A note on the Matlis dual of a certain injective hull

TL;DR

This paper investigates the conditions under which the Matlis dual of certain injective hulls over a local ring is isomorphic to the completion of the localized ring, providing a characterization involving completeness.

Contribution

It establishes a precise criterion linking the isomorphism of the Matlis dual to the completeness of the quotient ring, extending understanding of injective modules in local algebra.

Findings

Matlis dual of injective hull is isomorphic to the completion if and only if the quotient is complete.

Provides a description of tensor products involving completions in one-dimensional domains.

Characterizes when the Matlis dual corresponds to the ring's completion in local algebra.

Abstract

Let $(R,\mathfrak{m})$ denote a local ring with $E = E_R(R/\mathfrak{m})$ the injective hull of the residue field. Let $\mathfrak{p} \in \Spec R$ denote a prime ideal with $\dim R/\mathfrak{p} = 1$, and let $E_R(R/\mathfrak{p})$ be the injective hull of $R/\mathfrak{p}$. As the main result we prove that the Matlis dual $\Hom_R(E_R(R/\mathfrak{p}), E)$ is isomorphic to $\hat{R_{\mathfrak{p}}}$, the completion of $R_{\mathfrak{p}}$, if and only if $R/\mathfrak{p}$ is complete. In the case of $R$ a one dimensional domain there is a complete description of $Q \otimes_R \hat{R}$ in terms of the completion $\hat{R}$.