What is the advantage of the approach of valuations to the Riemann-Roch Theorem for curves (a la Chevalley)? AKA theory of algebraic functions in one variable

Question

Hello, Some books and courses take the approach to Riemann-Roch for curves considering only the algebraic viewpoint of algebraic extensions of a field k(t) without going into any algebraic geometry (no varieties, no topology, no dimension, no sheafs etc') divisors are defined using equivalence classes of valuations of the field, differentials have some 'wierd' definition as well. and riemann roch is proved only in that language, with no regard to concepts such as bundles. This can be seen in books such as Chevalley, Introduction to the Theory of Algebraic Functions of One Variable. Or even in Neukirch Algebraic number theory.

Since I know of treatments of Riemann-Roch in the general setting for bundles over algebraic curves in books such as Kempf, Algebraic Varieties, or even in Vakil's notes: http://math.stanford.edu/~vakil/725/bagsrr.pdf

My question is, what is the advantage of the approach using only algebra and the language of valuations instead of using concepts from algebraic geometry (such as sheaf)

Thanks

Answer 1

Hello Blade.

Reading your question one is tempted to dive into the interesting and diverse history of the various approaches to the Riemann Roch Theorem. However, an answer to your question from the point of view of a mathematician of today in my opinion depends on the person who is asking.

The amount of concepts and theory you have to learn before being able to fully understand the proof of Riemann Roch is smaller for the valuation theoretic approach compared to the one taught in Algebraic Geometry. Of course this is not an advantage for somebody who is interested in geometry and wants to focus future activity in that area anyway. On the other hand if one is tending more towards Algebraic Number Theory, then it is a demand to learn valuation theoretic concepts, even for non-discrete valuations. Finally if one wants to study Arithmetic Geometry or Non-archimedian Analysis geometric and valuation theoretic concepts appear almost on an equal level.

Answer 2

You didn't mention Weil, Basic Number Theory, where the case of a finite field of constants is handled, really only using Pontryagin duality. There is an elegant theory of John Tate that seems somewhat magical. I think the right question would be, "what is the trade-off?" rather than "what is the advantage?". Technically simpler methods require fewer prerequisites; and life is finite.

There certainly is some advantage, if you are doing number theory, to have a version of Riemann-Roch that does not require you to work over an algebraically closed field. Other than that, I don't really know. Riemann-Roch has been understood to work for higher dimensions, in some form, for about a century now, I think. If you are interested in the generalisations, there is a good case for starting from sheaves because the theory is slick. (Probably also true for R-R for vector bundles on curves, which Weil himself handled with bare hands c.1938, but makes more sense in Hirzebruch's book). These days, I suppose, valuation theory looks like a technical tool for singularity theory.

For some people, we won't really be comfortable saying we understand R-R until the Riemann Hypothesis is sorted out. I.e. the function field-number field analogy is still potent, and the view from arithmetic is that the circle of ideas is not yet complete. For that reason it is hard to rule out any approach, in terms of heuristic value anyway.