Extensions of Truncated Discrete Valuation Rings

Hiranouchi, Toshiro; Taguchi, Yuichiro

doi:10.4310/pamq.2008.v4.n4.a9

Cited by 4 publications

(11 citation statements)

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“…Let B be a finite flat A-algebra. The aim of this section is to study ramification of the extension B/A, as in [8] and [11].…”

Section: Non-log Ramificationmentioning

confidence: 99%

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Ramification theory and perfectoid spaces

Hattori

2014

Compositio Math.

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Abstract. Let K1 and K2 be complete discrete valuation fields of residue characteristic p > 0. Let πK 1 and πK 2 be their uniformizers. Let L1/K1 and L2/K2 be finite extensions with compatible isomorphisms of rings) for some positive integer m which is no more than the absolute ramification indices of K1 and K2. Let j ≤ m be a positive rational number. In this paper, we prove that the ramification of L1/K1 is bounded by j if and only if the ramification of L2/K2 is bounded by j. As an application, we prove that the categories of finite separable extensions of K1 and K2 whose ramifications are bounded by j are equivalent to each other, which generalizes a theorem of Deligne to the case of imperfect residue fields. We also show the compatibility of Scholl's theory of higher fields of norms with the ramification theory of Abbes-Saito, and the integrality of small Artin and Swan conductors of abelian extensions of mixed characteristic.

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“…Let B be a finite flat A-algebra. The aim of this section is to study ramification of the extension B/A, as in [8] and [11].…”

Section: Non-log Ramificationmentioning

confidence: 99%

“…Acknowledgments. The author would like to thank Yuichiro Taguchi for stimulating discussions, answering questions on the paper [11], valuable comments on earlier drafts and the invitation to Korea Institute for Advanced Study (KIAS), where a part of this work was carried out. He is grateful for the hospitality provided by KIAS.…”

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confidence: 99%

Ramification theory and perfectoid spaces

Hattori

2014

Compositio Math.

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“…First we recall some basic notions on tdvr's from [3] and [4]. A tdvr is an Artinian local ring whose maximal ideal is generated by one element.…”

Section: Flatness Of Modules Over a Tdvrmentioning

confidence: 99%

“…If O is a complete discrete valuation ring and m is its maximal ideal, then O=m a is a tdvr for any integer a > 1. Conversely, it is known that any tdvr is a quotient of a complete discrete valuation ring ( [3], Prop. 2.2).…”

Section: Flatness Of Modules Over a Tdvrmentioning

confidence: 99%

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Flat Modules and Gröbner Bases over Truncated Discrete Valuation Rings

Hiranouchi¹,

Taguchi

2010

IIS

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Truncations of ordered abelian groups

2021

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We give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a $$+$$ + , but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments $$[0, \tau ]$$ [ 0 , τ ] of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group $${\mathbb {Z}}$$ Z or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.

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Extensions of Truncated Discrete Valuation Rings

Cited by 4 publications

References 0 publications

Ramification theory and perfectoid spaces

Ramification theory and perfectoid spaces

Flat Modules and Gröbner Bases over Truncated Discrete Valuation Rings

Truncations of ordered abelian groups

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