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 [3]). We prove
a certain ’dual’ version of the Hartshorne-Lichtenbaum vanishing (see Theorem 2.2).
There is 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 [6] generalizes naturally (see Theorem 4.4).