Let (R, m) be a Noetherian regular local ring containing a field of characteristic p > 0 and I a nonzero ideal of R. In this short note, we prove that if H i I (R) = 0, then Supp R (D(H i I (R))) = Spec(R).

Let R be a Noetherian ring, I an ideal of R and M an R-module with cd(I, M ) = c. In this article, we first show that there exists a descending chain of ideals Iis not Artinian. We then give sufficient conditions for a non-negative integer t to be a lower bound for cd(I, M ) and use this to conclude that in non-catenary Noetherian local integral domains, there exist prime ideals that are not set theoretic complete intersection. Finally, we set conditions which determine whether or not a top local cohomology module is Artinian.