On the structure of certain valued 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 be an unramified henselian valued field of characteristic zero, where is the residue field and is the valuation group respectively, for . if and only if and . Fact 1.2 [4] Let be finitely ramified henselian valued fields of mixed characteristic, where is the n-th residue ring and is the valuation group respectively for . The following are equivalent: . for each and .
Question 1.3
Given two complete discrete valued fields and of mixed characteristic with perfect residue fields, if the n-th residue rings of and are isomorphic for each , then are and isomorphic? Moreover, is there such that and are isomorphic if the N-th residue rings of and are isomorphic?
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 ?
Next we recall the following well-known fact on unramified complete discrete valuation rings. Fact 1.5 [15] 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 . Let and be complete discrete valuation rings of mixed characteristic with perfect residue fields and respectively. Suppose is unramified. Then for every homomorphism , there exists a unique homomorphism making the following diagram commutative: where two vertical maps are the canonical epimorphisms.
Fact 1.6
Let be the category of complete unramified discrete valuation rings of mixed characteristic with perfect residue fields and the category of perfect fields of characteristic p. Then is equivalent to . More precisely, there is a functor which satisfies:
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is equivalent to the identity functor where is the natural projection functor.
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is equivalent to .
Question 1.7 For a principal Artinian local ring of length n with a perfect residue field, is there a unique complete discrete valuation ring R which has 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 ? Given complete discrete valuation rings and of mixed characteristic with perfect residue fields, let and be the -th residue ring of and the -th residue ring of respectively. If and are large enough, is there a unique lifting homomorphism such that g induces a given homomorphism ? Moreover, can such lower bounds on and be effectively computed in terms of the ramification indices of and ?
Question 1.8 Let be the category of complete discrete valuation rings of mixed characteristic with perfect residue fields and ramification index e. For , let 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 be the natural projection functor. Is there a lifting functor which satisfies: is equivalent to . is equivalent to .
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 depending on K such that any finitely ramified henselian valued field of the same ramification index e is elementarily equivalent to K if their -th residue rings are elementarily equivalent and their value groups are elementarily equivalent?
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 is sufficiently large, then for a given homomorphism , there is a homomorphism 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, for any and . This defines a functor 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 and , it turns out that L satisfies a similar functorial property to that of . 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 explicitly for the tame case and get a lower bound for for the wild case. Surprisingly we show that can be 1 even when K is not unramified. As a special case, we conclude that is 1 or if , and if when K is a finitely ramified henselian subfield of 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 consisting of the following data: K is the underlying field, R is the valuation ring, 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 will be abbreviated in accordance with the situational need for the components. For any field L,
Lifting homomorphisms
From now on, if there is no comment, we consider only complete discrete valued fields of mixed characteristic with perfect residue fields, and we assume that valuation groups are so that for a valued field , for . Let be a valuation ring. Let π be a uniformizer of R. Let L and K be the fraction fields of R and respectively.
Definition 3.1 If L is ramified, we denote the maximal value by or .
Lemma 3.2 Let 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 be a category consisting of the following data:
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is the family of complete discrete valuation rings of mixed characteristic having perfect residue fields of characteristic p and the ramification index e; and
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for and in .
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For ,
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 so that a valuation group can be an arbitrary ordered abelian group. Recall that for a valued field , is the number of the positive elements of Γ less than or equal to for .
Remark 5.1 Let and be finitely ramified
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Cited by (4)
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2020, arXiv
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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.
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Current address: Department of Mathematical Sciences, KAIST, 291, Daehak-Ro, Yuseong-Gu, Daejeon, 34141, Republic of Korea.
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Current address: Department of Mathematical Science, Ulsan National Institute of Science and Technology, Unist-gil 50, Ulsan 44919, Republic of Korea.