Sequentially generalized Cohen-Macaulayness of bigraded modules (Q2788762)

Let \(K\) be a field and \(S=K[x_1,\dots ,x_m, y_1,\dots ,y_n]\) the standard bigraded polynomial ring over \(K\). Let \(M\) be a finitely generated bigraded \(S\)-module and \(Q=(y_1,\dots ,y_n)\). This paper examines the generalized Cohen-Macaulayness and sequentially generalized Cohen-Macaulayness of \(M\) with respect to \(Q\). Below, we recall the definitions of these notions.NEWLINENEWLINEThe cohomological dimension of \(M\) with respect to \(Q\), \(cd_R(Q,M)\), is the supremum of the non-negative integers \(n\) for which \(H_Q^n(M)\neq 0\). The \(R\)-module \(M\) is said to be \textit{generalized Cohen-Macaulay} with respect to \(Q\) if the \(R\)-module \(H_Q^i(M)\) is finitely generated for all \(i< cd_R(Q,M)\). Also, \(M\) is said to be \textit{sequentially generalized Cohen-Macaulay} with respect to \(Q\) if \(M\) possesses a filtration NEWLINE\[NEWLINE0=M_0 \subsetneqq M_1\subsetneqq \cdots \subsetneqq M_r=MNEWLINE\]NEWLINE of bigraded submodules of \(M\) such that each quotient module \(M_i/M_{i-1}\) is generalized Cohen-Macaulay with respect to \(Q\) and \(cd_R(Q,M_1/M_0)<cd_R(Q,M_2/M_1)<\cdots <cd_R(Q,M_r/M_{r-1})\).NEWLINENEWLINELet \(I\) be a monomial ideal of \(S\) with \(cd_R(Q,S/I)\leq 2\). As the main result of this paper, they authors show that \(S/I\) is sequentially generalized Cohen-Macaulay with respect to \(Q\).