Questions tagged [valuation-rings]

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15 votes
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Higher-rank Archimedean valuations of $\mathbb{Q}$, does it exist?

I was reading the proof of Ostrowski's theorem, with an eye toward the Zariski-Riemann space (as well as adic space, Berkovich space, etc.) In the proof, the value group is always assumed to be in $\...
zxx's user avatar
  • 179
10 votes
1 answer

Is it a valuation ring?

It is known that a one-dimensional Noetherian local ring is a discrete valuation ring if it is integrally closed. Then, even if it is not Noetherian, would a one-dimensional local ring become a ...
George's user avatar
  • 417
5 votes
0 answers

Is there a good notion of higher-rank archimedean norm?

Let $K$ be a field. I think I know what a norm (archimedean or not) $|-| : K \to \mathbb R_{\geq 0}$ is. In the case where the norm is nonarchimedean, it's equivalent to the data of a valuation of ...
Tim Campion's user avatar
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3 votes
0 answers

Frobenius and mixed characteristic valuation rings

Let $R$ be an $\mathbf{F}_p$-algebra. Kunz's theorem says that if $R$ is Noetherian, then the Frobenius of $R$ is flat iff $R$ is regular. Following the philosophy that valuation rings often behave ...
skd's user avatar
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1 vote
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Derivation for genus-degree formula from algebraic functions field theory

This is a copy of my question from math.stackexchange: I didn't get any ...
Konstantce's user avatar
12 votes
1 answer

Does every map from a noetherian ring to a valuation ring factor through a DVR?

Let R be a noetherian ring and V a valuation ring with maximal ideal $\mathfrak{m}_V$. Does every morphism of rings $\varphi: R \rightarrow V$ factor through a discrete valuation ring? One may ...
BnPrs's user avatar
  • 195
2 votes
0 answers

References on topological ringed spaces

This is a follow up to this question of mine. First of all, let me fix some terminologies, which may or may not be standard: Definition: A topological ringed space is a pair $X := (|X|, \mathcal{O}_X)...
Dat Minh Ha's user avatar
  • 1,432
1 vote
1 answer

Flatness criterion for $I$-adic ring: $I$-torsion free

Let $R$ be an $I$-adically separated and complete valuation ring, with $I$ finitely generated. It is used a few times in Bosch, Lectures on Formal and Rigid Geometry e.g. first lines of pg. 164, Cor. ...
Bryan Shih's user avatar
9 votes
1 answer

Is every field the residue field of a discretely valued field of characteristic 0?

Let $k$ be a field of positive characteristic $p$. Is there necessarily a discrete valuation ring of characteristic $0$ with maximal ideal $(p)$ and residue field isomorphic to $k$?
Alex Mennen's user avatar
  • 2,080
1 vote
0 answers

Approximating a scheme with irreducible fibers

Let $R$ be a (non-trivial) valuation ring of an algebraically closed field. Let $V$ be an integral affine scheme over $S=\mathrm{Spec} R$ and assume that $V(R)\neq \emptyset$ (so e.g. $V$ is ...
yatir's user avatar
  • 133
1 vote
1 answer

An example of a special $1$-dimensional non-Noetherian valuation domain

I am looking for a $1$-dimensional non-Noetherian valuation domain $R$ such that there exists a sequence $\{a_i\}_{i=1}^\infty$ of elements of $R$ such that $\langle a_1\rangle \subsetneqq\langle a_2\...
S. T. Stanly's user avatar
1 vote
1 answer

A question about Dedekind schemes and proper morphisms

The following is Exercise 15.3 of Görtz-Wedhorn Algebraic Geometry I: Let $S$ be a Dedekind scheme with function field $K$ and let $f: X\to S$ be a proper morphism of schemes. Then the canonical map $...
Lao-tzu's user avatar
  • 1,836
1 vote
1 answer

non-archimedean valuations on graded rings

Let $R$ be a commutative (non-trivially) graded ring. By a non-archimedean valuation I mean a map $v: R \to \Gamma \cup {0}$ such that for all $x,y \in R$, we have $v(x+y) \leq \max\{v(x),v(y)\}$, $v(...
Leray Jenkins's user avatar
3 votes
3 answers

What does "trait" mean?

