On the Riemann-Roch theorem for orders in the ring of valuation vectors of a function field

Abstract

In 1953 Kenkichi Iwasawa, following a suggestion of Artin, gave a characterisation of the ring of valuation vectors (also called repartitions) for function fields in simple topological algebraic terms. Using elementary properties of these rings a short and elegant proof of the Riemann-Roch theorem for smooth complete curves was given. In this paper the methods of linear topology and duality are used to study the Riemann-Roch problem for algebraic curves with singularities. Accordingly we study the linearly compact open modules associated with certain subrings of the ring of valuation vectors of the function field. By applying these methods the Riemann-Roch theorem for algebraic curves with singularities is extended to a larger class of modules than was usual in the literature.