A Few Comments On Matlis Duality

Abstract

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 $\mathfrak{p}{R}_{\mathfrak{p}}$. These results extend those of Matlis (see \cite{m}) shown in the case of the maximal ideal $\mathfrak{m}$.

arXiv comment9 pages,to be appeared in International Electronic Journal of Algebra