On the structure of certain valued fields

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Abstract

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields K1 and K2 of mixed characteristic with perfect residue fields, we show that if the n-th residue rings are isomorphic for each n1, then K1 and K2 are isometric and isomorphic. More generally, for n11, there is n2 depending only on the ramification indices of K1 and K2 such that any homomorphism from the n1-th residue ring of K1 to the n2-th residue ring of K2 can be lifted to a homomorphism between the valuation rings. Moreover, we get a functor from the category of certain principal Artinian local rings of length n to the category of certain complete discrete valuation rings of mixed characteristic with perfect residue fields, which naturally generalizes the functorial property of unramified complete discrete valuation rings. Our lifting result improves Basarab's relative completeness theorem for finitely ramified henselian valued fields, which solves a question posed by Basarab, in the case of perfect residue fields.

Introduction

In this paper, we are interested in finitely ramified mixed characteristic valued fields (see Definition 2.3). In model theory of valued fields, one of the most important theorems is the AKE-principle, proved by Ax and Kochen in [1], [2], and independently by Ershov in [7], [8]. The AKE-principle says that the theory of an unramified henselian valued field of characteristic 0 is determined by the theory of the residue field and the theory of the value group.

Fact 1.1 The Ax-Kochen-Ershov principle

[1], [2], [7], [8] Let (Ki,ki,Γi) be an unramified henselian valued field of characteristic zero, where ki is the residue field and Γi is the valuation group respectively, for i=1,2. K1K2 if and only if k1k2 and Γ1Γ2.

Basarab in [4] generalized the AKE-principle to the finitely ramified case. Actually, he showed that the theory of a finitely ramified henselian valued fields of mixed characteristic is determined by the theory of each n-th residue ring (see Definition 2.8), the quotient of the valuation ring by the n-th power of the maximal ideal and the theory of the valuation group.

Fact 1.2

[4] Let (Ki,Ri,(n),Γi) be finitely ramified henselian valued fields of mixed characteristic, where Ri,(n) is the n-th residue ring and Γi is the valuation group respectively for i=1,2. The following are equivalent:

  • (1)

    K1K2.

  • (2)

    R1,(n)R2,(n) for each n1 and Γ1Γ2.

Motivated by Fact 1.2, we ask the following related question on isomorphisms.

Question 1.3

Given two complete discrete valued fields K1 and K2 of mixed characteristic with perfect residue fields, if the n-th residue rings of K1 and K2 are isomorphic for each n1, then are K1 and K2 isomorphic? Moreover, is there N>0 such that K1 and K2 are isomorphic if the N-th residue rings of K1 and K2 are isomorphic?

We give a comment on Question 1.3. Macintyre in [16] raised the following question on the problem of lifting of homomorphisms of the n-th residue rings for more general rings.

Question 1.4

Are two complete local noetherian rings A and B isomorphic if the n-th residue rings of A and B are isomorphic for each n1?

In [16], van den Dries gave a positive answer to Question 1.4 in the case that the residue fields are algebraic over their prime fields. Furthermore, given complete local noetherian rings A and B, it is enough to check whether the N-th residue rings of A and B are isomorphic for some N=N(A,B) depending on A and B. Note that van den Dries showed the existence of a non explicit bound N, and in general, there is a counter example by Gabber in [16] for Question 1.4.

Next we recall the following well-known fact on unramified complete discrete valuation rings.

Fact 1.5

[15]

  • (1)

    Let k be a perfect field of characteristic p. Then there exists a complete discrete valuation ring of characteristic 0 which is unramified and has k as its residue field. Such a ring is unique up to isomorphism. This unique ring is called the ring of Witt vectors of k, denoted by W(k).

  • (2)

    Let R1 and R2 be complete discrete valuation rings of mixed characteristic with perfect residue fields k1 and k2 respectively. Suppose R1 is unramified. Then for every homomorphism ϕ:k1k2, there exists a unique homomorphism g:R1R2 making the following diagram commutative: where two vertical maps are the canonical epimorphisms.

