1
vote
1answer
28 views

Noetherianity of valuation ring and valuation being discrete

I need a hint for left to right part of the following: Let $K$ be a valued field with $\nu$ and $\mathcal{O}_\nu$ be its valuation ring. Then, $\mathcal{O}_\nu$ is Noetherian if and only ...
0
votes
0answers
30 views

Criterion of separability of function on a curve.

Let $K$ be finitely generated extension of an algebraically closed field $k$ of transcendence degree $1$, $\operatorname{char }k =p$. Let $C(K)$ be set of discrete valuation rings $(\mathcal ...
2
votes
0answers
64 views

A certain valuation of $k(X,Y)$ with value group $\mathbb{Z}+\mathbb{Z}\alpha$

Let $k$ be a field, $X$ and $Y$ indeterminate, and suppose that $\alpha$ is a positive irrational number. Then the map $\nu:k[X,Y]\rightarrow \mathbb{R}\cup \{\infty\}$ defined by taking $\sum ...
0
votes
1answer
34 views

Is the residue field of an algebraically closed field with respect to a non-trivial valuation infinite?

Let $K$ be an algebraically closed field with a non-trivial valuation. Let $\Bbb k$ be the residue field, i.e. $\Bbb k = R/\mathfrak m$ where $R$ is the valuation ring $R:=\{a\in K\mid \text{val} ...
1
vote
1answer
40 views

Subring of a field with a discrete valuation is a euclidean domain

I am confused by the following problem in Aluffi's Algebra Chapter Zero... A discrete valuation on a field $k$ is a surjective group homomorphism $v : k^* \to (\Bbb Z,+)$ such that $v(a + b) \geq ...
0
votes
1answer
43 views

Help in this proof on DVRs

I'm trying to understand this proof: Anyone could clarify the converse please? I really need help. Thanks a lot
4
votes
1answer
145 views

A question on valuation overrings of a PID

Let $A$ be a PID and let $K$ be its quotient field. Let $V$ be a valuation ring of $K$ containing $A$ and assume $V\neq K$. Show that $V$ is a local ring $A_{(p)}$ for some prime element $p$. I ...
2
votes
1answer
51 views

Localizations of the ring $A=\Bbb{C}[X,Y]/(Y^2-X^4+2X^2-1)$

Consider the ring $A=\Bbb{C}[X,Y]/(Y^2-X^4+2X^2-1)$. Let $A_M$ denote the localization of $A$ with respect to maximal ideal $M$. My question is: Is $A_N$ a DVR, where $N$ is the maximal ideal ...
4
votes
1answer
95 views

Definition of algebraic variety

In general, the definition of an algebraic variety differs from one reference to other. The definition that I was used to is to consider an algebraic variety as an integral scheme of finite type. ...
3
votes
0answers
70 views

Question about complete DVR.

Is there a simple proof that you know to the following statement: If the residue field $k$ of a complete DVR $R$ has the same characteristic as $R$, then $R$ contains a subfield isomorphic to $k$. ...
0
votes
0answers
57 views

Extending discrete valuation to a function field

Consider a field extension $Q$ over $L$, not necessarily finite. Let $R$ be a valuation ring in $Q$ and $A$ a DVR (discrete valuation ring) in $L$ such that $A \to R$ is local. Let $x \in Q$ be ...
4
votes
1answer
43 views

Discrete valuation on a field - equivalent statements

I have a question and I am stuck, although it should not be too difficult. We consider $K$ a field, $v$ a discrete valuation on $K$ and $O=\{x \in K:v(x)\geq 0\}$ the valuation ring of $v$. Let ...
-4
votes
1answer
103 views

Some question on localization of a polynomial ring and DVR

Let $A$ be a ring, $P$ be a prime ideal of the polynomial ring $A[x]$ and let $Q=P \cap A$. There are two questions... (1) $A[x]_P \cong A_Q[x]_{m_Q}$? (2) If $A_Q$ is a DVR then $ A_Q[x]_{m_Q}$ is ...
1
vote
1answer
119 views

Valuation Ring with value group $\Gamma=\mathbb{Z} \oplus \mathbb{Z}$

Let $\Gamma=\mathbb{Z}\oplus \mathbb{Z}$ be the free abelian group with two generators, lexicographically ordered. That is, $(a,b)\geq (a^{'},b^{'})$ iff either $a>a^{'}$ or $a=a^{'} \mathrm{and}\, ...
-2
votes
1answer
56 views

Discrete valuation rings are infinite.

Assume $F\supset k$ is a functional field. Assume $R\subset F$ is a discrete valuation ring with a quotient filed $F$, that contains the field of constants $k$. Assume $t$ is a local parameter for ...
0
votes
1answer
67 views

Discrete valuations of a functional field have discrete valuation rings.

