Weil divisors and symbolic algebras (Q1111632)

Let R be a normal analytic local ring, f: \(Y\to X=Spec(R)\) be a projective resolution of singularities, a divisor \(D\in Pic(Y)\) maps to zero in \(R^ 2f_*{\mathbb{Z}}\), and \(E=f_*(D)\). The author proves the following result: \(\oplus_{n\geq 0}{\mathcal O}_ X(nE) \) is finitely generated if and only if D is torsion in Pic(Y). In particular, if every Weil divisor \({\mathcal O}_ X(E)\) on X satisfies the condition that \(\oplus_{n\geq 0}{\mathcal O}_ X(nE) \) is finitely generated as an \({\mathcal O}_ X\)-algebra, then \(R^ 1f_*{\mathcal O}_ Y=0\). If R is algebraizable, then R is \(S_ 3.\) In addition, the author proves that there exists a 3-dimensional affine local ring R with a Weil divisor D on \(X=Spec(R)\) such that \(\oplus_{n\geq 0}{\mathcal O}_ X(nD) \) is not a finitely generated \({\mathcal O}_ X\)-algebra.