For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). It seems that codimension $1$ irreducible subschemes of these models yield valuations of $F$. My question is: which valuations are obtained this way? Would it be appropriate to call them 'geometric'; is there any canonical text that treats this question (an introduces some terms of sort)? The problem is that I do not want to restrict myself to the case when all of these subschemes are defined over a fixed subfield of $F$.
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asked Jul 25, 2014 at 13:07
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1$\begingroup$ Any closed irreducible subvariety gives rise to a discrete valuation, because you can blow it up and get a divisor. $\endgroup$– abxJul 25, 2014 at 15:48
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$\begingroup$ Perhaps I am misunderstanding your question, but don't all discrete valuations on $F$ arise this way? Namely, given a discrete valuation $v$ on $F$, one can consider the associated discrete valuation ring $A \subset F$. Then $\mbox{Spec } A$ is a model for $F$, whose point of codimension $1$ gives rise to the required discrete valuation $v$. $\endgroup$– Daniel LoughranJul 25, 2014 at 15:53
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1$\begingroup$ Possibly I am getting something wrong; yet in the 'geometrical' case there are 'bad' valuations; see mathoverflow.net/questions/135544/… $\endgroup$– Mikhail BondarkoJul 25, 2014 at 20:01
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$\begingroup$ I believe one wants to consider models of $F$ of finite type only. Then one needs $A$ to be a localization of a finitely generated subring of itself - this is clearly a necessary and sufficient condition. I don't know a better one. $\endgroup$– Will SawinJul 25, 2014 at 21:31
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$\begingroup$ This is probably true. Yet do you now any references for this (where some terms are introduced)? $\endgroup$– Mikhail BondarkoJul 26, 2014 at 6:35
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Suppose you are given a proper noetherian integral scheme $X$ whose field of rational functions is $F$. Let $A$ be a discrete valuation of $F$ with residue field $k_A$ and suppose it has a center $x\in X$. Then the existence of a model $Y$ (of finite type over $X$) of $F$ such that $A$ is induced by a codimension $1$ subscheme of $Y$ is equivalent to the condition
$$\dim O_{X,x} -1 =\mathrm{trdeg}_{k(x)}k_A$$
(transcendence degree), under the assumption that $X$ is excellent. The statement is apparently due to Zariski. You can find a proof in Artin: "Néron models", §5, in Arithmetic Geometry (ed. Cornell and Silverman). See also Liu : "Algebraic geometry and arithmetic curves", Theorem 8.3.26 (where the condition
$X$ is excellent is explicitly used).
