All Questions
Tagged with valuation-theory ag.algebraic-geometry
25 questions
1
vote
0
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78
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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 ...
- 41
1
vote
0
answers
218
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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 ...
- 328
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
2
votes
1
answer
151
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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 ...
- 5,359
3
votes
0
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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 $k$ ...
- 282
1
vote
0
answers
63
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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:...
- 11
12
votes
1
answer
535
views
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 ...
- 503
4
votes
1
answer
260
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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$...
- 565
4
votes
1
answer
256
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 ...
- 1,153
20
votes
8
answers
3k
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What are examples illustrating the usefulness of Krull (i.e., rank > 1) valuations?
In modern valuation theory, one studies not just absolute values on a field, but also Krull valuations. The motivation is easy enough:
If $k$ is a field, a valuation ring of $k$ is a subring $R$ ...
- 65.4k
1
vote
1
answer
197
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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}...
- 225
5
votes
2
answers
529
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
$$D=\sum_{\mathfrak p\;\...
- 1,237
2
votes
1
answer
139
views
Definition of model functions and their density in $C^0(X^\text{an})$
I am (still) working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10....
- 155
0
votes
1
answer
269
views
Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals
If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...
- 1,209
3
votes
1
answer
431
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Birational Group Law
Let $S$ be a scheme and $X$ a smooth separated faithfully flat over $S$.
An $S$-birational group law on $X$ is an $S$-rational map
$$m:X\times_S X\dashrightarrow X, (x,y)\mapsto xy$$
such that
a) the ...
- 119
13
votes
4
answers
2k
views
Replacing Spectrum with Valuations of a Field - An Alternative to Schemes?
A scheme is defined to be a sheaf which is locally isomorphic to the spectrum of a ring. The idea behind this is that given an affine coordinate ring of a variety over an algebraically closed field, ...
- 15.4k
1
vote
1
answer
747
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 ...
- 2,126
1
vote
1
answer
375
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 $R$ ...
- 503
9
votes
0
answers
339
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 ...
- 156k
7
votes
1
answer
2k
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The space of valuations of a function field
Hello, I'm looking for someone who can help me to understand Zariski's theory of valuations.
First I outline the theory: we take a field $K$ which is a finitely generated transcendent extension of ...
- 1,804
8
votes
1
answer
991
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Zariski-style valuation theory
I've been trying to read some of Zariski's older works, and I'm having some trouble getting into his mindset. I'd appreciate some help with this.
To quote Zariski (in "normal varieties and birational ...
- 5,929
4
votes
0
answers
536
views
Discrete valuations for which Abhyankar inequality is strict
The background to my question, in a nutshell, is: If $k$ is a field and $X$ a $k$-variety, i.e. an integral, separated, finite type $k$-scheme, which discrete rank $1$ valuations on $k(X)$ come from ...
- 4,450
2
votes
0
answers
190
views
Valuations given by flags on a variety and valuations of maximal rational rank
I am interested in valuations on a function field $K=k(X)$ of some say smooth, projective $k$-variety $X$ of dimension $n$, where $k$ is some (algebraically closed) field (that implies trdeg$(K/k) = n$...
- 43
2
votes
0
answers
764
views
Riemann-Roch for ARBITRARY Function Fields
I know that on an algebraic function field in one variable over any base field, there is a good divisor theory for it and a Riemann-Roch Theorem; in particular, there is a 'good' notion of 'genus'. (...
- 468
5
votes
0
answers
374
views
Sheaf Cohomology on Zariski-Riemann Spaces
Can sheaf cohomology on the Zariski-Riemann spaces give some sort of classification for field extensions (even just for function fields)? If not, are there any significant or useful results (e.g. for ...
- 468