Maximal Buchsbaum modules over Gorenstein local rings
Let \((A, \mathfrak{m})\) be a local Gorenstein ring of dimension \(d>0\), \(k=A/\mathfrak{m}\) and \(r=\dim_k (\mathfrak{m}/\mathfrak{m}^2).\) Let \(B\) be the set of all \(A\)-bases obtained permuting a fixed one. A noetherian \(A\)-module \(M\) is called maximal if \(\dim(M)=d\). \(M\) is called quasi-Buchsbaum if \(\mathfrak{m}H^i_{\mathfrak{m}}(M)=0\) for all \(i<\dim (M)\). The structure of maximal Buchsbaum modules over local Gorenstein rings is studied. It is proved that a maximal quasi-Buchsbaum module \(M\) is Buchsbaum if and only if \[ \mathfrak{m}\text{Ext}^i_A(M, A/\langle x_{r-n+1}, \dots, x_r\rangle)=0 \text{ for all }i>0, 0\leq n<d \] and \((x_1, \dots, x_r)\in B\).
scientific article; zbMATH DE number 2182062
scientific article