Questions tagged [valuation-theory]
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131 questions
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Hasse principle for Brauer groups of fields of transcendence degree 2
In his paper "A Hasse principle for function fields over PAC fields" (DOI link), Ido Efrat proves the following result: Let $F$ be an extension of a perfect PAC field $K$ of relative ...
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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 ...
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Discrepancy of general element of linear system
Let $X$ be a normal scheme and $|D|$ a linear system on $X$.
In "Singularity of Minimal Model Program" by Janos kollar p249, it says,
If $X$ is a variety over $\mathbb{C}$, and $E_j$ ...
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Krull dimension of affinoid algebra
Let $K$ be a complete field w.r.t. a valuation, with residue field $k$. Let $A$ be an affinoid algebra over $K$ with respect to a valuation $V$ (in the sense of Tate; in the terminology of Berkovich, $...
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Interpretation of model theory in algebraic geometry
I found a paper Some applications of a model theoretic fact to (semi-) algebraic geometry by Lou van den Dries. In this paper, the author uses model theoretical methods to prove the completeness of ...
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For an element in the integral closure of an ideal $I$ - which power is in $I$?
Consider an ideal $I$ in a ring $R$. If $f \in R$ belongs to the integral closure of $I$, then there is $k_0 \geq 0$ such that $f^k \in I^{k-k_0}$ for all $k \geq k_0$. Are there any known upper ...
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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 $\...
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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 ...
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119
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Semi-stable model over a totally ramified extension
Notation: Let $R$ be a DVR, $K=\text{Frac}(R)$ and $k=R/\mathfrak{m}$. Given an $R$-scheme $X$, write $X_K=X\times_{R} K$ for the generic fiber and $X_k=X\times_R k$ for the special fiber.
Suppose $k$ ...
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Does maximally incompleteness cause nonvanishing of the extension of maximal ideal of a valuation ring by rank 1 free module?
In B. Bhatt's lecture notes[1], Remark 4.2.5 says
... $\operatorname{Ext}_R^2(k,R)$ is non-zero if $K$ is not spherically complete.
which amounts to the following pure algebraic question.
Statement ...
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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
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How do I extend the $2$-adic absolute value to prove Monsky's Theorem?
In proving Monsky's Theorem, it is required that we define the $2$-adic absolute value on an arbitrary finitely generated extension of $\mathbb{Q}$ say $\mathbb{K}=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$...
user500557
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Space of valuations is spectral space and what does it mean to say that conditions are closed conditions
I am reading lecture 3 of Conrad notes (link : https://math.stanford.edu/~conrad/Perfseminar/ ), in which he proves space of valuations is a spectral space. Last theorem of lecture 3.
We have a map $j:...
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Valuations of coefficients of minimal polynomials for tuples
Suppose you are given two valued fields $(K,v) \subseteq (L,w)$ and a tuple $a \in L^n$. What kind of restrictions do we have on the valuation of the coefficients of polynomials $q \in K[x_1,\dots x_n]...
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Existence of invariant valuations
Given a field $K$, one can enrich it via a valuation, an automorphism or both structures at the same time in a compatible way. In all of these three cases, the model theory is well-understood (under ...
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References on function fields over imperfect fields in positive characteristic
There are many references (good books, papers, ...) available that treat global function fields (over finite fields) of one independent variable. To name a few, Stichtenoth's book "Algebraic ...
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Literature on non-Archimedean analogues of basic complex analysis results
It looks like there is some literature out there on what might be called 'non-Archimedean complex analysis' e.g. Benedetto - An Ahlfors Islands Theorem for non-archimedean meromorphic functions and ...
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Pair of recurrence relations with $a(2n+1)=a(2f(n))$
Let $f(n)$ be A053645, distance to largest power of $2$ less than or equal to $n$; write $n$ in binary, change the first digit to zero, and convert back to decimal.
Let $g(n)$ be A007814, the ...
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Formula from the recurrence relation
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then we have an integer sequence ...
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Generating function for partial sums of the sequence
Let $p$ and $q$ be integers.
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then ...
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Recurrence for the sum
Let $m\geq 2$ be a fixed integer.
Let
$$f(n):=\begin{cases}
mf\left(\frac{n}{m}\right),&\text{if $n\mod m = 0$;}\\
1,&\text{otherwise}
\end{cases}$$
then if we have
$$a(n):=\begin{cases}
1,&...
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Subsequence of the cubes
Let $p$ and $q$ be integers.
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then ...
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104
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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)...
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213
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$C^*$-algebras over an extension of $\mathbb{Q}_p$?
I'm wondering to what extent it might be possible for the theory of $C^*$-algebras to be translated into the $p$-adic context i.e. to define 'p-adic $C^*$-algebras' over some extension of $\mathbb{Q}...
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Open immersion of affinoid adic spaces
If $R$ and $S$ are complete Huber rings with $\varphi: R \to S$ a continuous map, then is it true in general that if $\mathrm{Spa}(S, S^\circ) \to \mathrm{Spa}(R, R^\circ)$ is an open immersion of ...
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Question about valuation and blow up (a lemma in GIT book)
I'm reading Mumford's book Geometric Invariant Theory and confused about the proof of a lemma on Page 91&92:
Lemma:
Let $V_0$ be a smooth surface over an algebraically closed field $k$
with char$...
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Request for bibliographic information
Greetings to everyone on this forum (I am a new-comer). I would like to ask the experienced members for suggestions on (as) comprehensive and systematic (as possible) bibliographic sources regarding:
...
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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$?
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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 $\...
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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 $\...
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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 ...
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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:
$...
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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}...
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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 $\...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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}$ ...
user30211
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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....
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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$ ...
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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 ...
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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 ...
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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 ...
- 64k
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145
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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 ...
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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 ?
user111492
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'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 ...
- 10.1k
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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
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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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