On Cohen-Macaulay local rings with embedding dimension \(e+d-2\)
The main result of this paper is the following theorem 3.1. If \((R, {\mathfrak m})\) is a \(d\)-dimensional local Cohen-Macaulay ring of embedding dimension \(e+d-2\) with infinite residue field, then \(\text{depth} G({\mathfrak m}) \geq d-1\), where \(G({\mathfrak m}) =R/{\mathfrak m} \oplus {\mathfrak m}/{\mathfrak m}^2 \oplus \dots\) is the associated graded ring of \(R\) with respect to \({\mathfrak m}\). -- This answers a question of \textit{J. D. Sally} [J. Algebra 83, 393-408 (1983; Zbl 0517.13013)]. The author uses properties of Hilbert coefficients and of the Sally module \[ S_J(I)= \bigoplus^\infty_{n=0} I^{n+1}/J^nI, \] where \(I\) is an \({\mathfrak m}\)-primary ideal with a minimal reduction \(J\); see also: \textit{H.-J. Wang}, Commun. Algebra 25, No. 1, 303-309 (1997; see the preceding review).
scientific article; zbMATH DE number 1013973