Nicht endlich erzeugte Primideale in Steinschen Algebren. (Not finitely generated prime ideals in Stein algebras) (Q1070393)

In this note the author constructs a closed prime ideal I in the ring \({\mathcal O}({\mathbb{C}}^ 3)\) which is not finitely generated. It is known that 3 is the smallest dimension in which this kind of phenomena can occur. I is the ideal of a certain irreducible curve \(Y\subset {\mathbb{C}}^ 3,\) obtained as the image of a proper holomorphic map \(f: {\mathbb{C}}\to {\mathbb{C}}^ 3.\) The construction is performed as follows: After some preliminar results on local ideal, the author considers the planes \(E_ n=\{(z_ 1,z_ 2,z_ 3)\in {\mathbb{C}}^ 3| z_ 1+nz_ 2+n^ 2z_ 3=0\}\) and the union of lines \(T_ k=\cup_{m<n\leq k}E_ m\cap E_ n;\) then he proves that the ideal in \({\mathcal O}_ 0\) of germs of holomorphic functions around the origin, vanishing on \(T_ k\) cannot be generated by less than k elements. Then an irreducible curve Y, obtained as the image of a proper holomorphic map \(f: {\mathbb{C}}\to {\mathbb{C}}^ 3\) is constructed in such a way in a discrete sequence of points \(p_ k\), the germs of Y coincides up to order k (in a suitable sense) with the model \(T_ k\).