Let $k$ be a field and $R=k[X,Y]_{(X,Y)}$. Describe a minimal injective resolution of $R$ as an $R$-module (in terms of the Bass numbers of $R$).
Solution: Let $k$ be a field and $R=k[X,Y]_{(X,Y)}$. Then $$ k(p) = \left.R_p\middle/pR_p\right. = \left(\left.R\middle/pR\right.\right)_p = \mathrm{Quot}\left(\left.R\middle/pR\right.\right) $$
Let $p\in \mathrm{Spec}(R)$. By assumption $R_p$ is Gorenstein, thus $$ \mathrm{Ext}_{R_p}^i\left(k(p),R_p\right) = \begin{cases} 0 & \text{ for } i\neq \dim R_p = \mathrm{ht}(p) \\ k(p) & \text{ if } i = \dim R_p = \mathrm{ht}(p). \end{cases} $$ Thus, $\mu_i(p,R)=\dim_{k(p)} \mathrm{Ext}_{R_p}^i \left(k(p),R_p\right) =\delta_{i,\mathrm{ht}(p)}$, which is the Bass number.
So I was trying to answer the question above and found the following answer. The problem though is that I am not sure that $R_p$ is Gorenstein. Can I get some help to show that (if it is indeed true)?