All Questions
Tagged with valuation-theory valuation-rings
9 questions
16
votes
2
answers
943
views
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 $\...
- 343
12
votes
1
answer
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 ...
- 19.1k
10
votes
1
answer
854
views
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 ...
- 328
9
votes
1
answer
782
views
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 $k$?
- 2,130
5
votes
0
answers
144
views
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 ...
- 64k
4
votes
1
answer
799
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 ...
- 41
4
votes
1
answer
348
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 $R \subseteq V \subsetneq Q(R)$ ...
user111524
3
votes
0
answers
174
views
What are non-archimedean norms on $\mathbb{R}$, whose restriction to $\mathbb{Q}$ is trivial?
I wonder if there is any classification result on non-archimedean norms on $\mathbb{R}$, with trivial restriction to $\mathbb{Q}$? Any references or examples would be welcomed!
Some examples of such ...
- 241
2
votes
0
answers
104
views
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)...
- 1,516