On finite generation of Rees rings defined by filtrations of ideals (Q1332792)

Let \(S\) be a Noetherian ring and \(T = S[V]\) the polynomial ring over \(S\) with the variable \(V\). The author gives sufficient conditions for finite generation of the Rees ring \(R({\mathcal F})\) associated with a filtration \({\mathcal F} = \{F_ i\}_{i \in \mathbb{Z}}\) of ideals of \(T\), when the Rees ring \(R({\mathcal G})\) is Noetherian. Here \({\mathcal G} = \{G_ i\}_{i \in \mathbb{Z}}\) is a filtration of \(S\) such that \(G_ i = F_ i \cap S\). Moreover, the author proves that to investigate whether \(R({\mathcal F})\) is Noetherian or not \(({\mathcal F}\) is a filtration in a base ring \(A)\) we may assume that \(A\) is local. Finally, he investigates the Noetherian property of the symbolic Rees algebra of the ideal defining the ordinary Rees ring associated with an ideal \(I\) of \(A\) such that \(\text{grade} (I) \geq \rho_ A(I) - 1\), where \(\rho_ A (I)\) is the minimal number of generators of \(I\).