Nilpotence theorems via homological residue fields

This chapter focuses on diamonds associated with adic spaces. The goal is to construct a functor which forgets the structure morphism to Spa Zp, but retains topological information. The chapter studies how much information is lost when applying this construction. The intuition is that only topological information is kept. A morphism of adic spaces is a universal homeomorphism if all pullbacks are homeomorphisms. As in the case of schemes, in characteristic 0 the map f is a universal homeomorphism if and only if it is a homeomorphism and induces isomorphisms on completed residue fields. In keeping with the intuition, universal homeomorphisms induce isomorphisms of diamonds. The chapter then considers the underlying topological space of diamonds, as well as the étale site of diamonds.

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On value groups and residue fields of some valued function fields

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500018897 ◽

1994 ◽

Vol 37 (3) ◽

pp. 445-454

Author(s):

Sudesh K. Khanduja

Keyword(s):

Cyclic Group ◽

Function Field ◽

Function Fields ◽

Residue Field ◽

Torsion Group ◽

Transcendence Degree ◽

Manuscripta Math ◽

Direct Sum ◽

Residue Fields ◽

The One

Let K = K0(x, y) be a function field of transcendence degree one over a field K0 with x, y satisfying y2 = F(x), F(x) being any polynomial over K0. Let υ0 be a valuation of K0 having a residue field k0 and υ be a prolongation of υ to K with residue field k. In the present paper, it is proved that if G0⊆G are the value groups of υ0 and υ, then either G/G0 is a torsion group or there exists an (explicitly constructible) subgroup G1 of G containing G0 with [G1:G0]<∞ together with an element γ of G such that G is the direct sum of G1 and the cyclic group ℤγ. As regards the residue fields, a method of explicitly determining k has been described in case k/k0 is a non-algebraic extension and char k0≠2. The description leads to an inequality relating the genus of K/K0 with that of k/k0: this inequality is slightly stronger than the one implied by the well-known genus inequality (cf. [Manuscripta Math.65 (1989), 357–376’, [Manuscripta Math.58 (1987), 179–214]).

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Degree Functions in Local Rings

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100034794 ◽

1961 ◽

Vol 57 (1) ◽

pp. 1-7 ◽

Cited By ~ 18

Author(s):

D. Rees

Keyword(s):

Maximal Ideal ◽

Zero Element ◽

Transcendence Degree ◽

Local Rings ◽

Local Domain ◽

Degree Function ◽

Finite Set ◽

Residue Fields ◽

Image Position ◽

The Ideal

Let Q be a local domain of dimension d with maximal ideal m and let q be an m-primary ideal. Then we define the degree function dq(x) to be the multiplicity of the ideal , where x; is a non-zero element of m. The degree function was introduced by Samuel (5) in the case where q = m. The function dq(x) satisfies the simple identityThe main purpose of this paper is to obtain a formulawhere vi(x) denotes a discrete valuation centred on m (i.e. vi(x) ≥ 0 if x ∈ Q, vi(x) > 0 if x ∈ m) of the field of fractions K of Q. The valuations vi(x) are assumed to have the further property that their residue fields Ki have transcendence degree d − 1 over k = Q/m. The symbol di(q) denotes a non-negative integer associated with vi(x) and q which for fixed q is zero for all save a finite set of valuations vi(x).

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NOTES ON EXTREMAL AND TAME VALUED FIELDS

Journal of Symbolic Logic ◽

10.1017/jsl.2016.6 ◽

2016 ◽

Vol 81 (2) ◽

pp. 400-416

Author(s):

SYLVY ANSCOMBE ◽

FRANZ-VIKTOR KUHLMANN

Keyword(s):

Approximation Property ◽

Elementary Theory ◽

Residue Field ◽

Axiom System ◽

Complete Characterization ◽

Valued Fields ◽

Mixed Characteristic ◽

Valued Field ◽

Residue Fields

AbstractWe extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.

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Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

Annals of Pure and Applied Logic ◽

10.1016/j.apal.2013.06.010 ◽

2013 ◽

Vol 164 (12) ◽

pp. 1236-1246 ◽

Cited By ~ 11

Author(s):

Raf Cluckers ◽

Jamshid Derakhshan ◽

Eva Leenknegt ◽

Angus Macintyre

Keyword(s):

Valued Fields ◽

Valuation Rings ◽

Henselian Valued Fields ◽

Residue Fields

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Ramification theory and perfectoid spaces

Compositio Mathematica ◽

10.1112/s0010437x1300763x ◽

2014 ◽

Vol 150 (5) ◽

pp. 798-834 ◽

Cited By ~ 1

Author(s):

Shin Hattori

Keyword(s):

Positive Integer ◽

Rational Number ◽

Positive Rational Number ◽

Ramification Theory ◽

Local Monodromy ◽

Residue Fields ◽

Perfectoid Spaces ◽

Residue Characteristic ◽

The Absolute ◽

Separable Extensions

AbstractLet $K_1$ and $K_2$ be complete discrete valuation fields of residue characteristic $p>0$. Let $\pi _{K_1}$ and $\pi _{K_2}$ be their uniformizers. Let $L_1/K_1$ and $L_2/K_2$ be finite extensions with compatible isomorphisms of rings $\mathcal{O}_{K_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{K_2}/(\pi _{K_2}^m)$ and $\mathcal{O}_{L_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{L_2}/(\pi _{K_2}^m)$ for some positive integer $m$ which is no more than the absolute ramification indices of $K_1$ and $K_2$. Let $j\leq m$ be a positive rational number. In this paper, we prove that the ramification of $L_1/K_1$ is bounded by $j$ if and only if the ramification of $L_2/K_2$ is bounded by $j$. As an application, we prove that the categories of finite separable extensions of $K_1$ and $K_2$ 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 $p$-adic representations with finite local monodromy.

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