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 p Supp H-p(t)(M), then H-m(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-m(n)(M) is finitely generated if and only if 0 n W, where W = t + dim R/p p Supp H-p(t)(M) V(m)}. Also, we show that if J I are 1-dimensional ideals of R, then H-I(t)(M) is J-cominimax, and H-I(t)(M) is finitely generated (resp., minimax) if and only if H-IRp(t) (M-p) is finitely generated for all p Spec R (resp., p Spec RMax R). Moreover, the concept of the J-cofiniteness dimension c(I)(J)(M) of M relative to I is introduced, and we explore an interrelation between c(m)I(M) and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM 0, then c(m)(I)(R) = infdepth M-p + dim R/p p Supp M/IM V (m)}.