If $(R,\mathfrak m,k)$ is a Gorenstein local ring, then show that $\textrm{inj dim}_R\ k$ is finite.
This was previously asked here as a second part of the question and remained unaswered, but I think it is independent of the first part and deserves a separate thread.
In my opinion this question is false, as can be seen using this result. There it is shown that a finite generated $R$-module over a Gorenstein ring has finite projective dimension iff it has finite injective dimension. In the question above, if the injective dimension of $k$ is finite, then its projective dimension is finite, that is, $R$ is regular. But there are Gorenstein local rings which are not regular.
