Questions tagged [valuation-rings]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0
votes
1answer
128 views

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\...
0
votes
1answer
160 views

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 $...
1
vote
1answer
67 views

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(...
3
votes
2answers
350 views

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 ...
6
votes
1answer
194 views

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 ...
12
votes
1answer
2k views

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 ...
0
votes
0answers
73 views

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 ...
25
votes
0answers
494 views

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 ...
4
votes
0answers
256 views

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 ...
4
votes
1answer
403 views

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 ...
6
votes
1answer
258 views

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 $K$. ...
3
votes
1answer
87 views

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 ...
1
vote
1answer
126 views

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 $F$. ...
4
votes
1answer
128 views

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 $R \subseteq V \subsetneq Q(R)$ ...
6
votes
2answers
314 views

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 $k$-...
0
votes
0answers
569 views

valuation ring of dimension 2

I was looking the valuation ring of dimension $2$. Then I found, it has two number of non-zero prime ideals and localization at prime is a valuation domain again. Moreover, there is a one-to-one ...
6
votes
6answers
3k views

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

Is there an easy example of valuation ring which is not noetherian?