Linear growth of primary decompositions of integral closures
Let \(R\) be an commutative noetherian ring and \(I\) a proper ideal of \(R\). \textit{I. Swanson} has shown [Math. Ann. 307, No. 2, 299-313 (1997; Zbl 0869.13001)] that \(I\) has linear growth of primary decompositions, that is there exists a positive integer \(h\) such that, for every positive integer \(n\), the ideal \(I^n\) has a minimal primary decompositin \(I^n=Q_{n1}\cap \dots\cap Q_{nk_n}\) such that \((\sqrt{Q_{ni}})^{hn}\subseteq Q_{ni}\) for all \(i=1,\dots,k_n\). In the paper under review, the author establishes a parallel result for integral closures of ideals, that is, there exists a positive integer \(h\) such that, for every positive integer \(n\), the integral closure \(\overline{I^n}\) of \(I^n\) has a minimal primary decomposition \(\overline{I^n}=Q_{n1}\cap \dots\cap Q_{nk_n}\) such that \((\sqrt{Q_{ni}})^{hn}\subseteq Q_{ni}\) for all \(i=1,\dots,k_n\). His proof uses the extended Rees algebra and some properties of the Nagata rings to reduce the assertion to the case in which \(I\) is principal generated by a non-zerodivisor and \(R\) is a reduced normal Nagata ring.
scientific article; zbMATH DE number 1213856