There is a short claim that I've met:
Let $(R,m)$ be a Noetherian local ring of dimension $d$ and $E=E_R(R/m)$ be the injective hull of $R/m$, $\widehat{R}$ is $m$-adic completion of $R$. Then we have: $Ext^{1}_{\widehat{R}}(\widehat{R} / \widehat{m}, \widehat{R} \otimes_{R} E) \cong \widehat{R} \otimes_{R} Ext^{1}_{R}(R/m,E)$.
In order to prove this claim, we also need to prove two statements: 1) $Ext^1_{R}(R/m,E)$ is finitely generated and 2) $\widehat{Ext^1_{R}(R/m,E)} = Ext^{1}_{\widehat{R}}(\widehat{R} / \widehat{m}, \widehat{R} \otimes_{R} E)$ (because if $Ext^1_{R}(R/m,E)$ is finitely generated, then $\widehat{R} \otimes_{R} Ext^{1}_{R}(R/m,E)=\widehat{Ext^1_{R}(R/m,E)}$. I wonder if $Ext^1_{R}(-,E)$ would preserve the inverse limmit or is there any way to do with base change ? I saw that $\widehat{R}$ is faithfully flat over $R$ but don't know how to use here.