Compatible valuations and generalized Milnor 𝐾-theory

Efrat, Ido

doi:10.1090/s0002-9947-07-04132-3

Cited by 4 publications

(1 citation statement)

References 22 publications

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“…The Milnor K-ring of F modulo S, denoted K M * (F )/S, was defined in [Efr06a] and further developed in [Efr06b, Part IV] and [Efr07]. It generalizes the classical Milnor K-ring of F [Mil70], which corresponds to the case S = {1}.…”

Section: Relative Milnor K-ringsmentioning

confidence: 99%

Valuations and diameters of milnor K-rings

Efrat

2009

Isr. J. Math.

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Given a field F and a subgroup S of F × containing −1, we define a graph on F × /S associated with the relative Milnor K-ring K M * (F )/S. We prove that if the diameter of this graph is at least 4, then there exists a valuation v on F such that S is v-open. This is done by adopting to our setting a construction in a noncommutative setting due to Rapinchuk, Segev and Seitz. We study the behavior of the diameter under important K-theoretic constructions, and relate it to the elementary type conjecture. Finally, we provide an example showing that the above bound 4 is sharp.

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Section: Relative Milnor K-ringsmentioning

confidence: 99%

Valuations and diameters of milnor K-rings

Efrat

2009

Isr. J. Math.

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On graphs and valuations

2014

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In the last two decades new techniques emerged to construct valuations on an infinite division ring D, given a normal subgroup N ⊆ D × of finite index. These techniques were based on the commuting graph of D × /N in the case where D is non-commutative, and on the Milnor K-graph on D × /N, in the case where D is commutative. In this paper we unify these two approaches and consider V-graphs on D × /N and how they lead to valuations. We furthermore generalize previous results to situations of finitely many valuations.

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Commuting-liftable subgroups of Galois groups II

Topaz

2015

Journal Für Die Reine Und Angewandte Mathematik

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Abstract. Let n denote either a positive integer or ∞, let ℓ be a fixed prime and let K be a field of characteristic different from ℓ. In the presence of sufficiently many roots of unity in K, we show how to recover some of the inertia/decomposition structure of valuations inside the maximal ℓ n -abelian Galois group of K using the maximal ℓ N -abelian-by-central Galois group of K, whenever N is sufficiently large relative to n.

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Compatible valuations and generalized Milnor 𝐾-theory

Cited by 4 publications

References 22 publications

Valuations and diameters of milnor K-rings

Valuations and diameters of milnor K-rings

On graphs and valuations

Commuting-liftable subgroups of Galois groups II

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