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

beforethe bundle/sheaf viewpoint became standard in algebraic geometry. – Donu Arapura Jan 21 '12 at 14:28