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



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
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.


