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Nicht endlich erzeugte Primideale in Steinschen Algebren. (Not finitely generated prime ideals in Stein algebras). (German) Zbl 0584.32028

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 \(T_ k\).

MSC:

32E10 Stein spaces
32A38 Algebras of holomorphic functions of several complex variables
32H35 Proper holomorphic mappings, finiteness theorems
13E15 Commutative rings and modules of finite generation or presentation; number of generators
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References:

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