Hilbert Nullstellensatz in global complex-analytic case

Abstract

It is well known that the Hilbert Nullstellensatz holds in the algebraic case [3, Theorem 14, p. 164] as well as in the local complexanalytic case [2, III.A.7]. The question arises whether it holds in the global complex-analytic case when we have a Stein space. In this paper this question is answered in the affirmative for most prime ideals, whereas the answer is negative in the general case. Thanks are due to Professor Wolfgang Rothstein who posed to the author the question whether the Hilbert Nullstellensatz holds in the global form in the case of polydiscs and prime ideals. Complex spaces here are all in the sense of Grauert [1, Section 1]. Suppose (X, C) is a complex space. By a holomorphic function on (X, JC) we mean an element of r (X, SC). Suppose f is a holomorphic function on (X, JC) and xCX. We say thatf vanishes at x if the germ fx is not a unit in the local ring C.. f vanishes on a subset E if f vanishes at every point of E. Suppose M is a set of holomorphic functions on (X, a C). By the analytic subvariety Z of X defined by M we mean the set