On the structure of certain valued fields

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In this article, we study the structure of finitely ramified mixed characteristic valued fields. 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 residue rings are isomorphic for each n >= 1, then K-1 and K-2 are isometric and isomorphic. More generally, for n(1) >= 1, there is n(2) depending only on the ramification indices of K-1 and K2 such that any homomorphism from the n(1)-th residue ring of K1 to the n(2)-th residue ring of K-2 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. (C) 2020 Elsevier B.V. All rights reserved.
Publisher
ELSEVIER
Issue Date
2021-04
Language
English
Article Type
Article
Citation

ANNALS OF PURE AND APPLIED LOGIC, v.172, no.4

ISSN
0168-0072
DOI
10.1016/j.apal.2020.102927
URI
http://hdl.handle.net/10203/282542
Appears in Collection
RIMS Journal Papers
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