ABSTRACT
This chapter discusses some of the results of K. Oka that do fit in quite naturally with the Weierstrass theorems and the local parametrization theorem. So far in the discussion of the local theory attention has been limited to the consideration of local rings at a single point. However it was recognized by Oka in his pioneering work on holomorphic functions of several variables that it is also essential to consider relations between local rings at nearby points; that leads to some subtle but very important concepts and results in the local theory. One of the fundamental discoveries of Oka was that it is not the case for arbitrary finitely generated families of modules, as distinct from finitely generated families of submodules of a family of free modules, that the module of relations is necessarily finitely generated, even locally.
