In his seminal paper, Some open problems on three-dimensional graded domains, M. Artin proposed a very small list of possible division rings of fractions that can appear as 'noncommutative function fields' of noncommutative surfaces. In the paper, Artin says that those division rings where shown to be nonisomorphic by Zhang, using a certain notion of valuation.
To be more precise, let $D$ be $k$-algebra division ring, where $k$ is algebraically closed of zero characteristic. We consider discrete valuations $\nu:D \rightarrow \mathbb{Z}$, that is, $\nu(x+y) \geq \min \{ \nu(x), \nu(y)\},$ $\nu(xy)=\nu(x)+\nu(y)$, $\nu(k)=0$, such that, calling $R=\{ x \in D\mid \nu(x) \geq 0 \},$ the residue field of $R$ is a function field of transcendence degree $1$ over $k$. Such valuations are called by Artin prime divisors.
Then in Proposition 5.3 of the above paper we have the following two statements:
i) If $D$ is the first Weyl field, given any prime divisor, the residue field is the rational function field $k(x)$.
ii) Let $C$ be a curve o positive genus, $\mathcal{D}(C)$ the ring of differential operators on $C$, and $\mathcal{F}(C)$ the division ring of fractions of $\mathcal{D}(C)$. Then each $\mathcal{F}(C)$ has a unique prime divisor whose function field is $k(C)$, and all other prime divisors have as function fields the rational function field $k(x)$.
i) appeared in the paper by Willaert, Discrete valuations of Weyl skew fields, but I could not find any proof, or any idea of proof, for item ii).
Do someone have a reference for this, or can roughly indicate the main ideas of the proof of this fact?
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