Questions tagged [valuation-theory]

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3
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2answers
355 views

In a CM field, must all conjugates of an algebraic integer lying outside the unit circle lie outside the same?

This question is inspired from the post linked below: Can an algebraic number on the unit circle have a conjugate with absolute value different from 1? What I am curious about is the following: let $\...
3
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1answer
135 views

Existence of algebraic integer with absolute value equal to reciprocal of maximum of $1$ and absolute value of a given algebraic number

Consider a number field $K$, and let $v_1, \cdots v_n$ ($n \in \mathbb N$) be some finite (i.e. non-archimedean) places of $K$. Is the following true? For every $\alpha \in K^\times$ there exists $\...
4
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1answer
168 views

What is the definable functor associated to an algebraic scheme (model theory of valued fields)

I have a very basic question regarding algebraic model theory. I am trying to read Espaces de Berkovich, polytopes, squelettes et théorie des modèles (MSN) by Antoine Ducros. The relevant section is ...
4
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1answer
166 views

Henselian valued fields for characteristic $0$: a characterization

Let $K=K(v)$ be a valued field of characteritic $0$ with non trivial valuation $v:K\rightarrow\mathbb{R}\cup\{\infty\}$. I'm looking for a proof of following characterization of Henselian property: $...
1
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1answer
131 views

Chain of closed irreducible sets on Zariski Riemann spaces

Let $A$ be a domain and $K=\mathrm{Frac}(A)$. The Zariski Riemann space $\mathrm{ZR}(K,A)$ is the set of all valuation rings of $K$ containing $A$. It comes with a natural center map \begin{align}...
6
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1answer
139 views

Is $\mathbb{F}_{p}(t)^{h}$ an elementary substructure of/existentially closed in $\mathbb{F}_{p}((t))$?

It is a well-known fact that the Henselization of the function field $\mathbb{F}_{p}(t)$ in regard to the $t$-adic valuation is $\mathbb{F}_{p}(t)^{alg} \cap \mathbb{F}_{p}((t))$, so of course $\...
3
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1answer
100 views

Rank 1 valuations that are not discrete on finite transcendental extensions of the rationals

Suppose $K=\mathbb{Q}(X_1,\dots,X_n)$ is a purely transcendental extension of the rationals on finitely many indeterminates. Can anyone give an example of a rank $1$ valuation on $K$ that fails to be ...
4
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0answers
155 views

Completeness of the field of fractions of a ring of formal power series

Let $k$ be a field and let $k[[X,Y]]$ be the ring of formal power series with coefficients in $k$. Let $k((X,Y))$ be its field of fractions. For $F\in k[[X,Y]]$, $F\neq 0$ define $v(F)$ as the least ...
1
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1answer
96 views

Valuation theory on semisimple algebras used in the paper of Cohen-Martinet: reference request

I'm currently reading the paper of Henri Cohen & Jacques Martinet "Etude heuristique des groupes de classes des corps de nombres" On the 2nd section, they recall some facts on valuations, ...
12
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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 ...
4
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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 ...
2
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0answers
88 views

Valuation Rings and Ultrafilters II

See my post here: Valuation Rings and Ultrafilters Let $K$ be a field, and let $\mathcal{S}$ be the set of pairs $(R, \mathfrak{p})$ of subrings $R$ of $K$ with designated prime ideals $\mathfrak{p}$ ...
2
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1answer
118 views

Definition of model functions and their density in $C^0(X^\text{an})$

I am (still) working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10....
1
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1answer
53 views

Valuation of congruent elements in a local division ring

Let $K$ be a complete local division ring (note $v$ its valuation). For $x,y\in K$ ($y\ne0$), one puts $x^y=yxy^{-1}$. Let $r\in\mathbb N$. Consider $x,y\in K$ and $a,b\in K^*$ such that $v(x-y)\ge r$ ...
1
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1answer
70 views

On the maximal value of the valuation at infinity of elements in the ring of integers of a global function field

Let $F$ be a global function field with full constant field $\mathbb{F}_q$. We fix a place $\infty$ and let $A$ be the ring of elements of $F$ regular away from $\infty$. We denote by $v_\infty$ the ...
0
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1answer
153 views

Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals

If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...
14
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2answers
377 views

Ostrowski's Theorem for topological rings?

