

• Mor_{R_{p,e}^{n}}(\overline{R_{1}}, \overline{R_{2}}) := Hom(\overline{R_{1}}, \overline{R_{2}}) for \overline{R_{1}} and \overline{R_{2}} in Ob(R_{p,e,}^{n}), Note that for e_{1}, e_{2} ≥ 1 and for n ≤ e_{1}, e_{2}, two categories R_{p,e_{1}}^{n}, R_{p,e_{2}}^{n} are the same. For each m > n, let Pr_{n} : C_{p,e} → R_{p,e}^{n} and Pr_{n}^{m} : R_{p,e}^{m} → R_{p,e}^{n} be the canonical projection functors respectively. Given a prime number p and a positive integer e, let l_{p,e} := e(1 + ν(e)) for some(every) R ∈ Ob(C_{p,e}). Note that for p |̸e, we have that l_{p,e} = e.

Fact 4.6. [14, Definition 4.2, Theorem 4.7] Fix a prime number p and a positive integer e. For every n > l_{p,e}, there is a functor L : R_{p,e}^{n} ⟶ C_{p,e}, called an n-th lifting functor, which satisfies the following:

(1) (Pr_{n} ∘ L)(\overline{R}) ≅ \overline{R} for each \overline{R} in Ob(R_{p,e}^{n}).

(2) Pr_{1} ∘ L is equivalent to Pr_{1}^{n}.
(3) L ∘ Pr_{n} is equivalent to Id_{C_{p,e}}.

Moreover, there is a unique n-th lifting functor L satisfying (4) For each g ∈ Mor_{R_{p,e}^{n}}(\overline{R}_{1}, \overline{R}_{2}) and for any x ∈ L(\overline{R}_{1}), there is a represen-

tative b_{x} of (Pr_{n} ∘ L)(g)(x + 𝔪_{1}^{n}) such that
ν_{2}(L(g)(x) − b_{x}) > M(L(\overline{R}_{1})).

Next we introduce two categories of valued hyperfields. Let H_{p,e}^{n} be a category consisting of the following data :

• Ob(H_{p,e}^{n}) is the family of discrete normalized valued hyperfield of length n and mixed characteristic (0, p) having perfect residue fields, and for n > e in addition, having ramification indices e; and

• For n ≤ e, Mor_{H_{p,e}^{n}}(H_{1}, H_{2}) := Iso(H_{1}, H_{2}) and for n > e, Mor_{H_{p,e}^{n}}(H_{1}, H_{2}) := Iso_{ℤ}(H_{1}, H_{2}), where Iso_{ℤ}(H_{1}, H_{2}) = Hom_{ℤ}(H_{1}, H_{2}) ∩ Iso(H_{1}, H_{2}) for H_{1} and H_{2} in Ob(H_{p,e}^{n}).

Let Ĥ_{p,e}^{n} be a category consisting of the following data :

• Ob(Ĥ_{p,e}^{n}) = Ob(H_{p,e}^{n}); and
• MorĤ_{p,e}^{n}(H_{1}, H_{2}) := Iso(H_{1}, H_{2}) for H_{1} and H_{2} in Ob(Ĥ_{p,e}^{n}).

By Remark/Definition 2.20, we have functors Res_{p,e}^{n} : H_{p,e}^{n} → R_{p,e}^{1} and R̂es : Ĥ_{p,e}^{n} → R_{p,e}^{1}. Next we introduce lifting functors for two categories H_{p,e}^{n} and Ĥ_{p,e}^{n}, which are analogous to the lifting functors of R_{p,e}^{n} in [14, Definition 4.2].

Definition 4.7. Fix a prime number p and a positive integer e. We say that the category C_{p,e} is n-H-liftable if there is a functor L^{H} : H_{p,e}^{n} → C_{p,e} which satisfies the following:

• (H_{n} ∘ L^{H})(\overline{R}) ≅ \overline{R} for each \overline{R} in Ob(R_{p,e}^{n}).

• Pr_{1} ∘ L^{H} is equivalent to Res_{p,e}^{n}.
• L^{H} ∘H_{n} is equivalent to Id_{C_{p,e}}.

We say that L^{H} is an n-th H-lifting functor of C_{p,e}.

