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Paper IPM / M / 8310  


Abstract:  
In this article we will give some of the ideas we consider
important and point out the directions taken by some recent
research on the set of associated primes of the local cohomology
modules. In addition, we prove the following result.
Let R be a Noetherian ring and \fraka be an ideal of R.
Let M be an Rmodule and s be a nonnegative integer. Then
the following hold:
(a) If Ext^{s−j}_{R}(R/\fraka, H^{j}_{\fraka}(M)) is
finitely generated for all j < s and if Hom_{R}(R/\fraka,H^{s}_{\fraka}(M)) is a finitely generated Rmodule, then
Ext^{s}_{R}(R/\fraka, M) is a finitely generated Rmodule.
(b) If Ext^{s+1−j}_{R}(R/\fraka, H^{j}_{\fraka}(M)) is
finitely generated for all j < s and if Ext^{s}_{R}(R/\fraka,M)
is a finitely generated Rmodule, then Hom_{R}(R/\fraka,H^{s}_{\fraka}(M)) is a finitely generated Rmodule.
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