Ostrowski's theorem classifies all absolute values on a number field $K$. Questions: More generally, can one classify all Hausdorff topologies on $K$ making $K$ into a topological field? In ...
0
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1answer
132 views

Understanding a valuation property of function fields

I came across this point in a paper recently and I'm having difficulty seeing why it's true. Any explanations or hints would be appreciated. For any prime $\mathfrak{p}$ of $\mathbb{F}_q [t]$ such ...
1
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1answer
273 views

Valuation ring whose maximal ideal and every ideal of finite height are principal

Let $(R, \mathfrak m)$ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?
3
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1answer
131 views

'Smallest' subfield of the Surreals which is isomorphic to the Surreals as an ordered group

What is the smallest subfield $F\subset N_0$ such that $$(F,+,\times,\leq)\ncong(N_0,+,\times,\leq)$$ but $$(F,+,\leq)\cong(N_0,+,\leq)?$$ Since these are all going to be proper classes cardinality is ...
4
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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 $R \subseteq V \subsetneq Q(R)$ ...
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0answers
124 views

Why is any non-archimedean field Huber?

Here a non-archimedean field means a field $k$ whose topology is induced from a non-archimedean norm $| \cdot |: k \to \mathbb{R}_{\geq 0}$. As a reminder, a ring $A$ is adic if there is an ideal $I \...
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0answers
80 views

Luroth's theorem for Discrete valuation rings?

Luroth's theorem states that if $k$ is a field and $L$ is a field extension of $k$ such that $k \subset L \subseteq k(X)$, then $L=k(f(X))$ for some $f(X) \in k(X) $ . My question is ; is there any ...
3
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0answers
206 views

Is the special case of Abhyankar's lemma is also considered as such?

Consider the following statement: Assume $E$ and $F$ are unramified (over some fixed prime) finite separable extensions of a field $K$. Then $EF$ is also unramified. I always thought that it is ...
4
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88 views

Valued fields with quantifier elimination in the Macintyre language

For which fields $k$ of characteristic $p$ does the Witt construction of a discretely valued field $W(k)$ of characteristic $0$ with residue field $k$ eliminate quantifiers in the language of rings ...
3
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0answers
131 views

Rational power series and extensions

Let $F$ be a field, let $F(x)$ the field of rational functions, and let $F((x))$ the field of Laurent series (which contains $F(x)$). One may ask: which series $\sum_i a_i x^i$ lie in $F(x)$? The ...
7
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1answer
264 views

Uniquely ordered commutative rings

I am wondering whether there are reasonable necessary and/or sufficient conditions to dedice whether a commutative ring can be uniquely ordered (like for instance $\mathbb{Z}$) or not. In the field ...
2
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0answers
53 views

Terminology for valuation-like functions on a vector space

Let $V$ be a vector space over a field $k$. I was wondering if there is a standard terminology for a function $v: V \setminus \{0\} \to \mathbb{R}$ which is invariant under multiplication by nonzero ...
4
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1answer
776 views

simple questions on topological rings arising in the context of Perfectoid Spaces

(I apologize in advance for these simple questions, I am a beginner trying to go through Scholze's paper Perfectoid Spaces). Let $(R, R^+)$ be an affinoid $k$-algebra as defined in Scholze's paper ...
2
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0answers
342 views

Valuation topology vs modified valuation topology

Let $K$ be a field with valuation $v:K\to G\cup\{\infty\}$ where $G$ is an ordered abelian group. In section 7.62 of the book "Foundations of analysis over surreal number fields." Vol. 141. Elsevier, ...
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0answers
171 views

Completions of $K(x)$

Let $K$ be a field. Are there books or articles discussing completions of $K(x)$ with respect to the metric induced by the $p$-adic valuation $|\;\;|_p$ where $p\in k[x]$ is irreducible and different ...
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0answers
164 views

How to prove that $k(x)$ is not complete in the $x$-adic metric [closed]

It is not hard to find proofs showing that $\mathbb{Q}$ is not complete with respect to the metric induced by the valuation $|\;\;|_p$. For example, it is enough to recall that every complete metric ...
3
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1answer
338 views

Birational Group Law

Let $S$ be a scheme and $X$ a smooth separated faithfully flat over $S$. An $S$-birational group law on $X$ is an $S$-rational map $$m:X\times_S X\dashrightarrow X, (x,y)\mapsto xy$$ such that a) the ...
5
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2answers
462 views

The notions of $H^0(\widehat{ D})$ and $h^0(\widehat{D})$ are not satisfactory

Let $K$ be a number field with ring of integers $O_K$. Moreover consider an Arakelov divisor $\widehat{D}\in\overline{\operatorname{Div }(\operatorname {Spec }O_K)}$, namely $$D=\sum_{\mathfrak p\;\...
2
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0answers
91 views

