We provide a formula (see Theorem 1.5) for the Matlis dual of the injective hull of R/p where p is a one dimensional prime ideal in a local complete Gorenstein domain (R, m). This is related to results of Enochs and Xu (see [4] and [5]). We prove a certain 'dual' version of the Hartshorne-Lichtenbaum vanishing (see Theorem 2.2). We prove a generalization of local duality to cohomologically complete intersection ideals I in the sense that for I = m we get back the classical Local Duality Theorem. We determine the exact class of modules to which a characterization of cohomologically complete intersection from [7] generalizes naturally (see Theorem 4.4).
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