On some asymptotic properties of finitely generated modules (Q651832)

Let \(I\) and \(J\) be two ideals of a noetherian local ring \((R,\mathfrak m)\) and \(M\) a finitely generated \(R\)-module. \textit{M. Brodmann} [Math. Proc. Camb. Philos. Soc. 86, No. 1, 35--39 (1979; Zbl 0413.13011)] has proved that the integers \(\text{depth}(I,J^nM/J^{n+1}M)\) and \(\text{depth}(I,M/J^{n}M)\) take constant values for large \(n\). Then Herzog and Hibi have proved that \[ \lim_{n\rightarrow \infty} \text{depth}(\mathfrak m,R/J^n)\leq \lim_{n\rightarrow \infty} \text{depth}(\mathfrak m,J^n/J^{n+1})= \lim_{n\rightarrow \infty} \text{depth}(\mathfrak m,J^n)-1; \] see [\textit{J. Herzog} and \textit{T. Hibi}, J. Algebra 291, No. 2, 534--550 (2005; Zbl 1096.13015)]. For each integer \(k>-1\), the notion of depth of \(M\) in dimension \(>k\) in I is defined by \[ \text{depth}_k(I,M):=\inf \{i\mid \dim_R(Ext_R^i(R/I,M))>k\}. \] The main achievement of the paper under review is extending the above mentioned result of Herzog and Hibi to the notion of depth in dimension \(>k\).