Group of units of a valuation

Let K be a field. Then a subring R of K is called a valuation ring if for all $x \in K^*,$ either $x \in R$ or $x^{-1} \in R$ (or both). It can be shown that for any valuation $v$ on $K,$ the ring $\...
7
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2answers
1k views

Completion and algebraic closure

Following this question: Given a valued field $K$, denote with $\bar{K}$ its algebraic closure and with $\hat{K}$ the completion. Then both $\hat{\bar{K}}$ and $\hat{\bar{\hat{K}}}$ are complete and ...
3
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0answers
98 views

Structure of valuations on $\mathbb{F}_q(X,Y)$?

I'm looking to construct all valuations on $\mathbb{Q}(X,Y)$ extending the p-adic valuation on $\mathbb{Q}$ and understand their structural properties. In doing this, to obtain 3 dimensional valuation ...
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0answers
295 views

Is every Polish ring topology on $\mathbb{C}$ defined by an absolute value?

There is a unique up to isomorphism algebraically closed field of characteristic 0 and cardinality of the continuum. Let's call it $K$. We usually call it $\mathbb{C}$, but by this we impose a ...
3
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0answers
329 views

Relation between ramification index and length of filtration of ramification groups

Given a complete valued field $K$ with a discrete value group $\mathbb{Z}$, consider a totally ramified finite Galois extension $L$ of $K$ with its Galois group $G$. Let $O_L$ be the valuation integer ...
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0answers
261 views

Extension of a valuation on a function field

Let $K$ be a field, and $K(x)$ the field of rational functions over $K$. Consider the degree valuation $v$ on $K(x)$, That is $v\left(\frac{f(x)}{g(x)}\right)=\deg(g)-\deg(f)$. So for every $f(x)\in ...
7
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2answers
622 views

Examples of NIP fields of characteristic $p$

Definition. According to Shelah, a field $K$ does not have the independence property (i.e. is NIP) if for every first order formula $\varphi(x, \bar y)$ in the language of fields $(+,\times,0,1)$, the ...
3
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1answer
1k views

Completion of a finite field extension is also finite?

Let $(L,w)/(K,v)$ be a finite extension of valuation fields, and let $L_w$, $K_v$ be the respective completions of $(L,w)$, $(K,v)$. Is the field extension $L_w/K_v$ finite? For nonarchimedean ...
1
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1answer
193 views

Separable extensions of henselian fields

Let $(k,v)$ be a henselian field, with $\mathcal{O}$ and $\bar{k}$ being respectively its valuation ring and its residue field. If $K/k$ a finite separable field extension (on which $v$ thus extends ...
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1answer
254 views

Model over DVR for smooth projective curves

Let $C$ be a smooth, projective, geometrically irreducible curve of genus at least $2$ over a complete discrete valued field $F$ of characteristic zero (not necessarily algebraically closed). Let $R$ ...
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2answers
369 views

Subgroup of Projective general linear group on complete discrete valuation ring

Let $R$ be a complete dvr and $k$ its residue field of positive characteristic. Let $H$ be a finite subgroup of $PGL_2(k)$ such that the order of $H$ is prime with $char(k)$. Is there some ...
1
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1answer
467 views

Completion of discrete valuation ring

Let $R$ be an excellent, Henselian, discrete valuation ring with algebraically closed residue field and $\hat{R}$ be the completion of $R$. If I understand correctly, the residue field of $\hat{R}$ is ...
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0answers
448 views

Extension of a complete discrete valuation ring

My question came when I was reading the famous Tate's paper on $p$-divisible groups. At the beginning of chapter $(2.4)$ he cites this fact as obvious. If you take a complete discrete valuation ring $...
3
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1answer
245 views

On conductors, levels and traces on quaternion algebras

I am currently working on level issues in the division central simple algebra case, say $D$ over a local non-archimedean field $F$ (e.g. $\mathbf{Q}_p$). Let say that $\mathcal{O}_D$ and $\mathcal{O}...
9
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0answers
288 views

Is it decidable whether a finite type scheme is proper?

Let $k$ be a field and let $X$ be a finite type scheme over $k$, explicitly given by finitely many affine patches which are $\mathrm{Spec}$ of finitely generated $k$-algebras, glued along other affine ...
9
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2answers
680 views

How did height in algeb. number theory/elliptic curves started?

Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people ...