Now we see relationships between the categories C_{p,e}, R_{p,e}^{n}, H_{p,e}^{n}, and Ĥ_{p,e}^{n}. Define a functor Û : R_{p,e}^{n} → Ĥ_{p,e}^{n} as follows:

• For \overline{R} ∈ Ob(R_{p,e}^{n}), Û(\overline{R}) := (U ∘T_{n} ∘ L)(\overline{R}) after rescaling θ_{(U ∘T_{n}∘L)(\overline{R})} = 1/e, where L is the n-th lifting in Fact 4.6; and

• For f ∈ Mor_{R_{p,e}^{n}}(\overline{R}_{1}, \overline{R}_{2}), Û(f) := (U ∘ T_{n})(f),

and define a functor T̂o : Ĥ_{p,e}^{n} → R_{p,e}^{n} as follows:

• For H ∈ Ob(Ĥ_{p,e}^{n}), T̂o(H) := \overline{R} where \overline{R} is a truncated DVR in Tr(H).

• For g ∈ MorĤ_{p,e}^{n}(H_{1}, H_{2}), T̂o(g) is a morphism in Mor_{R_{p,e}^{n}}(T̂o(H_{1}), T̂o(H_{1}))

such that Tr(f) = (1, T̂o(g), η). By Remark 2.21 and Proposition 4.2, we have the following result.

Theorem 4.8. Let p be a prime number and e be a positive integer. Fix n > e.

(1) Û ∘ T̂o is equivalent to IdĤ_{p,e}^{n}.
(2) T̂o ∘ Û is equivalent to Id_{R_{p,e}^{n}}

Therefore, two categories Ĥ_{p,e}^{n} and R_{p,e}^{n} are equivalent.

Remark 4.9. [14, Proposition 4.9] Let R_{1}/ W(k) and R_{2}/ W(k) be totally ramified extensions of degree e. Then R_{1,e} is isomorphic to R_{2,e} as W(k)-algebras. So, the assumption that n > e is natural in Theorem 4.8.

Each R ∈ Ob(C_{p,e}) gives a discrete valued hyperfield H(R) := H_{n}(K) in Ob(H_{p,e}^{n}) where K is the fraction field of R after rescaling θ_{H_{n}(R)} = 1/e. Also each f ∈ Mor_{C_{p,e}}(R_{1}, R_{2}) induces a morphism H(f) : H(R_{1}) → H(R_{2}). So we have a functor H : C_{p,e} → H_{p,e}^{n}. For n > l_{p,e}, we define a functor L^{H} : H_{p,e}^{n} → C_{p,e}. Since H_{p,e}^{n} is a subcategory of Ĥ_{p,e}^{n}, we have a functor To : H_{p,e}^{n} → R_{p,e}^{n} by restricting T̂o to H_{p,e}^{n}. For each H ∈ Ob(H_{p,e}^{n}), there is a unique( up to isomorphic) complete valued field K such that H ≅ H_{n}(K), where K is the fraction field of (L ∘ To)(H) by Corollary 3.19. For H_{1}, H_{2} ∈ Ob(H_{p,e}^{n}) and f ∈ Mor_{H_{p,e}^{n}}(H_{1}, H_{2}), we have a morphism L^{H}(f) ∈ Mor_{C_{p,e}}(R_{1}, R_{2}) where R_{1} = (L ∘ To)(H_{1}) and R_{2} = (L ∘ To)(H_{2}) by Theorem 3.18. By Proposition 3.20, L^{H} forms a functor. By Remark 3.3, Theorem 3.18, and Proposition 3.20, we have the following result which is analogous to Fact 4.6 in the case of valued hyperfields.

Theorem 4.10. Let p be a prime number and e be a positive integer. Fix n > l_{p,e}. Then H_{p,e}^{n} is n-H-liftable and there is an n-th H-lifting functor L^{H} of H_{p,e}^{n} satisfying the following:

(1) There is an isometric isomorphism, which is over p, between (H ∘ L^{H})(H) and H for each H in Ob(H_{p,e}^{n}).

(2) Pr_{1} ∘ L^{H} is equivalent to Res_{p,e}^{n}.

(3) L^{H} ∘H is equivalent to Id_{C_{p,e}}.

Moreover, there is a unique n-th lifting functor L^{H} satisfying

(4) For each g ∈ Mor_{H_{p,e}^{n}}(H_{1}, H_{2}) and for any x ∈ L^{H}(H_{1}), there is a repre-

sentative b_{x} of (H_{n} ∘ L)(g)([x]) such that
ν_{2}(L^{H}(g)(x) − b_{x}) > M(L^{H}(H_{1})).

If we take L satisfying (1)-(4) and L^{H} satisfying (1)-(4), then L^{H} = L ∘ To.

Moreover, by Theorem 3.7, we have the following theorem.

