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I did some work on a new article, valuation ring. I didn’t finish everything I set out to do, but one thing I wouldn’t mind understanding better is the bit I put in about compact Riemann surfaces, whose points correspond to valuation rings sitting inside their fields of meromorphic functions. There is some very classical algebraic geometry there which might be nice to put down in the Lab.
Yes, the classical approach to an abstract Riemann surface due Dedekind and Weber is via valuations. Modern generalizations are due Grauert and Remmert (key words: analytic local algebras).
Thanks for the tips, Zoran.
Meanwhile, I added some examples to valuation ring, basically examples of Hardy fields and rates of growth. This to me is fascinating stuff. Also mentioned Hahn series at the end (related to but more general than Puiseux series).
Added a little more to valuation ring, adding for example the fact they are Pruefer domains (for which an entry has yet to be written).
There is some constructive theory in case you are interested Valuations and Dedekind’s Prague Theorem
That looks interesting; thanks!
That’s a neat statement. (1) $\Leftrightarrow$ (2) may be my favorite among the many recent additions to the notion of flatness.
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