In categorical setting, Fact 1.5 is equivalent to the following statement.

Fact 1.6

Let Cp be the category of complete unramified discrete valuation rings of mixed characteristic (0,p) with perfect residue fields and Rp the category of perfect fields of characteristic p. Then Cp is equivalent to Rp. More precisely, there is a functor L:RpCp which satisfies:

  • PrL is equivalent to the identity functor IdRp where Pr:CpRp is the natural projection functor.

  • LPr is equivalent to IdCp.

Based on Question 1.3 and Fact 1.6, we ask the following generalized questions for the finitely ramified case.

Question 1.7

  • (1)

    For a principal Artinian local ring R of length n with a perfect residue field, is there a unique complete discrete valuation ring R which has R as its n-th residue ring? Moreover, if it has a positive answer, can a lower bound for such n be effectively computed in terms of the ramification index of R?

  • (2)

    Given complete discrete valuation rings R1 and R2 of mixed characteristic with perfect residue fields, let R1,(n1) and R2,(n2) be the n1-th residue ring of R1 and the n2-th residue ring of R2 respectively. If n1 and n2 are large enough, is there a unique lifting homomorphism g:R1R2 such that g induces a given homomorphism ϕ:R1,(n1)R2,(n2)? Moreover, can such lower bounds on n1 and n2 be effectively computed in terms of the ramification indices of R1 and R2?

Question 1.8

Let Cp,e be the category of complete discrete valuation rings of mixed characteristic (0,p) with perfect residue fields and ramification index e. For n>e, let Rp,en be the category of principal Artinian local rings of length n having ramification index e and perfect residue fields (see at the beginning of Section 4 for the precise definition). Let Prn:Cp,eRp,en be the natural projection functor. Is there a lifting functor L:Rp,enCp,e which satisfies:

  • PrnL is equivalent to IdRp,en.

  • LPrn is equivalent to IdCp,e.

In general, the answer for Question 1.7.(2) is not positive, that is, there is a homomorphism ϕ:R1,n1R2,n2 such that no homomorphism from R1 into R2 induces ϕ (see Example 3.5). Instead of finding a ‘usual’ lifting in the sense of Question 1.8, we will show that for sufficiently large n2, if there is a given homomorphism ϕ:R1,(n1)R2,(n2), then there is an ‘approximate’ lifting g:R1R2 of ϕ (see Definition 3.4).

Let us come back to the question of elementary equivalence. In [4], Basarab posed the following question (see [4, pp. 23–24]):

Question 1.9

For a finitely ramified henselian valued field K of ramification index e, is there a finite integer N1 depending on K such that any finitely ramified henselian valued field of the same ramification index e is elementarily equivalent to K if their N-th residue rings are elementarily equivalent and their value groups are elementarily equivalent?

Given a finitely ramified henselian valued field K, Basarab in [4] denoted the minimal number N, which satisfies the condition in Question 1.9, by λ(T) for the complete theory T of K. He showed that λ(T) for a local field K is finite but did not give any explicit value of λ(T).

