Index of reducibility of distinguished parameter ideals and sequentially Cohen-Macaulay modules (Q2838955)

Let \((R,\mathfrak{m},k)\) denote a local ring and let \(M\) be a finitely generated \(R\)-module. For a parameter ideal \(\mathfrak{q}\) of \(M\) the index of reducibilty \(N(\mathfrak{q},M)\) is defined by \(\dim_k(\mathfrak{q}M:_M \mathfrak{m}/\mathfrak{q}M)\). Note that, if \(M\) is a Cohen-Macaulay module then \(N(\mathfrak{q},M)\) is an invariant independent on \(\mathfrak{q}\). Moreover it was shown by \textit{S. Goto} and \textit{N. Suzuki} [J. Algebra 87, 53--88 (1984; Zbl 0538.13003)] that \(\sup_{\mathfrak{q}} N(\mathfrak{q}, M)\) is finite for a generalized Cohen-Macaulay module \(M\). Moreover, they gave examples of local rings \(R\) such that \(\sup_{\mathfrak{q}} N(\mathfrak{q}, R)\) is not finite. It turns out that these examples are sequentially Cohen-Macaulay, that is the dimension filtration as defined by the reviewer see [Lect. Notes Pure Appl. Math. 206, 245--264 (1999; Zbl 0942.13015)] is either zero or Cohen-Macaulay. In the same paper see [loc. cit.] the reviewer introduced the notion of a distinguished system of parameters as one that satisfies a certain annihilation on the dimension filtration. In the paper under review it is shown that in a sequentially Cohen-Macaulay module there exists an integer \(n \gg 0\) such that the index of irreduciblity \(N(\mathfrak{q},M)\) is constant for all distinguished parameter ideals \(\mathfrak{q} \subset \mathfrak{m}^n\). In particular there is an expression of \(N(\mathfrak{q},M)\) as the sum of the socle dimensions of all the local cohomology modules \(H^i_{\mathfrak{m}}(M)\).