A counterexample to a conjecture of Ding (Q5964505)

Let \((R, m, k)\) be a Cohen-Macaulay local ring of krull dimension \(d\). The index of \(R\) is defined to be index\((R):=\inf\{n | \delta(R/m^n) =1 \}\) where \(\delta(R/m^n)\) is the Auslander \(\delta\) invariant of \(R/m^n\). Let \(ll(M)\) be the Löewy length of a finite length module \(M\), which is the smallest integer \(n\) such that \(m^nM=0\). For a finitely generated \(R\)-module \(M\) of dimension \(c\), the generalized Löewy length \(gll(M)\) is defined by \(\mathrm{gll}(M):=\min\{ ll(M/(x_1, \ldots,x_c)M)|x_1 \ldots, x_c\) is a system of parameters for \(M \}\). In [Commun. Algebra 21, No. 1, 53--71 (1993; Zbl 0766.13003)], \textit{S. Ding} conjectured:NEWLINENEWLINE{ Conjecture.} Let \((R, m, k)\) be a Gorenstein local ring of dimension \(d\), then \(\mathrm{index}(R)= \mathrm{gll}(R)\).NEWLINENEWLINEIn the paper, the author gives examples of one dimensional complete intersection with index \(5\) and generalized Löewy length 6, which contradict Ding's conjecture. He also uses a new approach to recover a condition needed to apply Ding's conjecture in the one dimensional case.