Buchsbaumness in Rees modules associated to ideals of minimal multiplicity in the equi-\(\mathbb I\)-invariant case (Q1614650)

Let \((A,{\mathbf m})\) be a local ring. \textit{J. Stückrad} proved that if \(A\) is a Buchsbaum ring, then the Rees algebra \(R_{\mathbf m}(A)\) is also a Buchsbaum ring [Beitr. Algebra Geom. 19, 83-103 (1985; Zbl 0567.13008)]. There has been few progress since J. Stückrad's result. The paper under review is a major step in the study of Buchsbaumness of Rees algebras. Let \(E\) be a Buchsbaum \(A\)-module of dimension \(s > 0\) and \({\mathbf a} \subset A\) an ideal such that \(\ell(E/{\mathbf a}E) < \infty\). Let \(G_{\mathbf a}(E) = \bigoplus_{n\geq 0}{\mathbf a}^nE/{\mathbf a}^{n+1}E\) and \(R_{\mathbf a}(E) = \bigoplus_{n \geq 0}{\mathbf a}^nE\). Suppose that \(I(G_{\mathbf a}(E)) = I(E)\), where \(I(.)\) is the Stückrad-Vogel invariant. If \({\mathbf a}\) is of minimal multiplicity with respect to \(E\), i.e. \(e_{\mathbf a}(E) = \mu_A({\mathbf a}E) + \ell(E{\mathbf a}E) - \rho({\mathbf a},E) + I(E)\), where \(\rho({\mathbf a},E)\) denotes the supremum of the minimal number of generators of minimal reductions of \({\mathbf a}\) with respect to \(E\), the author shows that \(\bigoplus_{n > 0}{\mathbf a}^nE\) is Buchsbaum module over \(R_{\mathbf a}(A)\). Moreover, \(R_{\mathbf a}(E)\) is a Buchsbaum module if \(s \geq 2\). The proof is very complicated and occupies almost all this long paper.