Talk:Discrete valuation ring

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[edit] Algebraic geometry

I removed the following:

...that satisfies conditions that, in algebraic geometry, describe non-singularity of a point on an algebraic curve. There are also many examples of discrete valuation rings that are not geometric in nature.

This seemed to me to have several problems. One is that I don't think it is correct; a ramified point leads to another DVR, not out of the land of DVRs. Anyway there's no reason to mention algebraic geometry as if it was the only source of DVRs. Gene Ward Smith 21:26, 8 May 2006 (UTC)

Isn't it true that the local ring at a point of an algebraic curve is a DVR iff the point is non-singular? AxelBoldt 19:23, 10 May 2006 (UTC)

If it ramifies at the point it's still a DVR; the Puiseux series is in a fractional power of x and not x. It can also split, so you can get more than one valuation ring, for the different valuations. Anyway I'd think the usual method of describing nonsigularity would be the rank of the Jacobian; for a plane curve, it's singular if the partials with respect to x, y, and z are all zero, so I think it's kind of confusing to bring in smoothness. "Describing the local behavoir" might be a better way to put it. Gene Ward Smith 23:41, 10 May 2006 (UTC)

Ok, thanks! AxelBoldt 18:56, 11 May 2006 (UTC)