Rings of global sections in two-dimensional schemes (Q5948421)

Let \(B=\Gamma(U,\mathcal O_X)\) denote the ring of global sections of \(X =\text{Spec }A\) with respect to an open subset, where \(A\) denotes a local Noetherian two-dimensional ring. In the paper, there is given a criterion to decide the finiteness of \(B\) in case of \(U = D(I)\), \(I\) an ideal of height one. It says that in case that \(U\) is not affine the ring of global sections of \(U\) is not finitely generated if and only if there exists an irreducible formal component where \(U\) is affine and a second one where \(U\) is not affine such that their intersection is one-dimensional. This generalizes a result of \textit{P. M. Eakin jun., W. Heinzer, D. Katz} and \textit{L. J. Ratliff jun.} [J. Algebra 110, 407-419 (1987; Zbl 0631.13003)], where it is shown that \(D(I)\) is affine if and only if \(B\) is Noetherian if and only if \(B\) is of finite type over \(A,\) provided \(A\) is an excellent Cohen-Macaulay ring. Moreover it provides a counterexample to a statement of the reviewer [\textit{P. Schenzel} in: Commutative algebra, Proc. Workshop, Salvador/Brazil 1988, Lect. Notes Math. 1430, 88-97 (1990; Zbl 0719.13003)], where it was claimed that the result of the paper cited above is true without the Cohen-Macaulay assumption on \(A.\)