Theorem 4.11. Let p be a prime number and e be a positive integer. Suppose p does not divide e so that l_{p,e} = e. For any n > e, two categories H_{p,e}^{n} and C_{p,e} are equivalent

In summary, for n > l_{p,e}, we have the following diagram between categories of C_{p,e}, R_{p,e}^{n}, H_{p,e}^{n}, and Ĥ_{p,e}^{n}:

\begin{matrix}H_{p,e}^{n} & C_{p,e} \\ Ĥ_{p,e}^{n} & R_{p,e}^{n} \end{matrix} To L^{H} H Pr_{n} L Û T̂o

, and we can take L^{H} = L ∘ To.

Question 4.12. Let p be a prime number and e be a positive integer. Fix n > l_{p,e}. We know that two categories Ĥ_{p,e}^{n} and R_{p,e}^{n} are equivalent, and L^{H} ∘H is equivalent to Id_{C_{p,e}}. And for the tame case, that is, p |̸e, H_{p,e}^{n} and C_{p,e} are equivalent. Is H_{n} ∘ L^{H} equivalent to Id_{H_{p,e}^{n}} if p|e?

5. Relative completeness via hyper fields

In this section, we aim to prove an AKE-type relative completeness theorem in terms of valued hyperfields for finitely ramified valued fields. We first recall basic facts on coarsenings of valuations.

Remark/Definition 5.1. [15] Let (K, ν, k, Γ) be valued field. Let Γ^{∘} be a convex subgroup of Γ and ν̇ : K \{0} ⟶ Γ/Γ^{∘} be a map sending x(≠ 0) ∈ K to ν(x)+Γ^{∘} ∈ Γ/Γ^{∘}. The map ν̇ is a valuation, called a coarse valuation of ν with respect to Γ^{∘}. The residue field K^{∘}, called the core field of (K, ν) with respect to Γ^{∘}, of (K, ν̇) forms a valued field equipped with a valuation ν^{∘} induced from ν and the value groups Γ^{∘}. More precisely, the valuation ν^{∘} is defined as follows: Let prν̇ : Rν̇ ⟶ K^{∘} be the canonical projection map and let x ∈ Rν̇. If x^{∘} := prν̇(x) ∈ K^{∘} \ {0}, then ν^{∘}(x^{∘}) := ν(x). And x^{∘} = 0 ∈ K^{∘} if and only if ν(x) > Γ^{∘}, that is, ν(x) > γ for all γ ∈ Γ^{∘}. If K is of characteristic 0 and Γ^{∘} is non-trivial, then (K, ν̇) is always of equal characteristic (0, 0).

Fact 5.2. Let (K, ν, Γ) be valued field. Let ν̇ be the coarse valuation and K^{∘} be the core field with respect to Γ^{∘} for a non-trivial convex subgroup Γ^{∘} of Γ.

(1) Let R_{ν}, Rν̇, and R_{ν^{∘}} be the valuation rings of (K, ν), (K, ν̇), and (K^{∘}, ν^{∘}) respectively. Then (prν̇)^{−1}(R_{ν^{∘}}) = R_{ν}.

(2) If (K, ν) is finitely ramified, then (K^{∘}, ν^{∘}) is finitely ramified, and K and K^{∘} have the same ramification index.

(3) If (K, ν) is finitely ramified and ℵ_{1}-saturated, and Γ^{∘} is the smallest nontrivial convex subgroup, then (K^{∘}, ν^{∘}) is complete.

Definition 5.3. Let (K, ν, Γ) be valued field. Let Γ^{∘} ⊂ Γ be a non-trivial convex subgroup and γ ∈ (Γ^{∘})_{>0}. Let 0_{Γ^{∘}}^{γ} := {α ∈ H_{γ}(K)| ν_{H}(α) > Γ^{∘}}. If γ is obvious, we write 0_{Γ^{∘}}^{γ} = 0_{Γ^{∘}}. Define H_{γ}^{×}(K, Γ^{∘}) := {α ∈ H_{γ}(K) : ν_{H}(α) ∈ Γ^{∘}} and H_{γ}(K, Γ^{∘}) := H_{γ}^{×}(K, Γ^{∘}) ∪ {0_{Γ^{∘}}^{γ}}.

Remark 5.4. Let Γ^{∘} be a non-trivial convex subgroup of Γ. Then H_{γ}(K, Γ^{∘}) forms a valued hyperfield.

