Let \(\Bbbk\) be a field, and let \(S=\Bbbk[x_1, \dots, x_m, y_1, \dots, y_n]\) denote a standard bigraded polynomial ring over \(\Bbbk\) . Consider M, a finitely generated bigraded S-module, and set Q = 〈y1,…, yn〉. Assume that there exists \(\frak{p} \in \text{Ass}_{S}M\) such that \(\text {cd}(Q, S/\frak{p})=j>0\) . We demonstrate that H Q j (M) is not finitely generated. Furthermore, we explore a more general version of this result.
