Finiteness of Local Cohomology Modules Defined by a Pair of Ideals

Let R be a Noetherian ring and I, J two ideals of R. Let M be a ZD-module such that Hom R (R/I, M/L) is finitely generated for any submodule L of M. It is shown that if t is an integer such that is minimax for all i<t, then is finitely generated. Let M be a finitely generated R-module such that dim R M=n. It is proved that (1) has finite length; (2) if R is local with dimR/I+J=0 and dim R M/JM=d>0, then is not finitely generated.