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Keywords:
associated prime; bigraded module; cohomological dimension; finiteness dimension; maximal depth; local cohomology
associated prime; bigraded module; cohomological dimension; finiteness dimension; maximal depth; local cohomology
Summary:
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.
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