Proof. Let (H_{γ}(K), +_{H}, ·_{H}, ν_{H}) be the valued γ-hyperfield of K. First, H_{γ}^{×}(K, Γ^{∘}) is a multiplicative subgroup of (H_{γ}^{×}(K), ·_{H}) because Γ^{∘} is a subgroup of Γ. Note that for α, β ∈ H_{γ}^{×}(K), if there is x ∈ α+_{H}β such that ν(x) > Γ^{∘}, then 0_{Γ^{∘}} ⊂ α+_{H}β(†). For α = [a] and β = [b], ⋃ α +_{H} β = (a + b) + 𝔪^{γ+min{ν(a),ν(b)}} by Lemma 3.1(2). Suppose x ∈ ⋃ α +_{H} β such that ν(x) > Γ^{∘}. If ν(a + b) < γ + min{ν(a), ν(b)}, then ν(x) = ν(a + b) < γ + min{ν(a), ν(b)} ∈ Γ^{∘} because γ, ν(a), ν(b) ∈ Γ^{∘}. So we have that ν(a + b) ≥ γ + min{ν(a), ν(b)} and ⋃ α +_{H} β = 𝔪^{γ+min{ν(a),ν(b)} ⊃ 0_{Γ^{∘}}. We define a multivalued operation +_{H}^{Γ^{∘}} on H_{γ}(K, Γ^{∘}) as follows: For α, β ∈ H_{γ}^{×}(K),

• α +_{H}^{Γ^{∘}} 0_{Γ^{∘}} = α = 0_{Γ^{∘}} +_{H}^{Γ^{∘}} α.

• α+_{H}^{Γ^{∘}} β = \begin{cases}α +_{H} β & if ∀x ∈ ⋃ α +_{H} β, ν(x) ∈ Γ^{∘} \\ (α +_{H} β) ∩ (H_{n})^{×}(K) ∪ \{0_{Γ^{∘}}\} & if ∃x ∈ ⋃ α +_{H} β, ν(x) > Γ^{∘} \end{cases}.

By (†) and covexity of Γ^{∘}, +_{H}^{Γ^{∘}} is well-defined. Define a map ν_{H}^{Γ^{∘}} on H_{γ}(K, Γ^{∘}) as follows: For α ∈ H_{γ}(K, Γ^{∘}), ν_{H}^{Γ^{∘}}(α) = ν_{H}(α) if α ≠ 0_{Γ^{∘}} and ν_{H}^{Γ^{∘}}(α) = ∞ if α = 0_{Γ^{∘}}. Then (H_{γ}(K, Γ^{∘}), +_{H}^{Γ^{∘}}, ·_{H}, ν_{H}^{Γ^{∘}}) forms a valued hyperfield. □

Lemma 5.5. Let (K, ν, Γ) be valued field. Let ν̇ be the coarse valuation and K^{∘} be the core field with respect to Γ^{∘} for a non-trivial convex subgroup Γ^{∘} of Γ. For each γ ∈ Γ_{>0}^{∘}, H_{γ}(K, Γ^{∘}) and H_{γ}(K^{∘}) are isomorphic.

Proof. Consider a map f : H_{γ}(K, Γ^{∘}) → H_{γ}(K^{∘}) defined as follows: For α ∈ H_{γ}(K, Γ^{∘}), f(α) = [a^{∘}] if α = [a], and f(α) = 0 if α = 0_{Γ^{∘}}. The map f is welldefined. Suppose [a] = [b] ∈ H_{γ}(K, Γ^{∘}). Then a/b ∈ (1+𝔪^{γ}) and ν(a) = ν(b) ∈ Γ^{∘}. So, we have that (a/b)^{∘} = a^{∘}/b^{∘} ∈ (1 + (𝔪^{∘})^{γ}), and it implies [a^{∘}] = [b^{∘}]. And it is clear that for all x ∈ K with ν(x) > Γ^{∘}, x^{∘} = 0. It is routinely to check that f is a homomorphism. □

We recall the following facts before proving a relative completeness theorem.

Fact 5.6. (Keisler-Shelah Isomorphism Theorem) Let M and N be two first order structures. If M ≡ N, then there is a ultrafilter U on an infinite set I such that

M^{U} ≅ N^{U},

where M^{U} and N^{U} are the ultrapowers of M and N with respect to U.

Fact 5.7. [5, Lemma 1.5] Let (K, ν) be a finitely ramified henselian valued field of mixed characteristic (0, p). Then the valuation ring R(K) of (K, ν) is definable by the formula

ϕ_{q}(x) ≡ ∃y y^{q} = 1 + px^{q}

for some q > 0 such that p |̸q and q > e_{ν}(p).

Theorem 5.8. Let (K_{1}, ν_{1}, k_{1}, Γ_{1}) and (K_{2}, ν_{2}, k_{2}, Γ_{1}) be finitely ramified henselian valued fields of mixed characteristic (0, p). Suppose k_{1} and k_{2} are perfect fields. Let n > e_{ν_{2}}(p)(1 + e_{ν_{1}}^{2}(p)). The following are equivalent:

(1) K_{1} ≡ K_{2}.

