Czechoslovak Mathematical Journal, first online, pp. 1-11
Non-finitely generated bigraded local cohomology modules
Ahad Rahimi
Received September 18, 2024. Published online January 28, 2025.
Abstract: 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 $Q=\langle y_1, \dots, y_n \rangle$. Assume that there exists $\frak p \in{\rm Ass}_S M$ such that ${\rm cd}(Q, S/\frak p)=j>0$. We demonstrate that ${\rm H}^j_Q(M)$ is not finitely generated. Furthermore, we explore a more general version of this result.
