Abstract: In this paper we study the ring of global sections $\Gamma(U,{\cal O})$ of an open subset $U=D(I) \subseteq Spec A$, where $A$ is a two-dimensional noetherian ring. The main concern is to give a geometric criterion when these rings are finitely generated, in order to correct an invalid statement of Schenzel in [Sch]. \item{[Sch]} Schenzel, P.: Flatness and Ideal-Transforms of Finite Type. In: Commutative Algebra (ed. W. Bruns, A. Simis), 1988.
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