Ramification of valuations

Cutkosky, Steven Dale; Piltant, Olivier

doi:10.1016/s0001-8708(03)00082-3

Cited by 44 publications

(236 citation statements)

References 20 publications

(67 reference statements)

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“…The following result is [13] (7) and (8) are a rephrasing of the statement "gn = 1 for rt > > 0" in loc.cit. For 6 = 0, the inequality ni > 1 follows from the definitions in [13] and for 6 > 0, the inequality ni > p6 follows from ibid.…”

mentioning

confidence: 93%

“…In the last few years, de Jong's foundational paper [17] triggered renewed interest in Jung's method; some of Abhyankar's conjectures have been studied and proved recently by Cutkosky [9] [10] (see also [11] and [13]). A valuative version of de Jong's theorem is proved in [22] using ramification theoretic methods (see also [20]).…”

mentioning

confidence: 99%

“…For 6 = 0, the inequality ni > 1 follows from the definitions in [13] and for 6 > 0, the inequality ni > p6 follows from ibid. Lemma 7.29 (2).…”

mentioning

confidence: 99%

“…Equation (18) Case 2. a = p, ordy (v mod x) = 1. The pair (R,,, S8l) is defined by equations (13) and (14). That any regular model R' of V with R C R' c R,, satisfies the hypotheses of …”

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confidence: 99%

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On the Jung method in positive characteristic

Piltant¹

2003

Annales De L’institut Fourier

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mentioning

confidence: 93%

mentioning

confidence: 99%

“…For 6 = 0, the inequality ni > 1 follows from the definitions in [13] and for 6 > 0, the inequality ni > p6 follows from ibid. Lemma 7.29 (2).…”

mentioning

confidence: 99%

“…Equation (18) Case 2. a = p, ordy (v mod x) = 1. The pair (R,,, S8l) is defined by equations (13) and (14). That any regular model R' of V with R C R' c R,, satisfies the hypotheses of …”

mentioning

confidence: 99%

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On the Jung method in positive characteristic

Piltant¹

2003

Annales De L’institut Fourier

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“…The first author has shown with Olivier Piltant in [9] that a relative local uniformization theorem holds for the extension K * of K, which gives the strongest possible generalization of the classical ramification theory of Dedekind domains to general valuations. The following theorem is a summary of some of the conclusions of Theorem 6.3 [9].…”

Section: Introductionmentioning

confidence: 99%

Completions of valuation rings

Cutkosky

Ghezzi

2005

Recent Progress in Arithmetic and Algebraic Geometry

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Abstract. Let k be a field of characteristic zero, K an algebraic function field over k, and V a k-valuation ring of K. Zariski's theorem of local uniformization shows that there exist algebraic regular local rings R i with quotient field K which are dominated by V , and such that the direct limit ∪R i = V.We investigate the ring T = ∪R i . The ring T is Henselian and thus can be considered to be a "completion" of the valuation ring V . We give an example showing that T is in general not a valuation ring. Making use of a result of Heinzer and Sally, we give necessary and sufficient conditions for T to be a valuation ring.The essential obstruction to T being a valuation ring is the problem of the rank of the valuation increasing upon extending the valuation dominating a particular R to a valuation dominating its completion. In the case of rank 1 valuations, we show that this problem can be handled in a very satisfactory way.Finally, suppose that K * is a finite algebraic extension of K and V * is a rank 1 k-valuation ring of K * such that V = V * ∩ K. We obtain a relative local uniformization theorem for the extension K * of K, that generalizes previous results of Cutkosky and Piltant.

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Essentially finite generation of valuation rings in terms of classical invariants

Cutkosky

Novacoski

2020

Mathematische Nachrichten

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The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field and an extension ω of ν to a finite extension L of K. Then we study when the valuation ring of ω is essentially finitely generated over the valuation ring of ν. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).

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Ramification of valuations

Cited by 44 publications

References 20 publications

On the Jung method in positive characteristic

On the Jung method in positive characteristic

Completions of valuation rings

Essentially finite generation of valuation rings in terms of classical invariants

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