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 inte...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 p C Supp H^t_p （M）, then Hm^t＋dim R/p （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_m （M） is finitely generated if and only if 0 ≤ n ￠ W, where W ---- {t ＋ dimR/p丨p ∈ SuppH^t_p（M）／V（m）}. Also, we show that if J C 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}R, （Mp） is finitely generated for all p C Spec R （resp., p ∈ SpecR／MaxR）. Moreover, the concept of the J-cofiniteness dimension cJ（M） of M relative to I is introduced, and we explore an interrelation between c^I_m（M） and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM ≠ 0, then c^I_m （R） ---- inf{depth Mp ＋ dim R/p 丨 P ∈ Supp M/IM／V（m）}.展开更多
文摘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 p C Supp H^t_p （M）, then Hm^t＋dim R/p （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_m （M） is finitely generated if and only if 0 ≤ n ￠ W, where W ---- {t ＋ dimR/p丨p ∈ SuppH^t_p（M）／V（m）}. Also, we show that if J C 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}R, （Mp） is finitely generated for all p C Spec R （resp., p ∈ SpecR／MaxR）. Moreover, the concept of the J-cofiniteness dimension cJ（M） of M relative to I is introduced, and we explore an interrelation between c^I_m（M） and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM ≠ 0, then c^I_m （R） ---- inf{depth Mp ＋ dim R/p 丨 P ∈ Supp M/IM／V（m）}.