(2) R_{n}(K_{1}) ≡ R_{n}(K_{2}) and Γ_{1} ≡ Γ_{2}.
(3) H_{n}(K_{1}) ≡_{H_{n}({p})} H_{n}(K_{2}).

Proof. For (1) ⇔ (2), see Fact 2.27. It is clear that (1) implies (3). We show (3) implies (1). Suppose (3) holds. By Fact 5.6, we may assume that H_{n}(K_{1}) ≅_{H_{n}(ℤ)} H_{n}(K_{2}). By Remark 2.8, Γ_{1} ≅ Γ_{2} and we may assume that Γ_{1} = Γ_{2} = Γ. By taking ultrapowers of K_{1} and K_{2} with respect to a nonprincipal ultrafilter on ℕ, we may assume that K_{1} and K_{2} are ℵ_{1}-saturated. Let ν̇_{1} and ν̇_{2} be the coarse valuation of ν_{1} and ν_{2} with respect to the smallest non-trivial convex subgroup Γ^{∘}. We have two valued fields (K_{1}, ν̇_{1}) and (K_{2}, ν̇_{2}) of equal characteristic (0, 0) with residue fields K_{1}^{∘} and K_{2}^{∘} respectively. By Fact 5.2(2) and (3). (K_{1}^{∘}, ν_{1}^{∘}) and (K_{2}^{∘}, ν_{2}^{∘}) are complete discrete valued fields of mixed characteristic (0, p). Note that the value groups of (K_{1}, ν̇_{1}) and (K_{2}, ν̇_{2}) are Γ/Γ^{∘}. Since H_{n}(K_{1}) ≅_{H_{n}(ℤ)} H_{n}(K_{2}), we have that H_{n}(K_{1}^{∘}) ≅_{H_{n}(ℤ)} H_{n}(K_{1}^{∘}) by Lemma 5.5. Since e_{ν_{i}^{∘}}(p) = e_{ν_{i}}(p) for i = 1, 2, we have the K_{1}^{∘} ≅ K_{2}^{∘} by Corollary 3.19. By Fact 2.26, we have that (K_{1}, ν̇_{1}) ≡ (K_{2}, ν̇_{2}). To show that (K_{1}, ν_{1}) ≡ (K_{2}, ν_{2}), it is enough to show that the valuation rings R_{ν}(K_{1}) of (K_{1}, ν_{1}) and R_{ν}(K_{2}) of (K_{2}, ν_{2}) are definable in (K_{1}, ν̇_{1}) and (K_{2}, ν̇_{2}) by the same formula. Recall the following result on a definability of a residue ring. Take l > 0 large enough so that q := p^{l} + 1 > max{e_{ν_{1}}(p), e_{ν_{2}}(p)}.

By Fact 5.7, ϕ_{q}(x) defines the residue rings R_{ν^{∘}}(K_{1}) and R_{ν^{∘}}(K_{2}) of (K_{1}^{∘}, ν_{1}^{∘}) and (K_{2}^{∘}, ν_{2}^{∘}). By Fact 5.2(2), the valuation rings R_{ν}(K_{1}) and R_{ν}(K_{2}) are definable by the same formula in (K_{1}, ν̇_{1}) and (K_{2}, ν̇_{1}) so that (K_{1}, ν_{1}) ≡ (K_{2}, ν_{2}). □

By Theorem 3.7 and the proof of (3) ⇒ (1) of Theorem 5.8, we have the following result.

Corollary 5.9. Let (K_{1}, ν_{1}, k_{1}, Γ_{1}) and (K_{2}, ν_{2}, k_{2}, Γ_{1}) be finitely tamely ramified henselian valued fields of mixed characteristic (0, p). Suppose k_{1} and k_{2} are perfect fields. The following are equivalent:

(1) K_{1} ≡ K_{2}.

(2) H_{1}(K_{1}) ≡_{H_{1}({p})} H_{1}(K_{2}).

For local fields of mixed characteristic, they are elementary equivalent if and only if they are isomorphic. So we have the following corollary.

Corollary 5.10. Let K_{1} and K_{2} be local fields of mixed characteristic. Let n > e_{ν_{2}}(p)(1 + e_{ν_{1}}^{2}(p)). The followings are equivalent:

(1) H_{n}(K_{1}) ≡_{H_{n}({p})} H_{n}(K_{2}).

(2) H_{n}(K_{1}) ≅_{H_{n}({p})} H_{n}(K_{2}).

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Instytut Matematyczny, Uniwersytet Wrocawski, pl. Grunwaldzki 2/4, 50-384 Wrocaw, Poland

E-mail address: jlee@math.uni.wroc.pl