Let $R$ be a complete local Noetherian $\mathbb{Z}_p$-algebra with maximal ideal $m$ and residue field $k = R/m$. Assume $R$ is Cohen-Macaulay and there is a ring homomorphism $S \to R$ which makes $R$ a finitely-generated free $S$-module, where $S$ is a power series ring over $\mathbb{Z}_p$ with $\dim S = \dim R$. I want to show that the canonical module $\omega_R$ of $R$ is isomorphic to $\textrm{Hom}_S(R,S)$ (with $R$-module structure given by $(r \cdot \varphi)(x) = \varphi(rx)$ for $r,x \in R, \varphi \in \textrm{Hom}_S(R,S))$.
This should amount to showing that $\dim_k \textrm{Ext}^i_R(k,\textrm{Hom}_S(R,S))$ is equal to $1$ if $i = \dim R$ and $0$ otherwise. We can take a free resolution
$$\cdots \to R^n \to R \to k \to 0$$
of $k$ and take the cochain complex
$$0 \to \textrm{Hom}_R(R, \textrm{Hom}_S(R,S)) \to \textrm{Hom}_R(R^n, \textrm{Hom}_S(R,S)) \to \cdots.$$
Now
$$\textrm{Hom}_R(R^n, \textrm{Hom}_S(R,S)) \cong \textrm{Hom}_S(R,S)^n,$$
but I don't see where to proceed from here. For context I am trying to verify the first sentence in the proof of Lemma 3.9 in this paper: https://arxiv.org/pdf/2108.09729.pdf.
Best Answer
In general, we have the following
In your setting, $S$ is regular (in particular Gorenstein) so $\omega_S \cong S$ and we also have $\dim S=\dim R$, so the above result gives $\omega_R \cong \operatorname{Hom}_S(R,S)$.