Finite homological dimension and primes associated to integrally closed ideals. II.
[For part I of this paper see \textit{S. Goto} and \textit{F. Hayasaka}, Proc. Am. Math. Soc. 130, No. 11, 3159--3164 (2002; Zbl 0995.13009).] From the paper: Let \(R\) be a Noetherian local ring with the maximal ideal \({\mathfrak m}\). Assume that \(R\) contains ideals \(I\) and \(J\) satisfying the conditions (1) \(I\subseteq J\), (2) \(I:{\mathfrak m} \nsubseteq J\), and (3) \(J\) is \({\mathfrak m}\)-full, that is \({\mathfrak m} J:x=J\) for some \(x\in{\mathfrak m}\). Then the main theorem says that \(R\) is a regular local ring, if the projective dimension \(\text{pd}_RI\) of \(I\) is finite. Let \({\mathfrak q}=(a_1, a_2,\dots,a_t)R\) be an ideal in a Noetherian local ring \(R\) generated by a maximal \(R\)-regular sequence \(a_1,a_2,\dots,a_t\) and let \(\overline{\mathfrak q}\) denote the integral closure of \({\mathfrak q}\). Then, thanks to the theorem applied to the ideals \(I={\mathfrak q}:{\mathfrak m}\) and \(J= \overline{\mathfrak q}\), it follows that \(I^2={\mathfrak q}I\), unless \(R\) is a regular local ring. Consequences are discussed, namely: Theorem. Let \(R\) be a Noetherian ring and let \({\mathfrak q}\) be an ideal in \(R\) generated by an \(R\)-regular sequence. Let \({\mathfrak p}\in\text{Ass}_R R/{\mathfrak q}\) and assume that \({\mathfrak p}\) is maximal in \(\text{Ass}_R R/{\mathfrak q}\) with respect to inclusion. We put \(I={\mathfrak q}:{\mathfrak p}\). Then the local ring \(R_{\mathfrak p}\) is regular and \(\overline{{\mathfrak q}R_{\mathfrak p}}={\mathfrak q}R_{\mathfrak p}\), if \(I^2\neq{\mathfrak q}I\).
scientific article; zbMATH DE number 2059514