Theorem: If $\nu:F\to\mathbb R\cup\{\infty\}$ is a valuation of a functional field, then the set $$\mathfrak O_{\nu}=\{x\in F: \nu(x)\geq 0\}$$ is a local ring with maximal ideal $$\mathfrak ...
0
votes
0answers
25 views

A question about the proof that every valuation domain has infinitely many extensions in transcendental extensions of fields.

It is known that every valuation domain has infinitely many extensions to a transcendental extension field of its quotient field. This theorem is proved in [Multiplicative Ideal Theory, by Robert ...
2
votes
0answers
70 views

How to prove that a DVR is not complete

My question is inspired by a comment in this topic. How to prove that $R=\mathbb C[x]_{(x)}$ is not complete in the topology of its maximal ideal? One knows that $R$ is a DVR, and its field of ...
0
votes
1answer
38 views

How to show that $R_{\mathfrak{m}}$ is $R$?

Let $R$ be a discrete valuation ring and $\mathfrak{m}$ its unique non-zero maximal ideal. How to show that $R_{\mathfrak{m}}$ is $R$ using definition of a discrete valuation ring? I know that ...
3
votes
0answers
78 views

Question about localizations of discrete valuation rings.

Let $R$ be a discrete valuation ring. Then $R$ has only two prime ideals: $0$ and the maximal ideal $\mathfrak{m}$. It is said in Hartshorne, page 74, Example 2.3.2 that the localization of $R$ at $0$ ...
5
votes
1answer
105 views

Valuation ring in an algebraically closed field

Let $k$ be an algebraically closed field and $(R, \mathfrak{m})$ a valuation ring in $k$ i.e. the field of fractions of $R$ is $k$. Then Mumford (Red Book, page 127) claimed that the residue field ...
2
votes
1answer
57 views

extension of a valuation of $K$ to $K(X)$

Let $A$ be a Krull ring and $p$ a prime ideal of height 1. Then $A_p$ is a DVR with corresponding valuation $v$ on the field of fractions $K$ of $A$. Question: Can we extend this valuation to an ...
3
votes
1answer
121 views

Discrete Valuation Ring and Subring of the Fractions Field

Let $R$ be a Discrete Valuation Ring, and $K$ its fractions field. Now if $B\subseteq K$ is a subring with $R\subseteq B$ then we have $$B=R \text{ or } B=K.$$ Now this seems to be a very basic ...
2
votes
3answers
265 views

Examples of Non-Noetherian Valuation Rings

For valuation rings I know examples which are Noetherian. I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind? I am very ...
0
votes
1answer
158 views

Discrete Valuation Rings problem 2

An order function on a field $K$ is a function $\phi:K\to \mathbb{Z} \cup {\{\infty}\}$ satisfying: i) $\phi(a) = \infty$ if and only if $a=0$. ii) $\phi(ab) = \phi(a) + \phi(b)$. iii) ...
4
votes
1answer
77 views

About Henselization

I have some question about Henselization of valued field. If $(K_{1}, \nu_{1})$ is a Henselization of valued field $(K, \nu)$. Which one is true. $K_{1}/K$ is an algebraic extension. $K_{1}/K$ is a ...
2
votes
2answers
89 views

Valuation rings and domination

I was wondering if the following holds: if $A$ is a valuation ring with maximal ideal $m_A$ and $B$ is a ring extension of $A$ contained in $Frac(A)$ then the only maximal ideal of $B$ is still $m_A$. ...
2
votes
1answer
72 views

Properties of valuation map

It seems not so hard to prove but how can we prove by induction. Let $K$ be a field and $\nu$ be a valuation map. If $a_{1} + a_{2} + ... + a_{n} = 0$ then prove that $\nu(a_{i}) = \nu(a_{j})$ for ...
1
vote
1answer
113 views

A question in valuation theory

Let $R$ be an integral domain with quotient field $K$. Let $P$ be a prime ideal of $R$, and $L$ be a transcendental extension of $K$. I think there is more than one valuation domain $O'$ with quotient ...
2
votes
1answer
125 views

Construction of a Valuation Ring via Zorn's Lemma, except not

In Atiyah & MacDonald they provide an abstract way to construct valuation rings, I'm curious how easy it is to work this out in general. As a refresher let $K$ be a field, $L$ an algebraically ...
0
votes
2answers
85 views

Function field has infinitely many valuations.

