A Few Comments On Matlis Duality

For a Noetherian local ring $(R,{\mathfrak m})$ with $\mathfrak p\in \Spec(R)$ we denote $E_R(R/\mathfrak p)$ by the $R$-injective hull of $R/\mathfrak p$. We will show that it has an $\hat{R}^\mathfrak p$-module structure and there is an isomorphism $E_R(R/\mathfrak p)\cong E_{\hat{R}^\mathfrak p}(\hat{R}^\mathfrak p/\mathfrak p\hat{R}^\mathfrak p)$ where $\hat{R}^\mathfrak p$ stands for the $\mathfrak p$-adic completion of $R$. Moreover for a complete Cohen-Macaulay ring $R$ the module $D(E_R(R/\mathfrak p))$ is isomorphic to $\hat{R}_\mathfrak{p}$ provided that $\dim(R/\mathfrak p)=1$ and $D(\cdot)$ denotes the Matlis dual functor $\Hom_R(\cdot, E_R(R/\mathfrak m))$. Here $\hat{R}_\mathfrak{p}$ denotes the completion of ${R_\mathfrak p}$ with respect to the maximal ideal $\mathfrak pR_\mathfrak p$. These results extend those of Matlis (see \cite{m}) shown in the case of the maximal ideal ${\mathfrak m}$.