On the depth of the associated graded ring of an \(m\)-primary ideal of a Cohen-Macaulay local ring
Let \(I\) be an \(m\)-primary ideal of the \(d\)-dimensional Cohen-Macaulay local infinite ring \(R/m\), and let \(\text{gr}_ I (R)\) denote the associated graded ring \(\oplus_{n \geq 0} I^ n/I^{n+1}\). The starting point of this work is a result of \textit{P. Valabrega} and \textit{G. Valla} [Nagoya Math. J. 72, 93-101 (1978; Zbl 0362.13007)] which states that \(\text{depth} (\text{gr}_ I R) = d\) if and only if \(I\) has a minimal reduction \(J\) such that \(I^ n \cap J = I^{n-1} J\) for all \(n \geq 2\). Here the main result states that if such \(J\) exists with \(\text{length} (I^ n \cap J/I^{n-1} J) \leq 1\) for all \(n \geq 2\) and equality for exactly one such \(n\), then \(\text{depth} (\text{gr}_ IR) = d - 1\). The proof requires detailed computations with a minimal generating set of \(J\) regarded as a regular sequence.
scientific article; zbMATH DE number 638515