Looking at some French papers, it seems that the word "trait" is often used to refer to the spectrum of a discrete valuation ring $A$. Does anyone know what the translation of this should be? Is it ...
Kim's user avatar
  • 3,994
6 votes
1 answer

Classifying Space of "Valuation Ringed Spaces over a Topos"

The classifying topos for local rings is the big Zariski topos of $\text{Spec}(\mathbb{Z})$. Call this topos $T$. Geometric maps of topoi from a topos $T'$ to $T$ are in correspondence with sheaves of ...
Cayley-Hamilton's user avatar
12 votes
1 answer

Extension of 2-adic valuation to the real numbers

I just want to know what properties of valuations extend to $\mathbb R$... Denote an extension of the 2-adic valuation from $\mathbb Q$ to $\mathbb R$ by $\nu$. Suppose $\nu(x)=\nu(y)=0$. Is it true ...
domotorp's user avatar
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1 vote
0 answers

Does separability of residue fields implies separability of $L/K$?

Let $A$ be discrete valuation domain, and $K$ be quotient field of $A$. Let $L$ be a finite extension of $K$ and $B$ be the integral closure of $A$ in $L$.Does separability of residue fields implies ...
SUNIL PASUPULATI's user avatar
30 votes
0 answers

On the definition of regular (non-noetherian, commutative) rings

All rings are commutative with unit. A ring $R$ is called regular if it satisfies (Reg) Every finitely generated ideal of $R$ has finite projective dimension. Clearly this gives the usual ...
Laurent Moret-Bailly's user avatar
6 votes
0 answers

The Zariski Riemann Space, but with Local Rings

The Zariski Riemann space, while an abandoned approach, has lead to later developments and generalizations, including $\text{Spv}$ (the space of valuations) and Huber's work. In studying it, I would ...
Cayley-Hamilton's user avatar
4 votes
1 answer

Is the integral closure of a valuation ring in a finite separable extension of its fraction field étale?

Let $K$ be a field endowed with a rank (height) one valuation with completion $\hat{K}$, which is not discrete. Let $R$ be the valuation ring of $K$. Let $L \subset \hat{K}$ be a separable finite ...
userm's user avatar
  • 41
6 votes
1 answer

Valuation Rings and Ultrafilters

I notice there is a certain similarity between the definition of a valuation ring and the definition of an ultrafilter. To begin, take a field $K$ and let $\mathcal{A}$ be the set of subrings of $K$. ...
Cayley-Hamilton's user avatar
3 votes
1 answer

What is K+M structure?

In the following paper (Example 2.1), it has been mentioned to K+M to provide an example of a pseudo valuation domain which is not a valuation ring, and its reference is Gilmer's book, but I have no ...
Mahsa Pavenejad's user avatar
1 vote
1 answer

Lifting Lang-Steinberg to DVR's in Characteristic 0

Let $A$ be a compact DVR in characteristic $0$, uniformizer $\pi$ and residue field $k$. Let $A\subset B$ be a complete DVR with the same uniformizer $\pi$ and algebraicly closed residue field $F$. ...
Estus's user avatar
  • 273
4 votes
1 answer

Given a non-field local domain $R$, finding a dominating Valuation ring whose residue field is algebraic/finite extension of the residue field of $R$

Let $(R, \mathfrak m)$ be a non-field local domain with fraction field $Q(R)$ . Let $k_{R}:=R/m$. We know that there is a Valuation ring $(V,\mathfrak m_V)$ such that $R \subseteq V \subsetneq Q(R)$ ...
user avatar
6 votes
2 answers

Torsors over complete local fields

Let $G$ be a linear algebraic group scheme, and let $R$ be a complete discrete valuation ring, with quotient field $K$ and residue field $k$. If $T$ is an $R$-torsor, it yields by base change a $k$-...
GreginGre's user avatar
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5 votes
6 answers

an easy example of valuation ring which is not noetherian? [duplicate]

Is there an easy example of valuation ring which is not noetherian?
iff's user avatar
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