Skip to main content
MathOverflow

Questions tagged [valuation-theory]

Ask Question

The tag has no summary.

135 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
-2 votes
0 answers
158 views

Let $r$ be an integer greater than $1$, and consider the radix-$r$ numeral system. For any integer $a>1$ not divisible by $r$, we introduce the base-$r$ congruence speed of the (integer) tetration ...
Community wiki
6 revs, 3 users 90%
Marco Ripà
4 votes
0 answers
236 views

Let $\mathcal O$ be a discrete valuation ring with maximal ideal $\mathfrak m$, fraction field $\mathbb K:=\mathrm{Frac}\,\mathcal O$, and residue field $\mathbb F:=\mathcal O/\mathfrak m$. Suppose ...
王乱坤 Luankun Wang's user avatar
  • 41
2 votes
1 answer
158 views

Given a function field $K$ over $\Bbb C$ with a discrete valuation $v$, are there known criteria under which an automorphism of $K$ extends to the completion $K_v$? I’m aware of sufficient conditions ...
Anushka_Grace's user avatar
  • 139
3 votes
1 answer
252 views

I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of a subring of the hyperreals. It has a non-archimedean ...
4u9ust's user avatar
  • 131
1 vote
0 answers
142 views

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 ...
aspear's user avatar
  • 161
3 votes
0 answers
239 views

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 ...
Mathstudent's user avatar
  • 281
1 vote
0 answers
96 views

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 $E_j$ ...
George's user avatar
  • 378
1 vote
0 answers
144 views

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, $...
Adelhart's user avatar
  • 397
1 vote
0 answers
273 views

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 ...
George's user avatar
  • 378
2 votes
1 answer
208 views

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 ...
pinaki's user avatar
  • 5,489
16 votes
2 answers
1k views

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 $\...
zxx's user avatar
  • 363
10 votes
1 answer
909 views

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 ...
George's user avatar
  • 378
3 votes
0 answers
151 views

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 $k$ ...
Kostas Kartas's user avatar
  • 292
1 vote
0 answers
142 views

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 ...
XYC's user avatar
  • 541
5 votes
0 answers
181 views

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 ...
Tim Campion's user avatar
  • 66.8k
0 votes
1 answer
528 views

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 $\mathbb{K}=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$...
user avatar
1 vote
0 answers
86 views

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:...
Math_1729's user avatar
  • 11
1 vote
0 answers
56 views

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]...
Simone Ramello's user avatar
  • 171
6 votes
0 answers
145 views

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 ...
Simone Ramello's user avatar
  • 171
1 vote
1 answer
196 views

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 ...
Andry's user avatar
  • 103
8 votes
2 answers
591 views

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 ...
Very Forgetful Functor's user avatar
  • 409
2 votes
1 answer
217 views

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 ...
user avatar
1 vote
1 answer
331 views

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 ...
user avatar
0 votes
1 answer
368 views

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 ...
user avatar
1 vote
2 answers
558 views

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,&...
user avatar
3 votes
2 answers
513 views

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 ...
user avatar
3 votes
0 answers
164 views

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)...
Dat Minh Ha's user avatar
  • 1,728
4 votes
0 answers
257 views

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}...
Very Forgetful Functor's user avatar
  • 409
12 votes
1 answer
612 views

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 ...
Ashwin Iyengar's user avatar
  • 505
4 votes
1 answer
320 views

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$...
Kim's user avatar
  • 625
2 votes
0 answers
133 views

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: ...
ΑΘΩ's user avatar
  • 121
9 votes
1 answer
939 views

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 $k$?
Alex Mennen's user avatar
  • 2,275
3 votes
2 answers
838 views

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 $\...
asrxiiviii's user avatar
  • 749
3 votes
1 answer
364 views

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 $\...
asrxiiviii's user avatar
  • 749
4 votes
1 answer
313 views

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 ...
Jakob Werner's user avatar
  • 1,648
3 votes
1 answer
529 views

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: $...
user267839's user avatar
  • 3,878
1 vote
1 answer
233 views

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}...
nowhere dense's user avatar
  • 225
7 votes
1 answer
313 views

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 $\...
Florian Felix's user avatar
  • 311
3 votes
1 answer
171 views

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 ...
Harry Gindi's user avatar
  • 19.9k
4 votes
0 answers
427 views

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 ...
Carlos's user avatar
  • 1,738
1 vote
1 answer
170 views

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, ...
gualterio's user avatar
  • 1,143
12 votes
1 answer
2k views

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 ...
domotorp's user avatar
  • 19.6k
4 votes
1 answer
905 views

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 ...
userm's user avatar
  • 41
2 votes
0 answers
144 views

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 $\mathfrak{p}$ ...
user avatar
2 votes
1 answer
168 views

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....
Riquelme's user avatar
  • 155
1 vote
1 answer
88 views

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 $v(x-y)\ge r$ ...
joaopa's user avatar
  • 4,306
1 vote
1 answer
171 views

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 ...
Andry's user avatar
  • 103
0 votes
1 answer
315 views

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 ...
user521337's user avatar
  • 1,229
19 votes
2 answers
694 views

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 ...
Tim Campion's user avatar
  • 66.8k
0 votes
1 answer
184 views

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 ...
user221330's user avatar
  • 487

1
2 3