On the finiteness properties of Matlis duals of local cohomology modules

Let R be a complete semi-local ring with respect to the topology defined by its Jacobson radical, a an ideal of R, and M a finitely generated R-module. Let DR(-) := HOM(R)(-, E), where E is the injective hull of the direct sum of all simple R-modules. If n is a positive integer such that Ext(R)(j) (R/a, DR (H(a)(n)(M))) is finitely gene-R- rated for all t > n and all j >= 0, then we show that HOM(R) (R/a, D(R) (H(a)(n),(M))) is also finitely generated. Specially, the set of prime ideals in Coass(R)(H(a)(n) (M)) which contains a is finite. Next, assume that (R, m) is a complete local ring. We study the finiteness properties of D(R)(H(a)(r)(R)) where r is the least integer i such that H(a)(i)(R) is not Artinian.