Suppose we have a field $K$ which is a finite algebraic extension of the field $\mathbb{C}(X)$. Can you give me an argument that $K$ admits infinitly many discrete valuations?
7
votes
2answers
258 views

Non-trivial valuation of $\mathbb R$

In a valued fields book on page $82$ there is a question: "show that every non-trivial valuation of $\mathbb R$ has divisible value group and algebraically closed residue class field." How do I ...
5
votes
1answer
148 views

Ring of integers in a field of fractions

Let $R$ be ring with complete non archimedian absolute value. Let $Q$ be the associated field of fractions with the extended absolute value. Does the ring $O_Q = \{x\in Q | |x|\leq 1\}$ is complete ...
3
votes
1answer
77 views

Real valuations on Dedekind domains

Let $D$ be a Dedekind domain. Let $v:D \to \mathbb{R}$ a valuation. We know that for every prime ideal $\mathfrak p$ of $D$ the localization $D_{\mathfrak p}$ is DVR. Does every valuation on $D$ ...
4
votes
1answer
104 views

Henselization and immediate extension

I am reading about the henselization and immediate extension of valuation. I am getting confusion about some basic terminology. I have few question. \ $1)$ Is every hensilization extension of ...
2
votes
1answer
99 views

Valuation over the algebraically closed field of rational number

How do we define the valuation over the algebraically closed field of rational numbers say $\bar{\mathbb Q}$ as an extension of the valuation of $\mathbb Q$ ?
1
vote
1answer
100 views

Valuation rings of complete non-archimedean fields which are not local

I would like to know how are the valuation rings of complete non-archimedean fields which are not local. Take, for example, $\mathbb{C}_p$, and its valuation ring, ...
1
vote
0answers
69 views

Henselization and valued field

What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks
3
votes
2answers
225 views

What is a “normalized valuation” corresponding to a valuation ring?

I encountered the phrase "normalized valuation" similar to the following: Let $A_i$ be the valuation ring $k[x_1,...,x_n]_{\langle x_i\rangle}$ and $v_i$ be the normalized valuation defined by ...
1
vote
1answer
136 views

Formal power series ring over a valuation ring of dimension $\geq 2$ is not integrally closed.

I recently tried exercise 10.4 in Matsumura's Commutative Ring Theory, but got stuck. The question is: If $R$ is a valuation ring of Krull dimension $\geq 2$, then the formal power series ring ...
4
votes
1answer
286 views

Existence of valuation rings in a finite extension of the field of fractions of a weakly Artinian domain without Axiom of Choice

Can we prove the following theorem without Axiom of Choice? This is a generalization of this problem. Theorem Let $A$ be a weakly Artinian domain. Let $K$ be the field of fractions of $A$. Let $L$ ...
2
votes
1answer
171 views

Existence of an element of given orders at finitely many prime ideals of a Dedekind domain

Let $A$ be a Dedekind domain. Let $P$ be a non-zero prime ideal of $A$. Let $\alpha \in A$. Let $k$ be a non-negative integer. If $\alpha \in P^k$ and $\alpha\notin P^{k+1}$, we write $v_P(\alpha) = ...
4
votes
0answers
67 views

Automorphism of $L|K$ mapping 3 distinct rational points of $S_{L|K}$ to other 3 distinct ones

Let $K$ be a field and consider $L = K(x)$ the field of rational functions. Let $v_{1}, v_{2}, v_{3}$ rational points in the abstract Riemann surface $S_{L|K}$, distinct from each other, and $w_{1}, ...
5
votes
1answer
142 views

Discrete valuation ring extension such that $A[\pi]$ is not integrally closed

Let $B/A$ be a finite integral extension of discrete valuation rings of characteristic zero. Question. Is there a uniformizer $\pi$ in $B$ such that $A[\pi]$ is a discrete valuation ring? If not, ...
2
votes
1answer
104 views

A construction in the proof of “ any local ring is dominated by a DVR”

Let $O$ be a noetherian, local domain with maximal ideal $m$. I want to prove: for a suitable choice of generators $x_1,\dots,x_n$ of $m$, the ideal $(x_1)$ in $O'=O[x_2/x_1,\dots,x_n/x_1]$ is not ...
1
vote
2answers
251 views

A characterisation of tame ramification

The following is the statement from Algebraic Number Theory by Neukirch (Chapter 2 Proposition(7.7) p155) Blockquote Suppose $K$ is Henselian field, $p=char(\kappa)$ , the character of the ...
5
votes
1answer
144 views

Lang's “General Integrality Criterion”

Theorem 3.7 in the chapter on ring extension on page 352 of the latest edition of Lang's "Algebra" appears redundant in its phrasing to me. Specifically, if $g_s$ is a polynomial of total degree ...
6
votes
1answer
252 views

Valuation ring of $k(x, y)$ of dimension $2$

My question is as follows: Given a field $k$, is it always possible to find a valuation ring of $k(x, y)$ of dimension $2$?
9
votes
2answers
310 views

Concrete examples of valuation rings of rank two.

Let $A$ be a valuation ring of rank two. Then $A$ gives an example of a commutative ring such that $\mathrm{Spec}(A)$ is a noetherian topological space, but $A$ is non-noetherian. (Indeed, otherwise ...