Let $(R, \mathfrak m)$ be a local Gorenstein ring and $\hat R$ be its $\mathfrak m$-adic completion. So we have a canonical map $R \to \hat R$ which makes $\hat R$ into an $R$-module. My question is: Does $\hat R$ have finite injective dimension as $R$-module ?
This is related to the injective dimension question in here On homological dimensions of finitely generated modules over a local ring and its completion . Notice that since $\hat R$ is Gorenstein, we do have inj$\dim_{\hat R} \hat R <\infty$ and I'm asking whether it is true that inj$\dim_R \hat R <\infty$ or not.
If $R$ is Gorenstein and $M$ is an $R$-module then $$ \text{inj dim}_{R}\,M<\infty \iff \text{flat dim}_{R}\,M<\infty \iff \text{proj dim}_{R}\,M<\infty.$$ Over a commutative noetherian local ring $(R,\mathfrak{m})$, the completion $\hat{R}$ is a (faithfully) flat $R$-module.
Therefore over a Gorenstein local ring $R$ the module $\hat{R}$ has finite injective dimension, as Mohammad Bagheri stated.