Cofiniteness of Local Cohomology Modules

Let M be a non-zero finitely generated module over a commutative Noetherian local ring (R, 𝔪). In this paper we consider when the local cohomology modules are finitely generated. It is shown that if t ≥ 0 is an integer and $\mathfrak{p}\in {\rm Supp\,} H^{t}_\mathfrak{p}(M)$, then $H^{t+\dim R/\mathfrak{p}}_\mathfrak{m}(M)$ is not 𝔭-cofinite. Then we obtain a partial answer to a question raised by Huneke. Namely, if R is a complete local ring, then $H^{n}_\mathfrak{m}(M)$ is finitely generated if and only if 0 ≤ n ∉ W, where $W=\{t+\dim R/\mathfrak{p}\,|\,\mathfrak{p}\in {\rm Supp\,} H^t_\mathfrak{p}(M)\backslash V(\mathfrak{m})\}$. Also, we show that if J ⊆ I are 1-dimensional ideals of R, then $H^t_I(M)$ is J-cominimax, and $H^t_I(M)$ is finitely generated (resp., minimax) if and only if $H^t_{IR_\mathfrak{p}} (M_\mathfrak{p})$ is finitely generated for all $\mathfrak{p}\in {\rm Spec\,}R$ (resp., $\mathfrak{p}\in {\rm Spec\,}R\backslash {\rm Max\,}R$). Moreover, the concept of the J-cofiniteness dimensi...