On adjusted Hilbert-Samuel functions (Q746805)

Let \((R,\mathrm{m})\) be a Noetherian local ring and \(M\) a finitely generated \(R\)-module of dimension \(d\), and let \(\mathrm{q}\) be a parameter ideal of \(M\). The paper under reviewed considers the non-negativity of the following function in \(n\), called \textit{adjusted Hilbert-Samuel function} of \(M\) with respect to \(\mathrm{q}\): \[ f_{\mathrm{q},M}(n):=\ell(M/\mathrm{q}^{n+1}M)-\sum_{i=0}^d\mathrm{adeg}_i(\mathrm{q};M)C^{i}_{n+i} \] Here \(\mathrm{adeg}_i(\mathrm{q};M)\), the \textit{ith arithmetic degree} of \(M\) with respect to \(\mathrm{q}\), is defined by \[ \mathrm{adeg}_i(\mathrm{q};M):=\sum_{p\in \mathrm{Ass}_i(M)}\ell(H_{\mathrm{p}R_{\mathrm{p}}}^0(M_{\mathrm{p}}))e_0(\mathrm{q};R/\mathrm{p}). \] In general, \(f_{\mathrm{q},M}(n)\) may take negative value for all \(n\geq 0\). However the paper under reviewed proves that if \(\mathrm{q}\) is a distinguished parameter ideal then there exists an integer \(n_0\) such that \(f_{\mathrm{q},M}(n)\) takes non-negative values and is increasing for all \(n\geq n_0\). Moreover, if assuming further that \(M\) is sequentially generalized Cohen-Macauly and \(\mathcal{F}\) is a generalized Cohen-Macauly filtration of \(M\), then there is an integer \(N\) such that \(f_{\mathrm{q},M}\geq 0\) for all distinguished parameter ideal \(\mathrm{q}\) of \(M\) with respect to \(\mathcal{F}\) and all integer \(n\geq N\).