Let $R=\mathbb C[X,Y]/(Y^3-X^3)$, let $x,y$ be the images of $X,Y$ in $R$, and let $R_1$ be the localization of $R$ at the maximal ideal $(x,y)$. I want to prove that $R_1$ is a Cohen-Macaulay ring.
If we could see that $R_1$ is of Krull dimension $1$ then we are done since any domain of K-dim $1$ is Cohen-Macaulay. I think that $R_1=\mathbb C[X,Y]_{(X,Y)}/(x_0,y_0)$, where $x_0,y_0$ are the images of $X,Y$ in $\mathbb C[X,Y]_{(X,Y)}$. Is this latter true? Thanks for any cooperation!
In fact, $R$ is a one dimensional reduced (why?) ring, and such rings are Cohen-Macaulay. Therefore all its localizations are Cohen-Macaulay.