The goal of this paper is to answer these questions when the residue fields are perfect. Its organization is as follows. In Section 2, we recall basic definitions and facts. In Section 3, we answer Question 1.3 positively for the perfect residue field case in Theorem 3.7. Our main result shows that if n2 is sufficiently large, then for a given homomorphism ϕ:R1,(n1)R2,(n2), there is a homomorphism L(ϕ):R1R2 satisfying a lifting property similar to that of the unramified case. This provides an answer for Question 1.3. Also, the lifting map L provides an answer for Question 1.7.(2) and Question 1.7.(1). In Section 4, we concentrate on Question 1.8. We can show that L is compatible with the composition of homomorphisms between residue rings. More precisely, L(ϕ2ϕ1)=L(ϕ2)L(ϕ1) for any ϕ1:R1,(n1)R2,(n2) and ϕ2:R2,(n2)R3,(n3). This defines a functor L:Rp,enCp,e for sufficiently large n. We prove that a lower bound for n depends only on the ramification index e and the prime number p. Even though L does not give an equivalence between Rp,en and Cp,e, it turns out that L satisfies a similar functorial property to that of L:RpCp. This provides an answer for Question 1.8. In Section 5, we reduce the problem on elementary equivalence between finitely ramified henselian valued fields of mixed characteristic to the problem on isometricity between complete discrete valued fields of mixed characteristic. Using results in Section 3, we improve Basarab's result on the AKE-principle which gives a positive answer to Question 1.9 when the residue fields are perfect. Under certain conditions, we calculate λ(T) explicitly for the tame case and get a lower bound for λ(T) for the wild case. Surprisingly we show that λ(T) can be 1 even when K is not unramified. As a special case, we conclude that λ(T) is 1 or e+1 if pe, and λ(T)e+1 if p|e when K is a finitely ramified henselian subfield of Cp with the ramification index e.

Section snippets

Preliminaries

In this section, we introduce basic notations, terminologies, and several preliminary facts which will be used in this paper. We denote a valued field by a tuple (K,R,m,ν,k,Γ) consisting of the following data: K is the underlying field, R is the valuation ring, m is the maximal ideal of R, ν is the valuation, k is the residue field, and Γ is the value group. Hereafter, the full tuple (K,R,m,ν,k,Γ) will be abbreviated in accordance with the situational need for the components. For any field L, La

Lifting homomorphisms

From now on, if there is no comment, we consider only complete discrete valued fields of mixed characteristic (0,p) with perfect residue fields, and we assume that valuation groups are Z so that for a valued field (L,R,ν), ν(x)=eν(x) for xR. Let (R,ν,k) be a valuation ring. Let π be a uniformizer of R. Let L and K be the fraction fields of R and W(k) respectively.

Definition 3.1

If L is ramified, we denote the maximal valuemax{ν˜(πσ(π)):σHomK(L,Lalg),σ(π)π} by M(R)π or M(L)π.

Lemma 3.2

Let (Ri,mi,νi,ki) be a

Functoriality

The main purpose of this section is to give a generalized version of Fact 1.6 for the ramified case. For a prime number p and a positive integer e, let Cp,e be a category consisting of the following data:

  • Ob(Cp,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

  • MorCp,e(R1,R2):=Hom(R1,R2) for R1 and R2 in Ob(Cp,e).

Let Rp,en be a category consisting of the following data:
  • For ne, Ob(Rp,en)

Ax-Kochen-Ershov principle for finitely ramified valued fields

Our main goal in this section is to strengthen Basarab's result on relative completeness for finitely ramified henselian valued fields of mixed characteristic with perfect residue fields. In this section, we drop the restriction that a valuation group is Z so that a valuation group can be an arbitrary ordered abelian group. Recall that for a valued field (K,R,ν,Γ), eν(x) is the number of the positive elements of Γ less than or equal to ν(x) for xR.

Remark 5.1

Let (K1,ν1) and (K2,ν2) be finitely ramified

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Cited by (4)

The first author was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1301-03. The second author was supported by the Yonsei University Research Fund (Post Doc. Researcher Supporting Program) of 2017 (project no: 2017-12-0026). He was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1A2C1088609).

The authors thank the anonymous referee for valuable comments and suggestions, which were very helpful to reorganize our paper more effectively. The authors thank Piotr Kowalski for helpful comments. Most of all, the authors thank Thomas Scanlon for detailed and valuable suggestions and comments, which encouraged us to keep writing this article.

1

Current address: Department of Mathematical Sciences, KAIST, 291, Daehak-Ro, Yuseong-Gu, Daejeon, 34141, Republic of Korea.

2

Current address: Department of Mathematical Science, Ulsan National Institute of Science and Technology, Unist-gil 50, Ulsan 44919, Republic of Korea.

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