19
votes
Accepted
Does every map from a noetherian ring to a valuation ring factor through a DVR?
The answer is no. I give two examples, which are standard non-dvr points on the Riemann-Zariski space of the plane.
(1) Let $R=k[x,y]$ and let $V\subseteq k(x,y)$ be the subring consisting of rational ...
- 16.1k
17
votes
Accepted
Is it a valuation ring?
This is (essentially) a conjecture of Krull; a counter-example was given by P. Ribenboim: Sur une note de Nagata relative à un problème de Krull, Math. Zeit. 64, 159-168 (1956). Note that "...
- 38k
16
votes
Accepted
Extension of 2-adic valuation to the real numbers
No. The important thing to know is that, if $K \subseteq L$ is a field extension and $v: K \to \mathbb{R}$ is a valuation, then $v$ can be extended to $L$. So I can answer all of your questions by ...
- 156k
11
votes
Accepted
Is every field the residue field of a discretely valued field of characteristic 0?
Yes, by Hasse-Schmidt ("Die Struktur diskret bewerteter Koerper", Crelle's Journal, 1934) for any field $k$ of characteristic $p$ there exists a strict Cohen ring $A$, which is a Noetherian, ...
- 9,263
7
votes
Accepted
Classifying Space of "Valuation Ringed Spaces over a Topos"
Since the axioms describing what a valuation ring can be put as what's called geometric sequents [*], by the fundamental theorem on classifying toposes, there is a topos $T_{val}$ with precisely the ...
- 3,302
7
votes
Accepted
Valuation Rings and Ultrafilters
The similarity has nothing to do with boolean algebras, but with orders in general. Filters can be defined for every partial order: A subset $\Phi$ of a poset $\Lambda$ is a filter if
$\Phi\neq\...
- 9,650
6
votes
Is the integral closure of a valuation ring in a finite separable extension of its fraction field étale?
EDIT I may have overlooked the assumption that $L$ is contained in $\hat{K}$. In general, if $v$ is a valuation of $K$ and $L/K$ is finite separable then $L \otimes_K \hat{K}$ is reduced and thus ...
- 20.8k
6
votes
Torsors over complete local fields
The isomorphism classes of torsors for $\mu_{n}$ over $R$ and $k$ are
classified by the fppf cohomology groups $H^{1}(R,\mu_{n})$ and $H^{1}%
(k,\mu_{n})$. From the Kummer sequence, one sees that $H^{...
- 61
5
votes
Accepted
Torsors over complete local fields
I am rewriting my comment as an answer. That is false in characteristic $p$ for torsors for the finite, flat group scheme $\mu_p=\text{Spec}\ \mathbb{Z}[t]/\langle t^p -1 \rangle$ with the usual ...
4
votes
What does "trait" mean?
This means a little segment of line: not a point, but the smallest thing after a point.
This also means "feature", which could be another reason to use this for a spectrum.
See https://en.wiktionary....
- 3,587
4
votes
Accepted
Lifting Lang-Steinberg to DVR's in Characteristic 0
Yes, as long as you are careful about the hypotheses, this is a result of M. J. Greenberg [Schemata over local rings II, Section 3, Proposition 3]. You need to assume that $\mathbb{G}$ is smooth over $...
- 3,813
3
votes
Accepted
What is K+M structure?
A "valuation ring of the form $K+M$" is a valuation ring $V$ with maximal ideal $M$ such that $V$ contains a subring $K$ which is a field and one has $V=K+M:=\{k+m\,|\,k\in K,m\in M\}$.
In this case ...
- 2,928
3
votes
Given a non-field local domain $R$, finding a dominating Valuation ring whose residue field is algebraic/finite extension of the residue field of $R$
Yes, an algebraic extension is always possible. If one applies Zorn's lemma not to
$$\{R\subseteq V\subseteq Q(R) \mid V\,\text{local}\}$$
but to
$$\{R\subseteq V\subseteq Q(R) \mid V\,\text{local} \...
- 9,650
3
votes
What does "trait" mean?
Someone means something as $\text{Spec}(k[[x]])$: his spec is $0$ and $(x)$, so it is an enlargement of the point $(0)$ of $\text{Spec}(k)$, for this reason it could be named as a "trait". ...
- 109
3
votes
Accepted
Higher-rank Archimedean valuations of $\mathbb{Q}$, does it exist?
Consider the following valuation:
$|\cdot|:\mathbb{Q}\to[0,\infty)^\mathbb{N}$ defined as
$|q|=(|q|_{p_i})_i$ where $(p_i)_i$
is the increasing sequence of all the prime numbers and $|\cdot|_{p_i}$ is ...
- 596
2
votes
Accepted
Flatness criterion for $I$-adic ring: $I$-torsion free
In Section 7.3 it is assumed that $I$ is the ideal of definition or $R$. It follows that the $I$-adic topology is separated, so (because $R$ is a valuation ring) every nonzero ideal of $R$ contains ...
- 148k
2
votes
A question about Dedekind schemes and proper morphisms
Injectivity is because $X$ is separated. The locus where two morphisms $S \to X$ agree is a closed subscheme and if it contains the generic point, it's everything.
For surjectivity, we can "...
- 7,746
1
vote
An example of a special $1$-dimensional non-Noetherian valuation domain
This isn't possible in any $1$-dimensional quasi-local domain $D$. If $a,b\in D$ and $b$ is not a unit then consider the multiplicative set $S$ generated by $b$. Clearly $S$ is not disjoint from any ...
- 938
1
vote
Accepted
non-archimedean valuations on graded rings
Let $K$ be a field, and set $R=K[x]$ with the usual grading by degree. For each irreducible polynomial $p(x)\in R$, we get a valuation $v_p$ on $R$ given by
$$
v_p(f)=2^{-\operatorname{ord}_p(f)},
$$
...
- 9,061
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