Asymptotic prime ideals of \(S_{2}\)-filtrations (Q1643105)

Let \(A\) denote a Noetherian ring with \(Q(A)\) the full ring of quotients. If \(S = \bigoplus_{n \in \mathbb{Z}}I_n t^n\) is a Noetherian graded ring with \(A[t^{-1}] \subseteq S \subseteq Q(A)[t,t^{-1}]\) and \(S\) satisfies the Serre condition \((S_2)\) then the family of ideals \(\mathcal{F} = \{I_n \cap A\}_{n \geq 1}\) is called an \(S_2\)-filtration in \(A\). The main subject of the paper is the study of the set of prime ideals \(\operatorname{Ass} A/I_n \cap A\) for \(n \geq 1\). Among other things the following is shown: Let \(A\) be a formally equidimensional domain, \(I \subset A\) an ideal, \(S = \bigoplus_{n \in \mathbb{Z}}I_n t^n\) a Noetherian ring with \(A[It,t^{-1}] \subseteq S \subseteq A[t,t^{-1}]\) that satisfies \((S_2)\), \(\operatorname{height} I_1 \geq 1\), and \(S\) is a finite extension of \(A[It,t^{-1}]\). Then: (a) \(\operatorname{Ass} A/I_n\cap A \subseteq \operatorname{Ass} A/I_{n+1}\cap A\) for all \(n \geq 1\). (b) \(\cup_{n\geq 1} \operatorname{Ass} A/I_n\cap A = \{Q \cap A| Q \in \operatorname{Min} (S/t^{-1}S)\}.\) (c) \(\cup_{n \geq 1}\operatorname{Ass} A/I_n\cap A = \cup_{n\geq 1} \operatorname{Ass} A/\overline{I^n}\). As an application there are several consequences about the symbolic powers of prime ideals and the asymptotic behaviour of associated prime ideals of ideals of analytic deviation one.