Relative Buchsbaumness of bigraded modules (Q2905529)

Based on the well-known surjectivity criterion of Buchsbaum modules over regular rings [\textit{J. Stückrad} and \textit{W. Vogel}, Am. J. Math. 100, 727--746 (1978; Zbl 0429.14001)], the authors introduce a notion of \textit{relative Buchsbaum modules} as follows:NEWLINENEWLINEDefinition. Let \(S=k[x_1,\dots,x_m,y_1,\dots,y_n]\) with the grading given by \(\deg(x_i)=(1,0)\) and \(\deg(y_i)=(0,1)\). Set \(Q=(y_1,\dots,y_n)\). Let \(M\) be a finitely generated bigraded \(S\)-module and let \(H^i(Q,M)=H^i(\underline{y};M)\) denote the \(i\)-th Koszul cohomology. \(M\) is called \textit{relative Buchsbaum with respect to} \(Q\) if the canonical maps NEWLINE\[NEWLINE\lambda^i_M:H^i(Q;M)\to H^i_Q(M)NEWLINE\]NEWLINE are surjective for all \(i<\mathrm{cd}(Q,M)\), where \(\mathrm{cd}(Q,M)\) denotes the greatest integer \(t\) such that \(H^t_Q(M)\neq 0\).NEWLINENEWLINEA number of results on relative Buchsbaum modules, analogous to those on Buchsbaum modules, are proved. For example, it is proved that \(M\) is relative Buchsbaum with respect to \(Q\) if and only if the natural maps \(\mathrm{Ext}^i_S(S/Q,M)\to H^i_Q(M)\) are surjective for all \(i<\mathrm{cd}(Q,M)\).NEWLINENEWLINEAs stated in the introduction of the paper under review, `one cannot expect any connections between Buchbaumness and relative Buchsbaumness with respect to the irrelevant bigraded ideals \(P\) and \(Q\)'.