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Let $(R,\fm)$ be a relative Cohen-Macaulay local ring with respect to an ideal $\fa$ of $R$ and set $c:=\h\fa$. In this paper, we investigate some properties of the Matlis dual of $R$-module $\H_{\fa}^c(R)$ and we show that such modules treat like canonical modules over Cohen-Macaulay local rings. Also, we provide some duality and equivalence results with respect to the module $\H_{\fa}^c(R)^{\vee}$ and so these results lead to achieve generalizations of some known results, such as the Local Duality Theorem, which have been provided over a Cohen-Macaulay local ring which admits a canonical $R$-module.
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