Some Theorems about Formal Functions

Abstract

It is well-known that in algebraic geometry there are very few nontrivial problems about continuation of objects from open subsets, which is a big difference to analytic geometry. So we may expect that in the algebraic version of analytic geometry, formal geometry, there are some interesting problems of this kind, and this assumption is supported by the following paper. In there we take a local ring which is complete in some */-adic topology and consider the formal scheme obtained by completing the spectrum of our local ring along the closed subscheme defined by */. As all interesting formal objects on this formal scheme stem from objects on the usual spectrum of our local ring, we remove a closed subset. After that there may exist nonalgebraic formal objects, and such objects are characterized by the fact that there is no continuation of them to the whole formal scheme. The objects we are interested in are formal meromorphic functions and formal subsheaves of algebraic formal sheaves. As it is the case in analytic geometry these two cases are connected, and any result about one of them implies a corresponding result about the other. As application of our theory we prove certain results about connectedness, including a generalization of Zariski's connectedness-theorem. I express my gratitude to Professor Hironaka for some very stimulating discussion, and to the Deutsche Forschungsgemeinschaft for support during the last year.