On associated graded modules having a pure resolution (Q2816991)

Let \(A=K[[X_1,\dots,X_n]]\) and \(m=(X_1,\dots,X_n)\). Let \(G_m(A)=\bigoplus_{n\geq 0}m^n/m^{n+1}\) be the associated graded ring of \(A\), and \(M\) a finitely generated \(A\)-module. Let \(G_m(M) =\bigoplus_{n\geq 0} m^nM/m^{n+1}M\) be the associated graded module of \(M\) with respect to \(m\). In the paper, the author gives a necessary and suffiicent condition for \(G_m(M)\) to have a pure resolution. He proves in theorem 1.3 that, if \(G_m(M)\) has a pure resolution and \(F\) is a minimal free resolution of \(M\), then \(\mathrm{in}(F)\) is a minimal free resolution of \(G_m(M)\). Further, when \(M\) is Cohen-Macaulay, he shows the following:NEWLINENEWLINETheorem 1.4. Let \(M\) be a Cohen-Macaulay \(A\)-module and \(p=\mathrm{projdim }M\). Let \(\beta_i=\beta_i(M)\) and \(F\) a minimal resolution of \(M\). The following conditions are equivalent:{\parindent=0.6cm\begin{itemize}\item[i.] \(G_m(M)\) has a pure resolution. \item[ii.] The following hold: {\parindent=0.8cm\begin{itemize}\item[(a)] \(\mathrm{in}(F)\) is acyclic. \item[(b)] For \(i\geq 1\), \(\beta_i=(-1)^{i+1}\beta_0\prod_{\overset{1\leq j\leq p}{j\neq i}}\frac{dj}{d_j-d_i}\) \item[(c)] The multiplicity of \(M\), \(e_0(M) = \frac{\beta_0}{p!}\prod_{i=1}^p\).NEWLINENEWLINE\end{itemize}}NEWLINENEWLINE\end{itemize}}