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(−) := HomR(−,E), where E is the injective hull of the direct sum of all simple R-modules. If n is a positive integer such that Extj R(R/a,DR(Hta (M))) is finitely generated for all t > n and all j
0, then we show that HomR(R/a,DR(Hn a (M))) is also finitely generated. Specially, the set of prime ideals in CoassR(Hn a (M)) which contain a is finite. Next, assume that (R,m) is a complete local ring. We study the finiteness properties of DR(Hr a (R)) where r is the least integer i such that Hi a(R) is not Artinian.

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