Let \( \Bbbk \) be a field, and let \( S = \Bbbk [x_1, \dots , x_m, y_1, \dots , y_n] \) be a standard bigraded polynomial ring over \( \Bbbk \) . Define the ideals \( P = \langle x_1, \dots , x_m \rangle \) and \( Q = \langle y_1, \dots , y_n \rangle \) , and set \( \Bbbk [y] = \Bbbk [y_1, \dots , y_n] \) . Let \( M \) be a finitely generated bigraded \( S \) -module. Our goal is to characterize the Cohen–Macaulayness, generalized Cohen–Macaulayness and sequential Cohen–Macaulayness of \( M \) with respect to \( Q \) . This characterization is formulated in terms of the corresponding properties of the graded components \( M_k = M_{(k,*)} = \bigoplus _{j \in \mathbb {Z}} M_{(k,j)} \) , where each \( M_k \) is regarded as a finitely generated graded \( \Bbbk [y] \) -module for all \( k \) . Furthermore, we express this characterization in terms of the corresponding properties of \( M \) as a \( \Bbbk [y] \) -module. As a consequence, let \( I \) be a bihomogeneous ideal of \( S \) such that \( S/I \) is Cohen-Macaulay (respectively, generalized Cohen-Macaulay or sequentially Cohen-Macaulay) with respect to \( Q \) . Then, \( S/(P+I) \) is Cohen-Macaulay (respectively, generalized Cohen-Macaulay or sequentially Cohen-Macaulay).
